Singular Hopf Bifurcation in Systems with Two Slow Variables
Research Article Hopf Bifurcation and Global Periodic...
-
Upload
vuongthuan -
Category
Documents
-
view
218 -
download
0
Transcript of Research Article Hopf Bifurcation and Global Periodic...
Research ArticleHopf Bifurcation and Global Periodic Solutions in aPredator-Prey System with Michaelis-Menten Type FunctionalResponse and Two Delays
Yunxian Dai1 Yiping Lin1 and Huitao Zhao12
1 Department of Applied Mathematics Kunming University of Science and Technology Kunming Yunnan 650093 China2Department of Mathematics and Information Science Zhoukou Normal University Zhoukou Henan 466001 China
Correspondence should be addressed to Yiping Lin linyiping689163com
Received 27 November 2013 Accepted 31 March 2014 Published 11 May 2014
Academic Editor Yuming Chen
Copyright copy 2014 Yunxian Dai et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
We consider a predator-prey systemwithMichaelis-Menten type functional response and two delaysWe focus on the case with twounequal and non-zero delays present in the model study the local stability of the equilibria and the existence of Hopf bifurcationand then obtain explicit formulas to determine the properties of Hopf bifurcation by using the normal form method and centermanifold theorem Special attention is paid to the global continuation of local Hopf bifurcation when the delays 120591
1= 1205912
1 Introduction
In [1] Xu and Chaplain studied the following delayedpredator-prey model with Michaelis-Menten type functionalresponse
1198891199091
119889119905
= 1199091(119905) [1198861minus 119886111199091(119905 minus 12059111) minus
119886121199092(119905)
1198981+ 1199091(119905)
]
1198891199092
119889119905
= 1199092(119905) [minus119886
2+
119886211199091(119905 minus 12059121)
1198981+ 1199091(119905 minus 12059121)
minus119886221199092(119905 minus 12059122) minus
119886231199093(119905)
1198982+ 1199092(119905)
]
1198891199093
119889119905
= 1199093(119905) [minus119886
3+
119886321199092(119905 minus 12059132)
1198982+ 1199092(119905 minus 12059132)
minus 119886331199093(119905 minus 12059133)]
(1)
with initial conditions
119909119894(119905) = 120601
119894(119905) 119905 isin [minus120591 0] 120601
119894(0) gt 0 119894 = 1 2 3 (2)
where 1199091(119905) 119909
2(119905) and 119909
3(119905) denote the densities of the
prey predator and top predator population respectively
119886119894 119886119894119895(119894 119895 = 1 2 3) are positive constants 120591
11 12059121 12059122 12059132
and 12059133
are nonnegative constants 12059111 12059122 12059133
denote thedelay in the negative feedback of the prey predator andtop predator crowding respectively 120591
21 12059132 are constant
delays due to gestation that is mature adult predators canonly contribute to the production of predator biomass 120591 =
max12059111 12059121 12059122 12059132 12059133 120601119894(119905) (119894 = 1 2 3) are continuous
bounded functions in the interval [minus120591 0] The authorsproved that the system is uniformly persistent under someappropriate conditions By means of constructing suitableLyapunov functional sufficient conditions are derived for theglobal asymptotic stability of the positive equilibrium of thesystem
Time delays of one type or another have been incorpo-rated into systems by many researchers since a time delaycould cause a stable equilibrium to become unstable andfluctuation In [2ndash12] authors showed effects of two delayson dynamical behaviors of system
It is well known that periodic solutions can arise throughtheHopf bifurcation in delay differential equations Howeverthese periodic solutions bifurcating from Hopf bifurcationsare generally local Under some circumstances periodicsolutions exist when the parameter is far away from thecritical value Therefore global existence of Hopf bifurcation
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 835310 16 pageshttpdxdoiorg1011552014835310
2 Abstract and Applied Analysis
is a more interesting and difficult topic A great deal ofresearch has been devoted to the topics [13ndash21] In this paperlet 12059111
= 12059122
= 12059133
= 0 12059121
= 1205911 12059132
= 1205912in (1) we
considerHopf bifurcation and global periodic solutions of thefollowing system with two unequal and nonzero delays
1198891199091
119889119905
= 1199091(119905) [1198861minus 119886111199091(119905) minus
119886121199092(119905)
1198981+ 1199091(119905)
]
1198891199092
119889119905
= 1199092(119905) [minus119886
2+
119886211199091(119905 minus 1205911)
1198981+ 1199091(119905 minus 1205911)
minus 119886221199092(119905) minus
119886231199093(119905)
1198982+ 1199092(119905)
]
1198891199093
119889119905
= 1199093(119905) [minus119886
3+
119886321199092(119905 minus 1205912)
1198982+ 1199092(119905 minus 1205912)
minus 119886331199093(119905)]
(3)
with initial conditions
119909119894(119905) = 120601
119894(119905) 119905 isin [minus120591 0] 120601
119894(0) gt 0
119894 = 1 2 3 120591 = max 1205911 1205912
(4)
Our goal is to investigate the possible stability switches of thepositive equilibrium and stability of periodic orbits arisingdue to a Hopf bifurcation when one of the delays is treatedas a bifurcation parameter Special attention is paid to theglobal continuation of local Hopf bifurcation when the delays1205911
= 1205912
This paper is organized as follows In Section 2 byanalyzing the characteristic equation of the linearized systemof system (3) at positive equilibrium the sufficient conditionsensuring the local stability of the positive equilibrium andthe existence of Hopf bifurcation are obtained [22] Someexplicit formulas determining the direction and stabilityof periodic solutions bifurcating from Hopf bifurcationsare demonstrated by applying the normal form methodand center manifold theory due to Hassard et al [23] inSection 3 In Section 4 we consider the global existence ofthese bifurcating periodic solutions [24] with two differentdelays Some numerical simulation results are included inSection 5
2 Stability of the Positive Equilibrium andLocal Hopf Bifurcations
In this section we first study the existence and local stabilityof the positive equilibrium and then investigate the effect ofdelay and the conditions for existence of Hopf bifurcations
There are at most four nonnegative equilibria for system(3)
1198641= (0 0 0) 119864
2= (
1198861
11988611
0 0)
1198643= (1199091 1199092 0) 119864
lowast= (119909lowast
1 119909lowast
2 119909lowast
3)
(5)
where (1199091 1199092 0) and (119909
lowast
1 119909lowast
2 119909lowast
3) satisfy
1198861minus 119886111199091minus
119886121199092
1198981+ 1199091
= 0
minus1198862+
119886211199091
1198981+ 1199091
minus 119886221199092= 0
(6)
1198861minus 11988611119909lowast
1(119905) minus
11988612119909lowast
2(119905)
1198981+ 119909lowast
1(119905)
= 0
minus1198862+
11988621119909lowast
1(119905)
1198981+ 119909lowast
1(119905)
minus 11988622119909lowast
2(119905) minus
11988623119909lowast
3(119905)
1198982+ 119909lowast
2(119905)
= 0
minus1198863+
11988632119909lowast
2(119905)
1198982+ 119909lowast
2(119905)
minus 11988633119909lowast
3(119905) = 0
(7)
where 1198643is a nonnegative equilibrium point if there is a
positive solution of (6) and 119864lowastis a nonnegative equilibrium
point if there is a positive solution of (7)Let
(H1) 1198861(11988621
minus 1198862) gt 119898
1119886211988611
(H2) (11988632minus1198863)[1198861(11988621minus1198862)minus11989811198862]minus1198982119886311988622(1198861+119898111988611) gt 0
(H3) 1199092(11988632
minus 1198863) minus 11989821198863gt 0
(H4) 11988622(11988611
minus (11988611198981)) minus (119886
12119886211198982
1) gt 0
From [1 25] we know that if (1198671) (1198672) (1198673) and (119867
4)
hold 1198643and 119864
lowastalways exist as nonnegative equilibria
Let119864 = (11990910 11990920 11990930) be the arbitrary equilibrium point
and let 1199091(119905) = 119909
1(119905) minus 119909
10 1199092(119905) = 119909
2(119905) minus 119909
20 1199093(119905) =
1199093(119905)minus119909
30 still denote 119909
1(119905) 1199092(119905) 1199093(119905) by 119909
1(119905) 1199092(119905) 1199093(119905)
respectively then the linearized system of the correspondingequations at 119864 is as follows
(119905) = 119861119906 (119905) + 119862119906 (119905 minus 1205911) + 119863119906 (119905 minus 120591
2) (8)
where
119906 (119905) = (1199091(119905) 1199092(119905) 1199093(119905))119879
119861 = (119887119894119895)3times3
119862 = (119888119894119895)3times3
119863 = (119889119894119895)3times3
11988711
= 1198861minus 21198861111990910
minus
11988612119898111990920
(1198981+ 11990910)2 119887
12= minus
1198861211990910
1198981+ 11990910
11988722
= minus1198862+
1198862111990910
1198981+ 11990910
minus 21198862211990920
minus
11988623119898211990930
(1198982+ 11990920)2
11988723
= minus
1198862311990920
1198982+ 11990920
11988733
= minus1198863+
1198863211990920
1198982+ 11990920
minus 21198863311990930
11988821
=
11988621119898111990920
(1198981+ 11990910)2 119889
32=
11988632119898211990930
(1198982+ 11990920)2
(9)
all the others of 119887119894119895 119888119894119895 and 119889
119894119895are 0
The characteristic equation for system (8) is
1205823+11990121205822+1199011120582 + 1199010+ (1199021120582 + 1199020) 119890minus1205821205911+ (1199031120582 + 1199030) 119890minus1205821205912= 0
(10)
Abstract and Applied Analysis 3
where
1199012= minus (119887
11+ 11988722
+ 11988733)
1199011= 1198871111988722
+ 1198872211988733
+ 1198871111988733
1199010= minus119887111198872211988733
1199021= minus1198871211988821 1199020= 119887121198882111988733
1199031= minus1198872311988932 1199030= 119887111198872311988932
(11)
We consider the following cases
(1) 119864 = 1198641 The characteristic equation reduces to
(120582 minus 1198861) (120582 + 119886
2) (120582 + 119886
3) = 0 (12)
There are always a positive root 1198861and two negative roots
1198862 1198863of (12) hence 119864
1is a saddle point
(2) 119864 = 1198642 Equation (10) takes the form
(120582 + 1198861) (120582 + 119886
2minus
119886111988612
119898111988611
+ 1198861
) (120582 + 1198863) = 0 (13)
There is a positive root 120582 = (119886111988612(119898111988611
+ 1198861)) minus 119886
2if
119886111988612(119898111988611
+ 1198861) gt 119886
2 hence 119864
2is a saddle point If
119886111988612(119898111988611
+ 1198861) lt 119886
2 1198642is locally asymptotically stable
(3) 119864 = 1198643 The characteristic equation is
(120582 minus 11988733) [1205822minus (11988711
+ 11988722) 120582 + 119887
1111988722
minus 1198871211988821119890minus1205821205911] = 0 (14)
We will analyse the distribution of the characteristic root of(14) from Ruan and Wei [26] which is stated as follows
Lemma 1 Consider the exponential polynomial
119875 (120582 119890minus1205821205911 119890
minus120582120591119898)
= 120582119899+ 119901(0)
1120582119899minus1
+ sdot sdot sdot + 119901(0)
119899minus1120582 + 119901(0)
119899
+ [119901(1)
1120582119899minus1
+ sdot sdot sdot + 119901(1)
119899minus1120582 + 119901(1)
119899] 119890minus1205821205911
+ sdot sdot sdot + [119901(119898)
1120582119899minus1
+ sdot sdot sdot + 119901(119898)
119899minus1120582 + 119901(119898)
119899] 119890minus120582120591119898
(15)
where 120591119894⩾ 0 (119894 = 1 2 119898) and 119901
(119894)
119895(119894 = 0 1 119898 119895 =
1 2 119899) are constants As (1205911 1205912 120591
119898) vary the sum of
the order of the zeros of 119875(120582 119890minus1205821205911 119890minus120582120591119898) on the open righthalf plane can change only if a zero appears on or crosses theimaginary axis
By using Lemma 1 we can easily obtain the followingresults
Lemma 2 If 1198643is a nonnegative equilibrium point then
(1) 1198643is unstable if 119887
33gt 0
(2) 1198643is locally asymptotically stable if 119887
33lt 0 11988711
+ 11988722
lt
0 1198871111988722
minus 1198871211988821
gt 0 and 1198871111988722
+ 1198871211988821
gt 0
Proof (1) 120582 = 11988733
is a root of (14) if 11988733
gt 0 then 1198643is
unstable(2) Clearly 120582 = 0 is not a root of (14) we should discuss
the following equation instead of (14)
1205822minus (11988711
+ 11988722) 120582 + 119887
1111988722
minus 1198871211988821119890minus1205821205911= 0 (16)
Assume that 119894120596 with 120596 gt 0 is a solution of (16) Substituting120582 = 119894120596 into (16) and separating the real and imaginary partsyield
minus1205962+ 1198871111988722
= 1198871211988821cos120596120591
1
120596 (11988711
+ 11988722) = 1198871211988821sin120596120591
1
(17)
which implies
1205964+ (1198872
11+ 1198872
22) 1205962+ 1198872
111198872
22minus 1198872
121198882
21= 0 (18)
If 1198872111198872
22minus 1198872
121198882
21gt 0 that is (119887
1111988722
+ 1198871211988821)(1198871111988722
minus 1198871211988821) gt
0 there is no real root of (16) Hence there is no purelyimaginary root of (18) When 120591
1= 0 (16) reduces to
1205822minus (11988711
+ 11988722) 120582 + 119887
1111988722
minus 1198871211988821
= 0 (19)
If 11988711+ 11988722
lt 0 and 1198871111988722minus 1198871211988821
gt 0 both roots of (19) havenegative real parts Thus by using Lemma 1 when 119887
33lt 0
11988711
+ 11988722
lt 0 1198871111988722
minus 1198871211988821
gt 0 and 1198871111988722
+ 1198871211988821
gt 0 1198643is
locally asymptotically stable
(4)119864 = 119864lowastThe characteristic equation about119864
lowastis (10) In the
following we will analyse the distribution of roots of (10)Weconsider four cases
Case a Consider1205911= 1205912= 0
The associated characteristic equation of system (3) is
1205823+ 11990121205822+ (1199011+ 1199021+ 1199031) 120582 + (119901
0+ 1199020+ 1199030) = 0 (20)
Let
(H5) 1199012gt 0 119901
2(1199011+1199021+1199031)minus(1199010+1199020+1199030) gt 0 119901
0+1199020+1199030gt
0
By Routh-Hurwitz criterion we have the following
Theorem 3 For 1205911= 1205912= 0 assume that (119867
1)ndash(1198675) hold
Thenwhen 1205911= 1205912= 0 the positive equilibrium119864
lowast(119909lowast
1 119909lowast
2 119909lowast
3)
of system (3) is locally asymptotically stable
Case b Consider1205911= 0 1205912gt 0
The associated characteristic equation of system (3) is
1205823+ 11990121205822+ (1199011+ 1199021) 120582 + (119901
0+ 1199020) + (1199031120582 + 1199030) 119890minus1205821205912= 0
(21)
4 Abstract and Applied Analysis
We want to determine if the real part of some rootincreases to reach zero and eventually becomes positive as 120591varies Let 120582 = 119894120596 (120596 gt 0) be a root of (21) then we have
minus 1198941205963minus 11990121205962+ 119894 (1199011+ 1199021) 120596 + (119901
0+ 1199020)
+ (1199031120596119894 + 119903
0) (cos120596120591
2minus 119894 sin120596120591
2) = 0
(22)
Separating the real and imaginary parts we have
minus1205963+ (1199011+ 1199021) 120596 = 119903
0sin120596120591
2minus 1199031120596 cos120596120591
2
minus11990121205962+ (1199010+ 1199020) = minus119903
1120596 sin120596120591
2minus 1199030cos120596120591
2
(23)
It follows that
1205966+ 119898121205964+ 119898111205962+ 11989810
= 0 (24)
where 11989812
= 1199012
2minus 2(119901
1+ 1199021) 11989811
= (1199011+ 1199021)2minus 21199012(1199010+
1199020) minus 1199032
1 11989810
= (1199010+ 1199020)2minus 1199032
0
Denoting 119911 = 1205962 (24) becomes
1199113+ 119898121199112+ 11989811119911 + 119898
10= 0 (25)
Let
ℎ1(119911) = 119911
3+ 119898121199112+ 11989811119911 + 119898
10 (26)
we have
119889ℎ1(119911)
119889119911
= 31199112+ 211989812119911 + 119898
11 (27)
If 11989810
= (1199010+ 1199020)2minus 1199032
0lt 0 then ℎ
1(0) lt
0 lim119911rarr+infin
ℎ1(119911) = +infin We can know that (25) has at least
one positive rootIf 11989810
= (1199010+ 1199020)2minus 1199032
0ge 0 we obtain that when Δ =
1198982
12minus 311989811
le 0 (25) has no positive roots for 119911 isin [0 +infin)On the other hand when Δ = 119898
2
12minus 311989811
gt 0 the followingequation
31199112+ 211989812119911 + 119898
11= 0 (28)
has two real roots 119911lowast11
= (minus11989812
+ radicΔ)3 119911lowast
12= (minus119898
12minus
radicΔ)3 Because of ℎ101584010158401(119911lowast
11) = 2radicΔ gt 0 ℎ
10158401015840
1(119911lowast
12) = minus2radicΔ lt
0 119911lowast
11and 119911lowast12are the local minimum and the localmaximum
of ℎ1(119911) respectively By the above analysis we immediately
obtain the following
Lemma 4 (1) If 11989810
ge 0 and Δ = 1198982
12minus 311989811
le 0 (25) hasno positive root for 119911 isin [0 +infin)
(2) If 11989810
ge 0 and Δ = 1198982
12minus 311989811
gt 0 (25) has at leastone positive root if and only if 119911lowast
11= (minus119898
12+ radicΔ)3 gt 0 and
ℎ1(119911lowast
11) le 0
(3) If 11989810
lt 0 (25) has at least one positive root
Without loss of generality we assume that (25) has threepositive roots defined by 119911
11 11991112 11991113 respectively Then
(24) has three positive roots
12059611
= radic11991111 120596
12= radic11991112 120596
13= radic11991113 (29)
From (23) we have
cos120596111989612059121119896
=
11990311205964
1119896+ [11990121199030minus (1199021+ 1199011) 1199031] 1205962
1119896minus 1199030(1199020+ 1199010)
1199032
0+ 1199032
11205962
1119896
(30)
Thus if we denote
120591(119895)
21119896
=
1
1205961119896
times arccos ((11990311205964
1119896+ [11990121199030minus (1199021+ 1199011) 1199031] 1205962
1119896
minus 1199030(1199020+ 1199010))
times (1199032
0+ 1199032
11205962
1119896)
minus1
) + 2119895120587
(31)
where 119896 = 1 2 3 119895 = 0 1 2 then plusmn1198941205961119896is a pair of purely
imaginary roots of (21) corresponding to 120591(119895)
21119896
Define
120591210
= 120591(0)
211198960
= min119896=123
120591(0)
21119896
12059610
= 12059611198960
(32)
Let 120582(1205912) = 120572(120591
2)+119894120596(120591
2) be the root of (21) near 120591
2= 120591(119895)
21119896
satisfying
120572 (120591(119895)
21119896
) = 0 120596 (120591(119895)
21119896
) = 1205961119896 (33)
Substituting 120582(1205912) into (21) and taking the derivative with
respect to 1205912 we have
31205822+ 21199012120582 + (119901
1+ 1199021) + 1199031119890minus1205821205912minus 1205912(1199031120582 + 1199030) 119890minus1205821205912
119889120582
1198891205912
= 120582 (1199031120582 + 1199030) 119890minus1205821205912
(34)
Therefore
[
119889120582
1198891205912
]
minus1
=
[31205822+ 21199012120582 + (119901
1+ 1199021)] 1198901205821205912
120582 (1199031120582 + 1199030)
+
1199031
120582 (1199031120582 + 1199030)
minus
1205912
120582
(35)
When 1205912
= 120591(119895)
21119896
120582(120591(119895)
21119896
) = 1198941205961119896
(119896 = 1 2 3) 120582(1199031120582 +
1199030)|1205912=120591(119895)
21119896
= minus11990311205962
1119896+ 11989411990301205961119896 [31205822
+ 21199012120582 + (119901
1+
1199021)]1198901205821205912|1205912=120591(119895)
21119896
= [minus31205962
1119896+ (1199011+ 1199021)] cos(120596
1119896120591(119895)
21119896
) minus
211990121205961119896sin(1205961119896120591(119895)
21119896
) + 119894211990121205961119896cos(120596
1119896120591(119895)
21119896
) + [minus31205962
1119896+ (1199011+
1199021)] sin(120596
1119896120591(119895)
21119896
)
Abstract and Applied Analysis 5
According to (35) we have
[
Re 119889 (120582 (1205912))
1198891205912
]
minus1
1205912=120591(119895)
21119896
= Re[[31205822+ 21199012120582 + (119901
1+ 1199021)] 1198901205821205912
120582 (1199031120582 + 1199030)
]
1205912=120591(119895)
21119896
+ Re[ 1199031
120582 (1199031120582 + 1199030)
]
1205912=120591(119895)
21119896
=
1
Λ1
minus 11990311205962
1119896[minus31205962
1119896+ (1199011+ 1199021)] cos (120596
1119896120591(119895)
21119896
)
+ 2119903111990121205963
1119896sin (120596
1119896120591(119895)
21119896
) minus 1199032
11205962
1119896
+ 2119903011990121205962
1119896cos (120596
1119896120591(119895)
21119896
)
+ 1199030[minus31205962
1119896+ (1199011+ 1199021)] 1205961119896sin (120596
1119896120591(119895)
21119896
)
=
1
Λ1
31205966
1119896+ 2 [119901
2
2minus 2 (119901
1+ 1199021)] 1205964
1119896
+ [(1199011+ 1199021)2
minus 21199012(1199010+ 1199020) minus 1199032
1] 1205962
1119896
=
1
Λ1
1199111119896(31199112
1119896+ 2119898121199111119896
+ 11989811)
=
1
Λ1
1199111119896ℎ1015840
1(1199111119896)
(36)
where Λ1= 1199032
11205964
1119896+ 1199032
01205962
1119896gt 0 Notice that Λ
1gt 0 1199111119896
gt 0
sign
[
Re 119889 (120582 (1205912))
1198891205912
]
1205912=120591(119895)
21119896
= sign
[
Re 119889 (120582 (1205912))
1198891205912
]
minus1
1205912=120591(119895)
21119896
(37)
then we have the following lemma
Lemma 5 Suppose that 1199111119896
= 1205962
1119896and ℎ
1015840
1(1199111119896) = 0 where
ℎ1(119911) is defined by (26) then 119889(Re 120582(120591(119895)
21119896
))1198891205912has the same
sign with ℎ1015840
1(1199111119896)
From Lemmas 1 4 and 5 and Theorem 3 we can easilyobtain the following theorem
Theorem6 For 1205911= 0 120591
2gt 0 suppose that (119867
1)ndash(1198675) hold
(i) If 11989810
ge 0 and Δ = 1198982
12minus 311989811
le 0 then all rootsof (10) have negative real parts for all 120591
2ge 0 and the
positive equilibrium 119864lowastis locally asymptotically stable
for all 1205912ge 0
(ii) If either 11989810
lt 0 or 11989810
ge 0 Δ = 1198982
12minus 311989811
gt
0 119911lowast
11gt 0 and ℎ
1(119911lowast
11) le 0 then ℎ
1(119911) has at least one
positive roots and all roots of (23) have negative realparts for 120591
2isin [0 120591
210
) and the positive equilibrium 119864lowast
is locally asymptotically stable for 1205912isin [0 120591
210
)
(iii) If (ii) holds and ℎ1015840
1(1199111119896) = 0 then system (3)undergoes
Hopf bifurcations at the positive equilibrium119864lowastfor 1205912=
120591(119895)
21119896
(119896 = 1 2 3 119895 = 0 1 2 )
Case c Consider1205911gt 0 1205912= 0
The associated characteristic equation of system (3) is
1205823+ 11990121205822+ (1199011+ 1199031) 120582 + (119901
0+ 1199030) + (119902
1120582 + 1199020) 119890minus1205821205911= 0
(38)
Similar to the analysis of Case 119887 we get the followingtheorem
Theorem7 For 1205911gt 0 120591
2= 0 suppose that (119867
1)ndash(1198675) hold
(i) If 11989820
ge 0 and Δ = 1198982
22minus 311989821
le 0 then all rootsof (38) have negative real parts for all 120591
1ge 0 and the
positive equilibrium 119864lowastis locally asymptotically stable
for all 1205911ge 0
(ii) If either 11989820
lt 0 or 11989820
ge 0 Δ = 1198982
22minus 311989821
gt 0119911lowast
21gt 0 and ℎ
2(119911lowast
21) le 0 then ℎ
2(119911) has at least one
positive root 1199112119896 and all roots of (38) have negative real
parts for 1205911isin [0 120591
120
) and the positive equilibrium 119864lowast
is locally asymptotically stable for 1205911isin [0 120591
120
)
(iii) If (ii) holds and ℎ1015840
2(1199112119896) = 0 then system (3) undergoes
Hopf bifurcations at the positive equilibrium119864lowastfor 1205911=
120591(119895)
12119896
(119896 = 1 2 3 119895 = 0 1 2 )
where
11989822
= 1199012
2minus 2 (119901
1+ 1199031)
11989821
= (1199011+ 1199031)2
minus 21199012(1199010+ 1199030) minus 1199022
1
11989820
= (1199010+ 1199030)2
minus 1199022
0
ℎ2(119911) = 119911
3+ 119898221199112+ 11989821119911 + 119898
20 119911lowast
21=
minus11989822
+ radicΔ
3
120591(119895)
12119896
=
1
1205962119896
times arccos ((11990211205964
2119896+ [11990121199020minus (1199031+ 1199011) 1199021] 1205962
2119896
minus 1199020(1199030+ 1199010) )
times (1199022
0+ 1199022
11205962
2119896)
minus1
) + 2119895120587
(39)
where 119896 = 1 2 3 119895 = 0 1 2 then plusmn1198941205962119896is a pair of purely
imaginary roots of (38) corresponding to 120591(119895)
12119896
Define
120591120
= 120591(0)
121198960
= min119896=123
120591(0)
12119896
12059610
= 12059611198960
(40)
6 Abstract and Applied Analysis
Case d Consider1205911gt 0 1205912gt 0 1205911
= 1205912
The associated characteristic equation of system (3) is
1205823+11990121205822+1199011120582+1199010+ (1199021120582 + 1199020) 119890minus1205821205911+ (1199031120582+1199030) 119890minus1205821205912=0
(41)
We consider (41) with 1205912= 120591lowast
2in its stable interval [0 120591
210
)Regard 120591
1as a parameter
Let 120582 = 119894120596 (120596 gt 0) be a root of (41) then we have
minus 1198941205963minus 11990121205962+ 1198941199011120596 + 1199010+ (1198941199021120596 + 1199020) (cos120596120591
1minus 119894 sin120596120591
1)
+ (1199030+ 1198941199031120596) (cos120596120591lowast
2minus 119894 sin120596120591
lowast
2) = 0
(42)
Separating the real and imaginary parts we have
1205963minus 1199011120596 minus 1199031120596 cos120596120591lowast
2+ 1199030sin120596120591
lowast
2
= 1199021120596 cos120596120591
1minus 1199020sin120596120591
1
11990121205962minus 1199010minus 1199030cos120596120591lowast
2minus 1199031120596 sin120596120591
lowast
2
= 1199020cos120596120591
1+ 1199021120596 sin120596120591
1
(43)
It follows that
1205966+ 119898331205964+ 119898321205963+ 119898311205962+ 11989830
= 0 (44)
where
11989833
= 1199012
2minus 21199011minus 21199031cos120596120591lowast
2
11989832
= 2 (1199030minus 11990121199031) sin120596120591
lowast
2
11989831
= 1199012
1minus 211990101199012minus 2 (119901
21199030minus 11990111199031) cos120596120591lowast
2+ 1199032
1minus 1199022
1
11989830
= 1199012
0+ 211990101199030cos120596120591lowast
2+ 1199032
0minus 1199022
0
(45)
Denote 119865(120596) = 1205966+ 119898331205964+ 119898321205963+ 119898311205962+ 11989830 If
11989830
lt 0 then
119865 (0) lt 0 lim120596rarr+infin
119865 (120596) = +infin (46)
We can obtain that (44) has at most six positive roots1205961 1205962 120596
6 For every fixed 120596
119896 119896 = 1 2 6 there exists
a sequence 120591(119895)1119896
| 119895 = 0 1 2 3 such that (43) holdsLet
12059110
= min 120591(119895)
1119896| 119896 = 1 2 6 119895 = 0 1 2 3 (47)
When 1205911
= 120591(119895)
1119896 (41) has a pair of purely imaginary roots
plusmn119894120596(119895)
1119896for 120591lowast2isin [0 120591
210
)In the following we assume that
(H6) ((119889Re(120582))119889120591
1)|120582=plusmn119894120596
(119895)
1119896
= 0
Thus by the general Hopf bifurcation theorem for FDEsin Hale [22] we have the following result on the stability andHopf bifurcation in system (3)
Theorem 8 For 1205911
gt 0 1205912
gt 0 1205911
= 1205912 suppose that
(1198671)ndash(1198676) is satisfied If 119898
30lt 0 and 120591
lowast
2isin [0 120591
210
] thenthe positive equilibrium 119864
lowastis locally asymptotically stable for
1205911
isin [0 12059110) System (3) undergoes Hopf bifurcations at the
positive equilibrium 119864lowastfor 1205911= 120591(119895)
1119896
3 Direction and Stability ofthe Hopf Bifurcation
In Section 2 we obtain the conditions underwhich system (3)undergoes the Hopf bifurcation at the positive equilibrium119864lowast In this section we consider with 120591
2= 120591lowast
2isin [0 120591
210
)
and regard 1205911as a parameter We will derive the explicit
formulas determining the direction stability and period ofthese periodic solutions bifurcating from equilibrium 119864
lowastat
the critical values 1205911by using the normal form and the center
manifold theory developed by Hassard et al [23] Withoutloss of generality denote any one of these critical values 120591
1=
120591(119895)
1119896(119896 = 1 2 6 119895 = 0 1 2 ) by 120591
1 at which (43) has a
pair of purely imaginary roots plusmn119894120596 and system (3) undergoesHopf bifurcation from 119864
lowast
Throughout this section we always assume that 120591lowast2lt 12059110
Let 1199061= 1199091minus 119909lowast
1 1199062= 1199091minus 119909lowast
2 1199063= 1199092minus 119909lowast
3 119905 = 120591
1119905
and 120583 = 1205911minus 1205911 120583 isin R Then 120583 = 0 is the Hopf bifurcation
value of system (3) System (3) may be written as a functionaldifferential equation inC([minus1 0]R3)
(119905) = 119871120583(119906119905) + 119891 (120583 119906
119905) (48)
where 119906 = (1199061 1199062 1199063)119879isin R3 and
119871120583(120601) = (120591
1+ 120583) 119861[
[
1206011(0)
1206012(0)
1206013(0)
]
]
+ (1205911+ 120583)119862[
[
1206011(minus1)
1206012(minus1)
1206013(minus1)
]
]
+ (1205911+ 120583)119863
[[[[[[[[[[
[
1206011(minus
120591lowast
2
1205911
)
1206012(minus
120591lowast
2
1205911
)
1206013(minus
120591lowast
2
1205911
)
]]]]]]]]]]
]
(49)
119891 (120583 120601) = (1205911+ 120583)[
[
1198911
1198912
1198913
]
]
(50)
Abstract and Applied Analysis 7
where 120601 = (1206011 1206012 1206013)119879isin C([minus1 0]R3) and
119861 = [
[
11988711
11988712
0
0 11988722
11988723
0 0 11988733
]
]
119862 = [
[
0 0 0
11988821
0 0
0 0 0
]
]
119863 = [
[
0 0 0
0 0 0
0 11988932
0
]
]
1198911= minus (119886
11+ 1198973) 1206012
1(0) minus 119897
11206011(0) 1206012(0) minus 119897
21206012
1(0) 1206012(0)
minus 11989751206013
1(0) minus 119897
41206013
1(0) 1206012(0) + sdot sdot sdot
1198912= minus11989761206012(0) 1206013(0) minus (119897
7+ 11988622) 1206012
2(0) + 119897
11206011(minus1) 120601
2(0)
+ 11989731206012
1(minus1) minus 119897
81206013
2(0) minus 119897
91206012
2(0) 1206013(0)
+ 11989721206012
1(minus1) 120601
2(0) + 119897
31206013
1(minus1) + 119897
41206013
1(minus1) 120601
2(0)
minus 119897101206013
2(0) 1206013(0) + sdot sdot sdot
1198913= 11989761206012(minus
120591lowast
2
1205911
)1206013(0) + 119897
71206012
2(minus
120591lowast
2
1205911
) minus 119886331206012
3(0)
+ 11989791206012
2(minus
120591lowast
2
1205911
)1206013(0) + 119897
81206013
2(minus
120591lowast
2
1205911
)
+ 119897101206013
2(minus
120591lowast
2
1205911
)1206013(0) + sdot sdot sdot
1199011(119909) =
1198861119909
1 + 1198871119909
1199012(119909) =
1198862119909
1 + 1198872119909
1198971= 1199011015840
1(119909lowast)
1198972=
1
2
11990110158401015840
1(119909lowast) 1198973=
1
2
11990110158401015840
1(119909lowast) 1199101lowast
1198974=
1
3
119901101584010158401015840
1(119909lowast) 1198975=
1
3
119901101584010158401015840
1(119909lowast) 1199101lowast
1198976= 1199011015840
2(1199101lowast) 1198977=
1
2
11990110158401015840
2(1199101lowast) 1199102lowast
1198978=
1
3
119901101584010158401015840
2(1199101lowast) 1199102lowast 1198979=
1
2
11990110158401015840
2(1199101lowast)
11989710
=
1
3
119901101584010158401015840
2(1199101lowast)
(51)
Obviously 119871120583(120601) is a continuous linear function mapping
C([minus1 0]R3) intoR3 By the Riesz representation theoremthere exists a 3 times 3 matrix function 120578(120579 120583) (minus1 ⩽ 120579 ⩽ 0)whose elements are of bounded variation such that
119871120583120601 = int
0
minus1
119889120578 (120579 120583) 120601 (120579) for 120601 isin C ([minus1 0] R3) (52)
In fact we can choose
119889120578 (120579 120583) = (1205911+ 120583) [119861120575 (120579) + 119862120575 (120579 + 1) + 119863120575(120579 +
120591lowast
2
1205911
)]
(53)
where 120575 is Dirac-delta function For 120601 isin C([minus1 0]R3)define
119860 (120583) 120601 =
119889120601 (120579)
119889120579
120579 isin [minus1 0)
int
0
minus1
119889120578 (119904 120583) 120601 (119904) 120579 = 0
119877 (120583) 120601 =
0 120579 isin [minus1 0)
119891 (120583 120601) 120579 = 0
(54)
Then when 120579 = 0 the system is equivalent to
119905= 119860 (120583) 119909
119905+ 119877 (120583) 119909
119905 (55)
where 119909119905(120579) = 119909(119905+120579) 120579 isin [minus1 0] For120595 isin C1([0 1] (R3)
lowast)
define
119860lowast120595 (119904) =
minus
119889120595 (119904)
119889119904
119904 isin (0 1]
int
0
minus1
119889120578119879(119905 0) 120595 (minus119905) 119904 = 0
(56)
and a bilinear inner product
⟨120595 (119904) 120601 (120579)⟩=120595 (0) 120601 (0)minusint
0
minus1
int
120579
120585=0
120595 (120585minus120579) 119889120578 (120579) 120601 (120585) 119889120585
(57)
where 120578(120579) = 120578(120579 0) Let 119860 = 119860(0) then 119860 and 119860lowast
are adjoint operators By the discussion in Section 2 weknow that plusmn119894120596120591
1are eigenvalues of 119860 Thus they are also
eigenvalues of 119860lowast We first need to compute the eigenvectorof 119860 and 119860
lowast corresponding to 1198941205961205911and minus119894120596120591
1 respectively
Suppose that 119902(120579) = (1 120572 120573)1198791198901198941205791205961205911 is the eigenvector of 119860
corresponding to 1198941205961205911 Then 119860119902(120579) = 119894120596120591
1119902(120579) From the
definition of 119860119871120583(120601) and 120578(120579 120583) we can easily obtain 119902(120579) =
(1 120572 120573)1198791198901198941205791205961205911 where
120572 =
119894120596 minus 11988711
11988712
120573 =
11988932(119894120596 minus 119887
11)
11988712(119894120596 minus 119887
33) 119890119894120596120591lowast
2
(58)
and 119902(0) = (1 120572 120573)119879 Similarly let 119902lowast(119904) = 119863(1 120572
lowast 120573lowast)1198901198941199041205961205911
be the eigenvector of 119860lowast corresponding to minus119894120596120591
1 By the
definition of 119860lowast we can compute
120572lowast=
minus119894120596 minus 11988711
119888211198901198941205961205911
120573lowast=
11988723(minus119894120596 minus 119887
11)
11988821(119894120596 minus 119887
33) 1198901198941205961205911
(59)
From (57) we have⟨119902lowast(119904) 119902 (120579)⟩
= 119863(1 120572lowast 120573
lowast
) (1 120572 120573)119879
minus int
0
minus1
int
120579
120585=0
119863(1 120572lowast 120573
lowast
) 119890minus1198941205961205911(120585minus120579)
119889120578 (120579)
times (1 120572 120573)119879
1198901198941205961205911120585119889120585
= 1198631 + 120572120572lowast+ 120573120573
lowast
+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573
lowast
120591lowast
2119890minus119894120596120591lowast
2
(60)
8 Abstract and Applied Analysis
Thus we can choose
119863 = 1 + 120572120572lowast+ 120573120573
lowast
+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573
lowast
120591lowast
2119890minus119894120596120591lowast
2
minus1
(61)
such that ⟨119902lowast(119904) 119902(120579)⟩ = 1 ⟨119902lowast(119904) 119902(120579)⟩ = 0
In the remainder of this section we follow the ideas inHassard et al [23] and use the same notations as there tocompute the coordinates describing the center manifold 119862
0
at 120583 = 0 Let 119909119905be the solution of (48) when 120583 = 0 Define
119911 (119905) = ⟨119902lowast 119909119905⟩ 119882 (119905 120579) = 119909
119905(120579) minus 2Re 119911 (119905) 119902 (120579)
(62)
On the center manifold 1198620 we have
119882(119905 120579) = 119882 (119911 (119905) 119911 (119905) 120579)
= 11988220(120579)
1199112
2
+11988211(120579) 119911119911 +119882
02(120579)
1199112
2
+11988230(120579)
1199113
6
+ sdot sdot sdot
(63)
where 119911 and 119911 are local coordinates for center manifold 1198620in
the direction of 119902 and 119902 Note that 119882 is real if 119909119905is real We
consider only real solutions For the solution 119909119905isin 1198620of (48)
since 120583 = 0 we have
= 1198941205961205911119911 + ⟨119902
lowast(120579) 119891 (0119882 (119911 (119905) 119911 (119905) 120579)
+ 2Re 119911 (119905) 119902 (120579))⟩
= 1198941205961205911119911+119902lowast(0) 119891 (0119882 (119911 (119905) 119911 (119905) 0) +2Re 119911 (119905) 119902 (0))
= 1198941205961205911119911 + 119902lowast(0) 1198910(119911 119911) ≜ 119894120596120591
1119911 + 119892 (119911 119911)
(64)
where
119892 (119911 119911) = 119902lowast(0) 1198910(119911 119911) = 119892
20
1199112
2
+ 11989211119911119911
+ 11989202
1199112
2
+ 11989221
1199112119911
2
+ sdot sdot sdot
(65)
By (62) we have 119909119905(120579) = (119909
1119905(120579) 1199092119905(120579) 1199093119905(120579))119879= 119882(119905 120579) +
119911119902(120579) + 119911 119902(120579) and then
1199091119905(0) = 119911 + 119911 +119882
(1)
20(0)
1199112
2
+119882(1)
11(0) 119911119911
+119882(1)
02(0)
1199112
2
+ 119900 (|(119911 119911)|3)
1199092119905(0) = 119911120572 + 119911 120572 +119882
(2)
20(0)
1199112
2
+119882(2)
11(0) 119911119911 +119882
(2)
02(0)
1199112
2
+ 119900 (|(119911 119911)|3)
1199093119905(0) = 119911120573 + 119911120573 +119882
(3)
20(0)
1199112
2
+119882(3)
11(0) 119911119911
+119882(3)
02(0)
1199112
2
+ 119900 (|(119911 119911)|3)
1199091119905(minus1) = 119911119890
minus1198941205961205911+ 1199111198901198941205961205911+119882(1)
20(minus1)
1199112
2
+119882(1)
11(minus1) 119911119911 +119882
(1)
02(minus1)
1199112
2
+ 119900 (|(119911 119911)|3)
1199092119905(minus1) = 119911120572119890
minus1198941205961205911+ 119911 120572119890
1198941205961205911+119882(2)
20(minus1)
1199112
2
+119882(2)
11(minus1) 119911119911 +119882
(2)
02(minus1)
1199112
2
+ 119900 (|(119911 119911)|3)
1199093119905(minus1) = 119911120573119890
minus1198941205961205911+ 119911120573119890
1205961205911+119882(3)
20(minus1)
1199112
2
+119882(3)
11(minus1) 119911119911 +119882
(3)
02(minus1)
1199112
2
+ 119900 (|(119911 119911)|3)
1199091119905(minus
120591lowast
2
12059110
) = 119911119890minus119894120596120591lowast
2+ 119911119890119894120596120591lowast
2+119882(1)
20(minus
120591lowast
2
1205911
)
1199112
2
+119882(1)
11(minus
120591lowast
2
1205911
)119911119911 +119882(1)
02(minus
120591lowast
2
1205911
)
1199112
2
+ 119900 (|(119911 119911)|3)
1199092119905(minus
120591lowast
2
1205911
) = 119911120572119890minus119894120596120591lowast
2+ 119911 120572119890
119894120596120591lowast
2+119882(2)
20(minus
120591lowast
2
1205911
)
1199112
2
+119882(2)
11(minus
120591lowast
2
1205911
)119911119911 +119882(2)
02(minus
120591lowast
2
1205911
)
1199112
2
+ 119900 (|(119911 119911)|3)
1199093119905(minus
120591lowast
2
1205911
) = 119911120573119890minus119894120596120591lowast
2+ 119911120573119890
120596120591lowast
2+119882(3)
20(minus
120591lowast
2
1205911
)
1199112
2
+119882(3)
11(minus
120591lowast
2
1205911
)119911119911 +119882(3)
02(minus
120591lowast
2
1205911
)
1199112
2
+ 119900 (|(119911 119911)|3)
(66)
Abstract and Applied Analysis 9
It follows together with (50) that
119892 (119911 119911)=119902lowast(0) 1198910(119911 119911) =119863120591
1(1 120572lowast 120573
lowast
) (119891(0)
1119891(0)
2119891(0)
3)
119879
= 1198631205911[minus (119886
11+ 1198973) 1206012
1(0) minus 119897
11206011(0) 1206012(0)
minus 11989721206012
1(0) 1206012(0) minus 119897
51206013
1(0) minus 119897
41206013
1(0) 1206012(0)
+ sdot sdot sdot ]
+ 120572lowast[11989761206012(0) 1206013(0) (1198977+ 11988622) 1206012
2(0)
+ 11989711206011(minus1) 120601
2(0)
+ 11989731206012
1(minus1) minus 119897
81206013
2(0) minus 119897
91206012
2(0) 1206013(0)
+ 11989721206012
1(minus1) 120601
2(0) + 119897
31206013
1(minus1) + sdot sdot sdot ]
+ 120573
lowast
[11989761206012(minus
120591lowast
2
1205911
)1206013(0) + 119897
71206012
2(minus
120591lowast
2
1205911
)
minus 119886331206012
3(0) + 119897
81206013
2(minus
120591lowast
2
1205911
) + sdot sdot sdot ]
(67)
Comparing the coefficients with (65) we have
11989220
= 1198631205911[minus2 (119886
11+ 1198973) minus 2120572119897
1]
+ 120572lowast[21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591
1
minus21198976120572120573 minus 2 (119897
7+ 11988622) 1205722]
+ 120573
lowast
[21198976120572120573119890minus119894120596120591lowast
2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591
lowast
2
minus2119886331205732]
11989211
= 1198631205911[minus2 (119886
11+ 1198973) minus 1198971(120572 + 120572)]
+ 120572lowast[1198971(1205721198901198941205961205911+ 120572119890minus1198941205961205911) minus 1198976(120572120573 + 120572120573)
minus2 (1198977+ 11988622) 120572120572 + 2119897
3]
+ 120573
lowast
[1198976(120573120572119890119894120596120591lowast
2+ 120572120573119890
minus119894120596120591lowast
2)
+ 21198977120572120572 minus 119886
33120573120573]
11989202
= 21198631205911[minus2 (119886
11+ 1198973) minus 21198971120572]
+ 120572lowast[minus21198976120572120573 minus 2 (119897
7+ 11988622) 1205722+ 211989711205721198901198941205961205911
+2119897311989021198941205961205911]
+ 120573
lowast
[21198976120572120573119890119894120596120591lowast
2+ 2119897712057221198902119894120596120591lowast
2minus 11988633120573
2
]
11989221
= 1198631205911[minus (119886
11+ 1198973) (2119882
(1)
20(0) + 4119882
(1)
11(0))
minus 1198971(2120572119882
(1)
11(0) + 120572119882
(1)
20(0) + 119882
(2)
20(0)
+ 2119882(2)
11(0))]
+ 120572lowast[minus1198976(2120573119882
(2)
11(0) + 120572119882
(3)
20(0) + 120573119882
(2)
20(0)
+ 2120572119882(3)
11(0)) minus (119897
7+ 11988622)
times (4120572119882(2)
11(0) + 2120572119882
(2)
20(0))
+ 1198971(2120572119882
(1)
11(minus1) + 120572119882
(1)
20(minus1)
+ 119882(2)
20(0) 1198901198941205961205911+ 2119882
(2)
11(0) 119890minus1198941205961205911)
+ 1198973(4119882(1)
11(minus1) 119890
minus1198941205961205911+2119882(1)
20(minus1) 119890
1198941205961205911)]
+ 120573
lowast
[1198976(2120573119882
(2)
11(minus
120591lowast
2
1205911
) + 120573119882(2)
20(minus
120591lowast
2
1205911
)
+ 120572119882(3)
20(0) 119890119894120596120591lowast
2+ 2120572119882
(3)
11(0) 119890minus119894120596120591lowast
2)
+ 1198977(4120572119882
(2)
11(minus
120591lowast
2
1205911
) 119890minus119894120596120591lowast
2
+2120572119882(2)
20(minus
120591lowast
2
1205911
) 119890119894120596120591lowast
2)
minus 11988633(4120573119882
(3)
11(0) + 2120573119882
(3)
20(0))]
(68)
where
11988220(120579) =
11989411989220
1205961205911
119902 (0) 1198901198941205961205911120579+
11989411989202
31205961205911
119902 (0) 119890minus1198941205961205911120579+ 119864111989021198941205961205911120579
11988211(120579) = minus
11989411989211
1205961205911
119902 (0) 1198901198941205961205911120579+
11989411989211
1205961205911
119902 (0) 119890minus1198941205961205911120579+ 1198642
1198641= 2
[[[[[
[
2119894120596 minus 11988711
minus11988712
0
minus11988821119890minus2119894120596120591
12119894120596 minus 119887
22minus11988723
0 minus11988932119890minus2119894120596120591
lowast
22119894120596 minus 119887
33
]]]]]
]
minus1
times
[[[[[
[
minus2 (11988611
+ 1198973) minus 2120572119897
1
21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591
1minus 21198976120572120573 minus 2 (119897
7+ 11988622) 1205722
21198976120572120573119890minus119894120596120591lowast
2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591
lowast
2minus 2119886331205732
]]]]]
]
10 Abstract and Applied Analysis
1198642= 2
[[[[
[
minus11988711
minus11988712
0
minus11988821
minus11988722
minus11988723
0 minus11988932
minus11988733
]]]]
]
minus1
times
[[[[
[
minus2 (11988611
+ 1198973) minus 1198971(120572 + 120572)
1198971(1205721198901198941205961205911+ 120572119890minus1198941205961205911) minus 1198976(120572120573 + 120572120573) minus 2 (119897
7+ 11988622) 120572120572 + 2119897
3
1198976(120573120572119890119894120596120591lowast
2+ 120572120573119890
minus119894120596120591lowast
2) + 21198977120572120572 minus 119886
33120573120573
]]]]
]
(69)
Thus we can determine11988220(120579) and119882
11(120579) Furthermore
we can determine each 119892119894119895by the parameters and delay in (3)
Thus we can compute the following values
1198881(0) =
119894
21205961205911
(1198922011989211
minus 2100381610038161003816100381611989211
1003816100381610038161003816
2
minus
1
3
100381610038161003816100381611989202
1003816100381610038161003816
2
) +
1
2
11989221
1205832= minus
Re 1198881(0)
Re 1205821015840 (1205911)
1198792= minus
Im 1198881(0) + 120583
2Im 120582
1015840(1205911)
1205961205911
1205732= 2Re 119888
1(0)
(70)
which determine the quantities of bifurcating periodic solu-tions in the center manifold at the critical value 120591
1 Suppose
Re1205821015840(1205911) gt 0 120583
2determines the directions of the Hopf
bifurcation if 1205832
gt 0(lt 0) then the Hopf bifurcationis supercritical (subcritical) and the bifurcation exists for120591 gt 120591
1(lt 1205911) 1205732determines the stability of the bifurcation
periodic solutions the bifurcating periodic solutions arestable (unstable) if 120573
2lt 0(gt 0) and119879
2determines the period
of the bifurcating periodic solutions the period increases(decreases) if 119879
2gt 0(lt 0)
4 Numerical Simulation
We consider system (3) by taking the following coefficients1198861
= 03 11988611
= 58889 11988612
= 1 1198981
= 1 1198862
=
01 11988621
= 27 11988622
= 12 11988623
= 12 1198982
= 1 1198863
=
02 11988632
= 25 11988633
= 12 We have the unique positiveequilibrium 119864
lowast= (00451 00357 00551)
By computation we get 11989810
= minus00104 12059611
= 0516411991111
= 02666 ℎ10158401(11991111) = 03402 120591
20
= 32348 FromTheorem 6 we know that when 120591
1= 0 the positive
equilibrium 119864lowastis locally asymptotically stable for 120591
2isin
[0 32348) When 1205912crosses 120591
20
the equilibrium 119864lowastloses its
stability and Hopf bifurcation occurs From the algorithm inSection 3 we have 120583
2= 56646 120573
2= minus31583 119879
2= 5445
whichmeans that the bifurcation is supercritical and periodicsolution is stable The trajectories and the phase graphs areshown in Figures 1 and 2
Regarding 1205911as a parameter and let 120591
2= 29 isin
[0 32348) we can observe that with 1205911increasing the pos-
itive equilibrium 119864lowastloses its stability and Hopf bifurcation
occurs (see Figures 3 and 4)
5 Global Continuationof Local Hopf Bifurcations
In this section we study the global continuation of peri-odic solutions bifurcating from the positive equilibrium(119864lowast 120591(119895)
1119896) (119896 = 1 2 6 119895 = 0 1 ) Throughout this
section we follow closely the notations in [24] and assumethat 120591
2= 120591lowast
2isin [0 120591
210
) regarding 1205911as a parameter For
simplification of notations setting 119911119905(119905) = (119909
1119905 1199092119905 1199093119905)119879 we
may rewrite system (3) as the following functional differentialequation
(119905) = 119865 (119911119905 1205911 119901) (71)
where 119911119905(120579) = (119909
1119905(120579) 1199092119905(120579) 1199093119905(120579))119879
= (1199091(119905 + 120579) 119909
2(119905 +
120579) 1199093(119905 + 120579))
119879 for 119905 ge 0 and 120579 isin [minus1205911 0] Since 119909
1(119905) 1199092(119905)
and 1199093(119905) denote the densities of the prey the predator and
the top predator respectively the positive solution of system(3) is of interest and its periodic solutions only arise in the firstquadrant Thus we consider system (3) only in the domain1198773
+= (1199091 1199092 1199093) isin 1198773 1199091gt 0 119909
2gt 0 119909
3gt 0 It is obvious
that (71) has a unique positive equilibrium 119864lowast(119909lowast
1 119909lowast
2 119909lowast
3) in
1198773
+under the assumption (119867
1)ndash(1198674) Following the work of
[24] we need to define
119883 = 119862([minus1205911 0] 119877
3
+)
Γ = 119862119897 (119911 1205911 119901) isin X times R times R+
119911 is a 119901-periodic solution of system (71)
N = (119911 1205911 119901) 119865 (119911 120591
1 119901) = 0
(72)
Let ℓ(119864lowast120591(119895)
11198962120587120596
(119895)
1119896)denote the connected component pass-
ing through (119864lowast 120591(119895)
1119896 2120587120596
(119895)
1119896) in Γ where 120591
(119895)
1119896is defined by
(43) We know that ℓ(119864lowast120591(119895)
11198962120587120596
(119895)
1119896)through (119864
lowast 120591(119895)
1119896 2120587120596
(119895)
1119896)
is nonemptyFor the benefit of readers we first state the global Hopf
bifurcation theory due to Wu [24] for functional differentialequations
Lemma 9 Assume that (119911lowast 120591 119901) is an isolated center satis-
fying the hypotheses (A1)ndash(A4) in [24] Denote by ℓ(119911lowast120591119901)
theconnected component of (119911
lowast 120591 119901) in Γ Then either
(i) ℓ(119911lowast120591119901)
is unbounded or(ii) ℓ(119911lowast120591119901)
is bounded ℓ(119911lowast120591119901)
cap Γ is finite and
sum
(119911120591119901)isinℓ(119911lowast120591119901)capN
120574119898(119911lowast 120591 119901) = 0 (73)
for all 119898 = 1 2 where 120574119898(119911lowast 120591 119901) is the 119898119905ℎ crossing
number of (119911lowast 120591 119901) if119898 isin 119869(119911
lowast 120591 119901) or it is zero if otherwise
Clearly if (ii) in Lemma 9 is not true then ℓ(119911lowast120591119901)
isunbounded Thus if the projections of ℓ
(119911lowast120591119901)
onto 119911-spaceand onto 119901-space are bounded then the projection of ℓ
(119911lowast120591119901)
onto 120591-space is unbounded Further if we can show that the
Abstract and Applied Analysis 11
0 200 400 60000451
00451
00452
t
x1(t)
t
0 200 400 60000356
00357
00358
x2(t)
t
0 200 400 6000055
00551
00552
x3(t)
00451
00451
00452
00356
00357
00357
003570055
0055
00551
00551
00552
x 1(t)
x2 (t)
x3(t)
Figure 1 The trajectories and the phase graph with 1205911= 0 120591
2= 29 lt 120591
20= 32348 119864
lowastis locally asymptotically stable
0 500 1000004
0045
005
0 500 1000002
004
006
0 500 10000
005
01
t
t
t
x1(t)
x2(t)
x3(t)
0044
0046
0048
002
003
004
005003
004
005
006
007
008
x 1(t)
x2 (t)
x3(t)
Figure 2 The trajectories and the phase graph with 1205911= 0 120591
2= 33 gt 120591
20= 32348 a periodic orbit bifurcate from 119864
lowast
12 Abstract and Applied Analysis
0 500 1000 1500
00451
00452
0 500 1000 15000035
0036
0037
0 500 1000 15000054
0055
0056
t
t
t
x1(t)
x2(t)
x3(t)
0045100451
0045200452
0035
00355
003600546
00548
0055
00552
00554
00556
00558
x1(t)
x2 (t)
x3(t)
Figure 3 The trajectories and the phase graph with 1205911= 09 120591
2= 29 119864
lowastis locally asymptotically stable
0 500 1000 15000044
0045
0046
0 500 1000 1500002
003
004
0 500 1000 1500004
006
008
t
t
t
x1(t)
x2(t)
x3(t)
00445
0045
00455
0032
0034
0036
00380052
0054
0056
0058
006
0062
0064
x 1(t)
x2 (t)
x3(t)
Figure 4 The trajectories and the phase graph with 1205911= 12 120591
2= 29 a periodic orbit bifurcate from 119864
lowast
Abstract and Applied Analysis 13
projection of ℓ(119911lowast120591119901)
onto 120591-space is away from zero thenthe projection of ℓ
(119911lowast120591119901)
onto 120591-space must include interval[120591 infin) Following this ideal we can prove our results on theglobal continuation of local Hopf bifurcation
Lemma 10 If the conditions (1198671)ndash(1198674) hold then all nontriv-
ial periodic solutions of system (71) with initial conditions
1199091(120579) = 120593 (120579) ge 0 119909
2(120579) = 120595 (120579) ge 0
1199093(120579) = 120601 (120579) ge 0 120579 isin [minus120591
1 0)
120593 (0) gt 0 120595 (0) gt 0 120601 (0) gt 0
(74)
are uniformly bounded
Proof Suppose that (1199091(119905) 1199092(119905) 1199093(119905)) are nonconstant peri-
odic solutions of system (3) and define
1199091(1205851) = min 119909
1(119905) 119909
1(1205781) = max 119909
1(119905)
1199092(1205852) = min 119909
2(119905) 119909
2(1205782) = max 119909
2(119905)
1199093(1205853) = min 119909
3(119905) 119909
3(1205783) = max 119909
3(119905)
(75)
It follows from system (3) that
1199091(119905) = 119909
1(0) expint
119905
0
[1198861minus 119886111199091(119904) minus
119886121199092(119904)
1198981+ 1199091(119904)
] 119889119904
1199092(119905) = 119909
2(0) expint
119905
0
[minus1198862+
119886211199091(119904 minus 1205911)
1198981+ 1199091(119904 minus 1205911)
minus 119886221199092(119904) minus
119886231199093(119904)
1198982+ 1199092(119904)
] 119889119904
1199093(119905) = 119909
3(0) expint
119905
0
[minus1198863+
119886321199092(119904 minus 120591
lowast
2)
1198982+ 1199092(119904 minus 120591
lowast
2)
minus 119886331199093(119904)] 119889119904
(76)
which implies that the solutions of system (3) cannot crossthe 119909119894-axis (119894 = 1 2 3) Thus the nonconstant periodic orbits
must be located in the interior of first quadrant It followsfrom initial data of system (3) that 119909
1(119905) gt 0 119909
2(119905) gt
0 1199093(119905) gt 0 for 119905 ge 0
From the first equation of system (3) we can get
0 = 1198861minus 119886111199091(1205781) minus
119886121199092(1205781)
1198981+ 1199091(1205781)
le 1198861minus 119886111199091(1205781) (77)
thus we have
1199091(1205781) le
1198861
11988611
(78)
From the second equation of (3) we obtain
0 = minus 1198862+
119886211199091(1205782minus 1205911)
1198981+ 1199091(1205782minus 1205911)
minus 119886221199092(1205782)
minus
119886231199093(1205782)
1198982+ 1199092(1205782)
le minus1198862+
11988621(119886111988611)
1198981+ (119886111988611)
minus 119886221199092(1205782)
(79)
therefore one can get
1199092(1205782) le
minus1198862(119886111198981+ 1198861) + 119886111988621
11988622(119886111198981+ 1198861)
≜ 1198721 (80)
Applying the third equation of system (3) we know
0 = minus1198863+
119886321199092(1205783minus 120591lowast
2)
1198982+ 1199092(1205783minus 120591lowast
2)
minus 119886331199093(1205783)
le minus1198863+
119886321198721
1198982+1198721
minus 119886331199093(1205783)
(81)
It follows that
1199093(1205783) le
minus1198863(1198982+1198721) + 119886321198721
11988633(1198982+1198721)
≜ 1198722 (82)
This shows that the nontrivial periodic solution of system (3)is uniformly bounded and the proof is complete
Lemma 11 If the conditions (1198671)ndash(1198674) and
(H7) 11988622
minus (119886211198982) gt 0 [(119886
11minus 11988611198981)11988622
minus
(11988612119886211198982
1)]11988633
minus (11988611
minus (11988611198981))[(119886211198982)11988633
+
(11988623119886321198982
2)] gt 0
hold then system (3) has no nontrivial 1205911-periodic solution
Proof Suppose for a contradiction that system (3) has non-trivial periodic solution with period 120591
1 Then the following
system (83) of ordinary differential equations has nontrivialperiodic solution
1198891199091
119889119905
= 1199091(119905) [1198861minus 119886111199091(119905) minus
119886121199092(119905)
1198981+ 1199091(119905)
]
1198891199092
119889119905
= 1199092(119905) [minus119886
2+
119886211199091(119905)
1198981+ 1199091(119905)
minus 119886221199092(119905) minus
119886231199093(119905)
1198982+ 1199092(119905)
]
1198891199093
119889119905
= 1199093(119905) [minus119886
3+
119886321199092(119905 minus 120591
lowast
2)
1198982+ 1199092(119905 minus 120591
lowast
2)
minus 119886331199093(119905)]
(83)
which has the same equilibria to system (3) that is
1198641= (0 0 0) 119864
2= (
1198861
11988611
0 0)
1198643= (1199091 1199092 0) 119864
lowast= (119909lowast
1 119909lowast
2 119909lowast
3)
(84)
Note that 119909119894-axis (119894 = 1 2 3) are the invariable manifold of
system (83) and the orbits of system (83) do not intersect each
14 Abstract and Applied Analysis
other Thus there are no solutions crossing the coordinateaxes On the other hand note the fact that if system (83) hasa periodic solution then there must be the equilibrium in itsinterior and that 119864
1 1198642 1198643are located on the coordinate
axis Thus we conclude that the periodic orbit of system (83)must lie in the first quadrant If (119867
7) holds it is well known
that the positive equilibrium 119864lowastis globally asymptotically
stable in the first quadrant (see [1])Thus there is no periodicorbit in the first quadrant too The above discussion meansthat (83) does not have any nontrivial periodic solution It isa contradiction Therefore the lemma is confirmed
Theorem 12 Suppose the conditions of Theorem 8 and (1198677)
hold let120596119896and 120591(119895)1119896
be defined in Section 2 then when 1205911gt 120591(119895)
1119896
system (3) has at least 119895 minus 1 periodic solutions
Proof It is sufficient to prove that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is [120591
1 +infin) for each 119895 ge 1 where
1205911le 120591(119895)
1119896
In following we prove that the hypotheses (A1)ndash(A4) in[24] hold
(1) From system (3) we know easily that the followingconditions hold
(A1) 119865 isin 1198622(1198773
+times119877+times119877+) where 119865 = 119865|
1198773
+times119877+times119877+
rarr
1198773
+
(A3) 119865(120601 1205911 119901) is differential with respect to 120601
(2) It follows from system (3) that
119863119911119865 (119911 120591
1 119901) =
[[[[[[[[[[[
[
1198861minus 2119886111199091minus
1198861211989811199092
(1198981+ 1199091)2
minus
119886121199091
1198981+ 1199091
0
1198862111989811199092
(1198981+ 1199091)2
minus1198862+
119886211199091
1198981+ 1199091
minus 2119886221199092minus
1198862311989821199093
(1198982+ 1199092)2
minus
119886231199092
1198982+ 1199092
0
1198863211989821199093
(1198982+ 1199092)2
minus1198863+
119886321199092
1198982+ 1199092
minus 2119886331199093
]]]]]]]]]]]
]
(85)
Then under the assumption (1198671)ndash(1198674) we have
det119863119911119865 (119911lowast 1205911 119901)
= det
[[[[[[[[
[
minus11988611119909lowast
1+
11988612119909lowast
1119909lowast
2
(1198981+ 119909lowast
1)2
11988612119909lowast
1
1198981+ 119909lowast
1
0
119886211198981119909lowast
2
(1198981+ 119909lowast
1)2
minus11988622119909lowast
2+
11988623119909lowast
2119909lowast
3
(1198982+ 119909lowast
2)2
minus
11988623119909lowast
2
1198982+ 119909lowast
2
0
119886321198982119909lowast
3
(1198982+ 119909lowast
2)2
minus11988633119909lowast
3
]]]]]]]]
]
= minus
1198862
21199101lowast1199102lowast
(1 + 11988721199101lowast)3[minus119909lowast+
11988611198871119909lowast1199101lowast
(1 + 1198871119909lowast)2] = 0
(86)
From (86) we know that the hypothesis (A2) in [24]is satisfied
(3) The characteristic matrix of (71) at a stationarysolution (119911 120591
0 1199010) where 119911 = (119911
(1) 119911(2) 119911(3)) isin 119877
3
takes the following form
Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863
120601119865 (119911 120591
1 119901) (119890
120582119868) (87)
that is
Δ (119911 1205911 119901) (120582)
=
[[[[[[[[[[[[[[[[[
[
120582 minus 1198861+ 211988611119911(1)
+
119886121198981119911(2)
(1198981+ 119911(1))
2
11988612119911(1)
1198981+ 119911(1)
0
minus
119886211198981119911(2)
(1198981+ 119911(1))
2119890minus1205821205911
120582 + 1198862minus
11988621119911(1)
1198981+ 119911(1)
+ 211988622119911(2)
+
119886231198982119911(3)
(1198982+ 119911(2))
2
11988623119911(2)
1198982+ 119911(2)
0 minus
119886321198982119911(3)
(1198982+ 119911(2))
2119890minus120582120591lowast
2120582 + 1198863minus
11988632119911(2)
1198982+ 119911(2)
+ 211988633119911(3)
]]]]]]]]]]]]]]]]]
]
(88)
Abstract and Applied Analysis 15
From (88) we have
det (Δ (119864lowast 1205911 119901) (120582))
= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902
0+ 1199030)] 119890minus1205821205911
(89)
Note that (89) is the same as (20) from the discussion inSection 2 about the local Hopf bifurcation it is easy to verifythat (119864
lowast 120591(119895)
1119896 2120587120596
119896) is an isolated center and there exist 120598 gt
0 120575 gt 0 and a smooth curve 120582 (120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575) rarr C
such that det(Δ(120582(1205911))) = 0 |120582(120591
1) minus 120596119896| lt 120598 for all 120591
1isin
[120591(119895)
1119896minus 120575 120591(119895)
1119896+ 120575] and
120582 (120591(119895)
1119896) = 120596119896119894
119889Re 120582 (1205911)
1198891205911
1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)
1119896
gt 0 (90)
Let
Ω1205982120587120596
119896
= (120578 119901) 0 lt 120578 lt 120598
10038161003816100381610038161003816100381610038161003816
119901 minus
2120587
120596119896
10038161003816100381610038161003816100381610038161003816
lt 120598 (91)
It is easy to see that on [120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575] times 120597Ω
1205982120587120596119896
det(Δ(119864
lowast 1205911 119901)(120578 + (2120587119901)119894)) = 0 if and only if 120578 = 0
1205911= 120591(119895)
1119896 119901 = 2120587120596
119896 119896 = 1 2 3 119895 = 0 1 2
Therefore the hypothesis (A4) in [24] is satisfiedIf we define
119867plusmn(119864lowast 120591(119895)
1119896
2120587
120596119896
) (120578 119901)
= det(Δ (119864lowast 120591(119895)
1119896plusmn 120575 119901) (120578 +
2120587
119901
119894))
(92)
then we have the crossing number of isolated center(119864lowast 120591(119895)
1119896 2120587120596
119896) as follows
120574(119864lowast 120591(119895)
1119896
2120587
120596119896
) = deg119861(119867minus(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
minus deg119861(119867+(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
= minus1
(93)
Thus we have
sum
(1199111205911119901)isinC
(119864lowast120591(119895)
11198962120587120596119896)
120574 (119911 1205911 119901) lt 0
(94)
where (119911 1205911 119901) has all or parts of the form
(119864lowast 120591(119896)
1119895 2120587120596
119896) (119895 = 0 1 ) It follows from
Lemma 9 that the connected component ℓ(119864lowast120591(119895)
11198962120587120596
119896)
through (119864lowast 120591(119895)
1119896 2120587120596
119896) is unbounded for each center
(119911lowast 1205911 119901) (119895 = 0 1 ) From the discussion in Section 2
we have
120591(119895)
1119896=
1
120596119896
times arccos(((11990121205962
119896minus1199010) (1199030+1199020)
+ 1205962(1205962minus1199011) (1199031+1199021))
times ((1199030+1199020)2
+ (1199031+1199021)2
1205962
119896)
minus1
) +2119895120587
(95)
where 119896 = 1 2 3 119895 = 0 1 Thus one can get 2120587120596119896le 120591(119895)
1119896
for 119895 ge 1Now we prove that the projection of ℓ
(119864lowast120591(119895)
11198962120587120596
119896)onto
1205911-space is [120591
1 +infin) where 120591
1le 120591(119895)
1119896 Clearly it follows
from the proof of Lemma 11 that system (3) with 1205911
= 0
has no nontrivial periodic solution Hence the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is away from zero
For a contradiction we suppose that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is bounded this means that the
projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is included in a
interval (0 120591lowast) Noticing 2120587120596119896lt 120591119895
1119896and applying Lemma 11
we have 119901 lt 120591lowast for (119911(119905) 120591
1 119901) belonging to ℓ
(119864lowast120591(119895)
11198962120587120596
119896)
This implies that the projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 119901-
space is bounded Then applying Lemma 10 we get thatthe connected component ℓ
(119864lowast120591(119895)
11198962120587120596
119896)is bounded This
contradiction completes the proof
6 Conclusion
In this paper we take our attention to the stability and Hopfbifurcation analysis of a predator-prey systemwithMichaelis-Menten type functional response and two unequal delaysWe obtained some conditions for local stability and Hopfbifurcation occurring When 120591
1= 1205912 we derived the explicit
formulas to determine the properties of periodic solutionsby the normal form method and center manifold theoremSpecially the global existence results of periodic solutionsbifurcating from Hopf bifurcations are also established byusing a global Hopf bifurcation result due to Wu [24]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to thank the anonymous referees fortheir careful reading and constructive suggestions which leadto truly significant improvement of the paper This researchis supported by the National Natural Science Foundation ofChina (no 11061016)
16 Abstract and Applied Analysis
References
[1] R Xu and M A J Chaplain ldquoPersistence and global stabilityin a delayed predator-prey system with Michaelis-Menten typefunctional responserdquo Applied Mathematics and Computationvol 130 no 2-3 pp 441ndash455 2002
[2] Y L Song Y H Peng and J JWei ldquoBifurcations for a predator-prey system with two delaysrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 466ndash479 2008
[3] G-P Hu W-T Li and X-P Yan ldquoHopf bifurcations in apredator-prey system with multiple delaysrdquo Chaos Solitons ampFractals vol 42 no 2 pp 1273ndash1285 2009
[4] M Liao X Tang and C Xu ldquoBifurcation analysis for a three-species predator-prey systemwith two delaysrdquoCommunicationsin Nonlinear Science and Numerical Simulation vol 17 no 1 pp183ndash194 2012
[5] C Xu and P Li ldquoDynamical analysis in a delayed predator-prey model with two delaysrdquo Discrete Dynamics in Nature andSociety vol 2012 Article ID 652947 22 pages 2012
[6] J-F Zhang ldquoStability and bifurcation periodic solutions ina Lotka-Volterra competition system with multiple delaysrdquoNonlinear Dynamics vol 70 no 1 pp 849ndash860 2012
[7] J Xia Z Yu and R Yuan ldquoStability and Hopf bifurcation ina symmetric Lotka-Volterra predator-prey system with delaysrdquoElectronic Journal of Differential Equations vol 2013 no 9 pp1ndash16 2013
[8] H Wan and J Cui ldquoA malaria model with two delaysrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 601265 8pages 2013
[9] M Zhao ldquoHopf bifurcation analysis for a semiratio-dependentpredator-prey system with two delaysrdquo Abstract and AppliedAnalysis vol 2013 Article ID 495072 13 pages 2013
[10] S Feyissa and S Banerjee ldquoDelay-induced oscillatory dynamicsin humoral mediated immune response with two time delaysrdquoNonlinear Analysis Real World Applications vol 14 no 1 pp35ndash52 2013
[11] G Zhang Y Shen andB Chen ldquoHopf bifurcation of a predator-prey systemwith predator harvesting and twodelaysrdquoNonlinearDynamics vol 73 no 4 pp 2119ndash2131 2013
[12] X-Y Meng H-F Huo and H Xiang ldquoHopf bifurcation in athree-species systemwith delaysrdquo Journal of AppliedMathemat-ics and Computing vol 35 no 1-2 pp 635ndash661 2011
[13] J-F Zhang ldquoGlobal existence of bifurcated periodic solutions ina commensalism model with delaysrdquo Applied Mathematics andComputation vol 218 no 23 pp 11688ndash11699 2012
[14] Y Wang W Jiang and H Wang ldquoStability and global Hopfbifurcation in toxic phytoplankton-zooplankton model withdelay and selective harvestingrdquoNonlinear Dynamics vol 73 no1-2 pp 881ndash896 2013
[15] Y Ma ldquoGlobal Hopf bifurcation in the Leslie-Gower predator-prey model with two delaysrdquo Nonlinear Analysis Real WorldApplications vol 13 no 1 pp 370ndash375 2012
[16] X-Y Meng H-F Huo and X-B Zhang ldquoStability and globalHopf bifurcation in a delayed food web consisting of a preyand two predatorsrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4335ndash4348 2011
[17] Y Song J Wei andM Han ldquoLocal and global Hopf bifurcationin a delayed hematopoiesis modelrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol14 no 11 pp 3909ndash3919 2004
[18] Y Song and J Wei ldquoLocal Hopf bifurcation and global periodicsolutions in a delayed predator-prey systemrdquo Journal of Mathe-matical Analysis and Applications vol 301 no 1 pp 1ndash21 2005
[19] J Wei and M Y Li ldquoGlobal existence of periodic solutions in atri-neuron network model with delaysrdquo Physica D vol 198 no1-2 pp 106ndash119 2004
[20] T Zhao Y Kuang and H L Smith ldquoGlobal existence of peri-odic solutions in a class of delayed Gause-type predator-preysystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 28 no 8 pp 1373ndash1394 1997
[21] S J Gao L S Chen and Z D Teng ldquoHopf bifurcation andglobal stability for a delayed predator-prey system with stagestructure for predatorrdquo Applied Mathematics and Computationvol 202 no 2 pp 721ndash729 2008
[22] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977
[23] B Hassard D Kazarinoff and Y WanTheory and Applicationsof Hopf Bifurcation Cambridge University Press CambridgeUK 1981
[24] J H Wu ldquoSymmetric functional-differential equations andneural networks with memoryrdquo Transactions of the AmericanMathematical Society vol 350 no 12 pp 4799ndash4838 1998
[25] V Hutson ldquoThe existence of an equilibrium for permanentsystemsrdquo The Rocky Mountain Journal of Mathematics vol 20no 4 pp 1033ndash1040 1990
[26] S G Ruan and J J Wei ldquoOn the zeros of transcendentalfunctions with applications to stability of delay differentialequations with two delaysrdquo Dynamics of Continuous Discreteamp Impulsive Systems A vol 10 no 6 pp 863ndash874 2003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Abstract and Applied Analysis
is a more interesting and difficult topic A great deal ofresearch has been devoted to the topics [13ndash21] In this paperlet 12059111
= 12059122
= 12059133
= 0 12059121
= 1205911 12059132
= 1205912in (1) we
considerHopf bifurcation and global periodic solutions of thefollowing system with two unequal and nonzero delays
1198891199091
119889119905
= 1199091(119905) [1198861minus 119886111199091(119905) minus
119886121199092(119905)
1198981+ 1199091(119905)
]
1198891199092
119889119905
= 1199092(119905) [minus119886
2+
119886211199091(119905 minus 1205911)
1198981+ 1199091(119905 minus 1205911)
minus 119886221199092(119905) minus
119886231199093(119905)
1198982+ 1199092(119905)
]
1198891199093
119889119905
= 1199093(119905) [minus119886
3+
119886321199092(119905 minus 1205912)
1198982+ 1199092(119905 minus 1205912)
minus 119886331199093(119905)]
(3)
with initial conditions
119909119894(119905) = 120601
119894(119905) 119905 isin [minus120591 0] 120601
119894(0) gt 0
119894 = 1 2 3 120591 = max 1205911 1205912
(4)
Our goal is to investigate the possible stability switches of thepositive equilibrium and stability of periodic orbits arisingdue to a Hopf bifurcation when one of the delays is treatedas a bifurcation parameter Special attention is paid to theglobal continuation of local Hopf bifurcation when the delays1205911
= 1205912
This paper is organized as follows In Section 2 byanalyzing the characteristic equation of the linearized systemof system (3) at positive equilibrium the sufficient conditionsensuring the local stability of the positive equilibrium andthe existence of Hopf bifurcation are obtained [22] Someexplicit formulas determining the direction and stabilityof periodic solutions bifurcating from Hopf bifurcationsare demonstrated by applying the normal form methodand center manifold theory due to Hassard et al [23] inSection 3 In Section 4 we consider the global existence ofthese bifurcating periodic solutions [24] with two differentdelays Some numerical simulation results are included inSection 5
2 Stability of the Positive Equilibrium andLocal Hopf Bifurcations
In this section we first study the existence and local stabilityof the positive equilibrium and then investigate the effect ofdelay and the conditions for existence of Hopf bifurcations
There are at most four nonnegative equilibria for system(3)
1198641= (0 0 0) 119864
2= (
1198861
11988611
0 0)
1198643= (1199091 1199092 0) 119864
lowast= (119909lowast
1 119909lowast
2 119909lowast
3)
(5)
where (1199091 1199092 0) and (119909
lowast
1 119909lowast
2 119909lowast
3) satisfy
1198861minus 119886111199091minus
119886121199092
1198981+ 1199091
= 0
minus1198862+
119886211199091
1198981+ 1199091
minus 119886221199092= 0
(6)
1198861minus 11988611119909lowast
1(119905) minus
11988612119909lowast
2(119905)
1198981+ 119909lowast
1(119905)
= 0
minus1198862+
11988621119909lowast
1(119905)
1198981+ 119909lowast
1(119905)
minus 11988622119909lowast
2(119905) minus
11988623119909lowast
3(119905)
1198982+ 119909lowast
2(119905)
= 0
minus1198863+
11988632119909lowast
2(119905)
1198982+ 119909lowast
2(119905)
minus 11988633119909lowast
3(119905) = 0
(7)
where 1198643is a nonnegative equilibrium point if there is a
positive solution of (6) and 119864lowastis a nonnegative equilibrium
point if there is a positive solution of (7)Let
(H1) 1198861(11988621
minus 1198862) gt 119898
1119886211988611
(H2) (11988632minus1198863)[1198861(11988621minus1198862)minus11989811198862]minus1198982119886311988622(1198861+119898111988611) gt 0
(H3) 1199092(11988632
minus 1198863) minus 11989821198863gt 0
(H4) 11988622(11988611
minus (11988611198981)) minus (119886
12119886211198982
1) gt 0
From [1 25] we know that if (1198671) (1198672) (1198673) and (119867
4)
hold 1198643and 119864
lowastalways exist as nonnegative equilibria
Let119864 = (11990910 11990920 11990930) be the arbitrary equilibrium point
and let 1199091(119905) = 119909
1(119905) minus 119909
10 1199092(119905) = 119909
2(119905) minus 119909
20 1199093(119905) =
1199093(119905)minus119909
30 still denote 119909
1(119905) 1199092(119905) 1199093(119905) by 119909
1(119905) 1199092(119905) 1199093(119905)
respectively then the linearized system of the correspondingequations at 119864 is as follows
(119905) = 119861119906 (119905) + 119862119906 (119905 minus 1205911) + 119863119906 (119905 minus 120591
2) (8)
where
119906 (119905) = (1199091(119905) 1199092(119905) 1199093(119905))119879
119861 = (119887119894119895)3times3
119862 = (119888119894119895)3times3
119863 = (119889119894119895)3times3
11988711
= 1198861minus 21198861111990910
minus
11988612119898111990920
(1198981+ 11990910)2 119887
12= minus
1198861211990910
1198981+ 11990910
11988722
= minus1198862+
1198862111990910
1198981+ 11990910
minus 21198862211990920
minus
11988623119898211990930
(1198982+ 11990920)2
11988723
= minus
1198862311990920
1198982+ 11990920
11988733
= minus1198863+
1198863211990920
1198982+ 11990920
minus 21198863311990930
11988821
=
11988621119898111990920
(1198981+ 11990910)2 119889
32=
11988632119898211990930
(1198982+ 11990920)2
(9)
all the others of 119887119894119895 119888119894119895 and 119889
119894119895are 0
The characteristic equation for system (8) is
1205823+11990121205822+1199011120582 + 1199010+ (1199021120582 + 1199020) 119890minus1205821205911+ (1199031120582 + 1199030) 119890minus1205821205912= 0
(10)
Abstract and Applied Analysis 3
where
1199012= minus (119887
11+ 11988722
+ 11988733)
1199011= 1198871111988722
+ 1198872211988733
+ 1198871111988733
1199010= minus119887111198872211988733
1199021= minus1198871211988821 1199020= 119887121198882111988733
1199031= minus1198872311988932 1199030= 119887111198872311988932
(11)
We consider the following cases
(1) 119864 = 1198641 The characteristic equation reduces to
(120582 minus 1198861) (120582 + 119886
2) (120582 + 119886
3) = 0 (12)
There are always a positive root 1198861and two negative roots
1198862 1198863of (12) hence 119864
1is a saddle point
(2) 119864 = 1198642 Equation (10) takes the form
(120582 + 1198861) (120582 + 119886
2minus
119886111988612
119898111988611
+ 1198861
) (120582 + 1198863) = 0 (13)
There is a positive root 120582 = (119886111988612(119898111988611
+ 1198861)) minus 119886
2if
119886111988612(119898111988611
+ 1198861) gt 119886
2 hence 119864
2is a saddle point If
119886111988612(119898111988611
+ 1198861) lt 119886
2 1198642is locally asymptotically stable
(3) 119864 = 1198643 The characteristic equation is
(120582 minus 11988733) [1205822minus (11988711
+ 11988722) 120582 + 119887
1111988722
minus 1198871211988821119890minus1205821205911] = 0 (14)
We will analyse the distribution of the characteristic root of(14) from Ruan and Wei [26] which is stated as follows
Lemma 1 Consider the exponential polynomial
119875 (120582 119890minus1205821205911 119890
minus120582120591119898)
= 120582119899+ 119901(0)
1120582119899minus1
+ sdot sdot sdot + 119901(0)
119899minus1120582 + 119901(0)
119899
+ [119901(1)
1120582119899minus1
+ sdot sdot sdot + 119901(1)
119899minus1120582 + 119901(1)
119899] 119890minus1205821205911
+ sdot sdot sdot + [119901(119898)
1120582119899minus1
+ sdot sdot sdot + 119901(119898)
119899minus1120582 + 119901(119898)
119899] 119890minus120582120591119898
(15)
where 120591119894⩾ 0 (119894 = 1 2 119898) and 119901
(119894)
119895(119894 = 0 1 119898 119895 =
1 2 119899) are constants As (1205911 1205912 120591
119898) vary the sum of
the order of the zeros of 119875(120582 119890minus1205821205911 119890minus120582120591119898) on the open righthalf plane can change only if a zero appears on or crosses theimaginary axis
By using Lemma 1 we can easily obtain the followingresults
Lemma 2 If 1198643is a nonnegative equilibrium point then
(1) 1198643is unstable if 119887
33gt 0
(2) 1198643is locally asymptotically stable if 119887
33lt 0 11988711
+ 11988722
lt
0 1198871111988722
minus 1198871211988821
gt 0 and 1198871111988722
+ 1198871211988821
gt 0
Proof (1) 120582 = 11988733
is a root of (14) if 11988733
gt 0 then 1198643is
unstable(2) Clearly 120582 = 0 is not a root of (14) we should discuss
the following equation instead of (14)
1205822minus (11988711
+ 11988722) 120582 + 119887
1111988722
minus 1198871211988821119890minus1205821205911= 0 (16)
Assume that 119894120596 with 120596 gt 0 is a solution of (16) Substituting120582 = 119894120596 into (16) and separating the real and imaginary partsyield
minus1205962+ 1198871111988722
= 1198871211988821cos120596120591
1
120596 (11988711
+ 11988722) = 1198871211988821sin120596120591
1
(17)
which implies
1205964+ (1198872
11+ 1198872
22) 1205962+ 1198872
111198872
22minus 1198872
121198882
21= 0 (18)
If 1198872111198872
22minus 1198872
121198882
21gt 0 that is (119887
1111988722
+ 1198871211988821)(1198871111988722
minus 1198871211988821) gt
0 there is no real root of (16) Hence there is no purelyimaginary root of (18) When 120591
1= 0 (16) reduces to
1205822minus (11988711
+ 11988722) 120582 + 119887
1111988722
minus 1198871211988821
= 0 (19)
If 11988711+ 11988722
lt 0 and 1198871111988722minus 1198871211988821
gt 0 both roots of (19) havenegative real parts Thus by using Lemma 1 when 119887
33lt 0
11988711
+ 11988722
lt 0 1198871111988722
minus 1198871211988821
gt 0 and 1198871111988722
+ 1198871211988821
gt 0 1198643is
locally asymptotically stable
(4)119864 = 119864lowastThe characteristic equation about119864
lowastis (10) In the
following we will analyse the distribution of roots of (10)Weconsider four cases
Case a Consider1205911= 1205912= 0
The associated characteristic equation of system (3) is
1205823+ 11990121205822+ (1199011+ 1199021+ 1199031) 120582 + (119901
0+ 1199020+ 1199030) = 0 (20)
Let
(H5) 1199012gt 0 119901
2(1199011+1199021+1199031)minus(1199010+1199020+1199030) gt 0 119901
0+1199020+1199030gt
0
By Routh-Hurwitz criterion we have the following
Theorem 3 For 1205911= 1205912= 0 assume that (119867
1)ndash(1198675) hold
Thenwhen 1205911= 1205912= 0 the positive equilibrium119864
lowast(119909lowast
1 119909lowast
2 119909lowast
3)
of system (3) is locally asymptotically stable
Case b Consider1205911= 0 1205912gt 0
The associated characteristic equation of system (3) is
1205823+ 11990121205822+ (1199011+ 1199021) 120582 + (119901
0+ 1199020) + (1199031120582 + 1199030) 119890minus1205821205912= 0
(21)
4 Abstract and Applied Analysis
We want to determine if the real part of some rootincreases to reach zero and eventually becomes positive as 120591varies Let 120582 = 119894120596 (120596 gt 0) be a root of (21) then we have
minus 1198941205963minus 11990121205962+ 119894 (1199011+ 1199021) 120596 + (119901
0+ 1199020)
+ (1199031120596119894 + 119903
0) (cos120596120591
2minus 119894 sin120596120591
2) = 0
(22)
Separating the real and imaginary parts we have
minus1205963+ (1199011+ 1199021) 120596 = 119903
0sin120596120591
2minus 1199031120596 cos120596120591
2
minus11990121205962+ (1199010+ 1199020) = minus119903
1120596 sin120596120591
2minus 1199030cos120596120591
2
(23)
It follows that
1205966+ 119898121205964+ 119898111205962+ 11989810
= 0 (24)
where 11989812
= 1199012
2minus 2(119901
1+ 1199021) 11989811
= (1199011+ 1199021)2minus 21199012(1199010+
1199020) minus 1199032
1 11989810
= (1199010+ 1199020)2minus 1199032
0
Denoting 119911 = 1205962 (24) becomes
1199113+ 119898121199112+ 11989811119911 + 119898
10= 0 (25)
Let
ℎ1(119911) = 119911
3+ 119898121199112+ 11989811119911 + 119898
10 (26)
we have
119889ℎ1(119911)
119889119911
= 31199112+ 211989812119911 + 119898
11 (27)
If 11989810
= (1199010+ 1199020)2minus 1199032
0lt 0 then ℎ
1(0) lt
0 lim119911rarr+infin
ℎ1(119911) = +infin We can know that (25) has at least
one positive rootIf 11989810
= (1199010+ 1199020)2minus 1199032
0ge 0 we obtain that when Δ =
1198982
12minus 311989811
le 0 (25) has no positive roots for 119911 isin [0 +infin)On the other hand when Δ = 119898
2
12minus 311989811
gt 0 the followingequation
31199112+ 211989812119911 + 119898
11= 0 (28)
has two real roots 119911lowast11
= (minus11989812
+ radicΔ)3 119911lowast
12= (minus119898
12minus
radicΔ)3 Because of ℎ101584010158401(119911lowast
11) = 2radicΔ gt 0 ℎ
10158401015840
1(119911lowast
12) = minus2radicΔ lt
0 119911lowast
11and 119911lowast12are the local minimum and the localmaximum
of ℎ1(119911) respectively By the above analysis we immediately
obtain the following
Lemma 4 (1) If 11989810
ge 0 and Δ = 1198982
12minus 311989811
le 0 (25) hasno positive root for 119911 isin [0 +infin)
(2) If 11989810
ge 0 and Δ = 1198982
12minus 311989811
gt 0 (25) has at leastone positive root if and only if 119911lowast
11= (minus119898
12+ radicΔ)3 gt 0 and
ℎ1(119911lowast
11) le 0
(3) If 11989810
lt 0 (25) has at least one positive root
Without loss of generality we assume that (25) has threepositive roots defined by 119911
11 11991112 11991113 respectively Then
(24) has three positive roots
12059611
= radic11991111 120596
12= radic11991112 120596
13= radic11991113 (29)
From (23) we have
cos120596111989612059121119896
=
11990311205964
1119896+ [11990121199030minus (1199021+ 1199011) 1199031] 1205962
1119896minus 1199030(1199020+ 1199010)
1199032
0+ 1199032
11205962
1119896
(30)
Thus if we denote
120591(119895)
21119896
=
1
1205961119896
times arccos ((11990311205964
1119896+ [11990121199030minus (1199021+ 1199011) 1199031] 1205962
1119896
minus 1199030(1199020+ 1199010))
times (1199032
0+ 1199032
11205962
1119896)
minus1
) + 2119895120587
(31)
where 119896 = 1 2 3 119895 = 0 1 2 then plusmn1198941205961119896is a pair of purely
imaginary roots of (21) corresponding to 120591(119895)
21119896
Define
120591210
= 120591(0)
211198960
= min119896=123
120591(0)
21119896
12059610
= 12059611198960
(32)
Let 120582(1205912) = 120572(120591
2)+119894120596(120591
2) be the root of (21) near 120591
2= 120591(119895)
21119896
satisfying
120572 (120591(119895)
21119896
) = 0 120596 (120591(119895)
21119896
) = 1205961119896 (33)
Substituting 120582(1205912) into (21) and taking the derivative with
respect to 1205912 we have
31205822+ 21199012120582 + (119901
1+ 1199021) + 1199031119890minus1205821205912minus 1205912(1199031120582 + 1199030) 119890minus1205821205912
119889120582
1198891205912
= 120582 (1199031120582 + 1199030) 119890minus1205821205912
(34)
Therefore
[
119889120582
1198891205912
]
minus1
=
[31205822+ 21199012120582 + (119901
1+ 1199021)] 1198901205821205912
120582 (1199031120582 + 1199030)
+
1199031
120582 (1199031120582 + 1199030)
minus
1205912
120582
(35)
When 1205912
= 120591(119895)
21119896
120582(120591(119895)
21119896
) = 1198941205961119896
(119896 = 1 2 3) 120582(1199031120582 +
1199030)|1205912=120591(119895)
21119896
= minus11990311205962
1119896+ 11989411990301205961119896 [31205822
+ 21199012120582 + (119901
1+
1199021)]1198901205821205912|1205912=120591(119895)
21119896
= [minus31205962
1119896+ (1199011+ 1199021)] cos(120596
1119896120591(119895)
21119896
) minus
211990121205961119896sin(1205961119896120591(119895)
21119896
) + 119894211990121205961119896cos(120596
1119896120591(119895)
21119896
) + [minus31205962
1119896+ (1199011+
1199021)] sin(120596
1119896120591(119895)
21119896
)
Abstract and Applied Analysis 5
According to (35) we have
[
Re 119889 (120582 (1205912))
1198891205912
]
minus1
1205912=120591(119895)
21119896
= Re[[31205822+ 21199012120582 + (119901
1+ 1199021)] 1198901205821205912
120582 (1199031120582 + 1199030)
]
1205912=120591(119895)
21119896
+ Re[ 1199031
120582 (1199031120582 + 1199030)
]
1205912=120591(119895)
21119896
=
1
Λ1
minus 11990311205962
1119896[minus31205962
1119896+ (1199011+ 1199021)] cos (120596
1119896120591(119895)
21119896
)
+ 2119903111990121205963
1119896sin (120596
1119896120591(119895)
21119896
) minus 1199032
11205962
1119896
+ 2119903011990121205962
1119896cos (120596
1119896120591(119895)
21119896
)
+ 1199030[minus31205962
1119896+ (1199011+ 1199021)] 1205961119896sin (120596
1119896120591(119895)
21119896
)
=
1
Λ1
31205966
1119896+ 2 [119901
2
2minus 2 (119901
1+ 1199021)] 1205964
1119896
+ [(1199011+ 1199021)2
minus 21199012(1199010+ 1199020) minus 1199032
1] 1205962
1119896
=
1
Λ1
1199111119896(31199112
1119896+ 2119898121199111119896
+ 11989811)
=
1
Λ1
1199111119896ℎ1015840
1(1199111119896)
(36)
where Λ1= 1199032
11205964
1119896+ 1199032
01205962
1119896gt 0 Notice that Λ
1gt 0 1199111119896
gt 0
sign
[
Re 119889 (120582 (1205912))
1198891205912
]
1205912=120591(119895)
21119896
= sign
[
Re 119889 (120582 (1205912))
1198891205912
]
minus1
1205912=120591(119895)
21119896
(37)
then we have the following lemma
Lemma 5 Suppose that 1199111119896
= 1205962
1119896and ℎ
1015840
1(1199111119896) = 0 where
ℎ1(119911) is defined by (26) then 119889(Re 120582(120591(119895)
21119896
))1198891205912has the same
sign with ℎ1015840
1(1199111119896)
From Lemmas 1 4 and 5 and Theorem 3 we can easilyobtain the following theorem
Theorem6 For 1205911= 0 120591
2gt 0 suppose that (119867
1)ndash(1198675) hold
(i) If 11989810
ge 0 and Δ = 1198982
12minus 311989811
le 0 then all rootsof (10) have negative real parts for all 120591
2ge 0 and the
positive equilibrium 119864lowastis locally asymptotically stable
for all 1205912ge 0
(ii) If either 11989810
lt 0 or 11989810
ge 0 Δ = 1198982
12minus 311989811
gt
0 119911lowast
11gt 0 and ℎ
1(119911lowast
11) le 0 then ℎ
1(119911) has at least one
positive roots and all roots of (23) have negative realparts for 120591
2isin [0 120591
210
) and the positive equilibrium 119864lowast
is locally asymptotically stable for 1205912isin [0 120591
210
)
(iii) If (ii) holds and ℎ1015840
1(1199111119896) = 0 then system (3)undergoes
Hopf bifurcations at the positive equilibrium119864lowastfor 1205912=
120591(119895)
21119896
(119896 = 1 2 3 119895 = 0 1 2 )
Case c Consider1205911gt 0 1205912= 0
The associated characteristic equation of system (3) is
1205823+ 11990121205822+ (1199011+ 1199031) 120582 + (119901
0+ 1199030) + (119902
1120582 + 1199020) 119890minus1205821205911= 0
(38)
Similar to the analysis of Case 119887 we get the followingtheorem
Theorem7 For 1205911gt 0 120591
2= 0 suppose that (119867
1)ndash(1198675) hold
(i) If 11989820
ge 0 and Δ = 1198982
22minus 311989821
le 0 then all rootsof (38) have negative real parts for all 120591
1ge 0 and the
positive equilibrium 119864lowastis locally asymptotically stable
for all 1205911ge 0
(ii) If either 11989820
lt 0 or 11989820
ge 0 Δ = 1198982
22minus 311989821
gt 0119911lowast
21gt 0 and ℎ
2(119911lowast
21) le 0 then ℎ
2(119911) has at least one
positive root 1199112119896 and all roots of (38) have negative real
parts for 1205911isin [0 120591
120
) and the positive equilibrium 119864lowast
is locally asymptotically stable for 1205911isin [0 120591
120
)
(iii) If (ii) holds and ℎ1015840
2(1199112119896) = 0 then system (3) undergoes
Hopf bifurcations at the positive equilibrium119864lowastfor 1205911=
120591(119895)
12119896
(119896 = 1 2 3 119895 = 0 1 2 )
where
11989822
= 1199012
2minus 2 (119901
1+ 1199031)
11989821
= (1199011+ 1199031)2
minus 21199012(1199010+ 1199030) minus 1199022
1
11989820
= (1199010+ 1199030)2
minus 1199022
0
ℎ2(119911) = 119911
3+ 119898221199112+ 11989821119911 + 119898
20 119911lowast
21=
minus11989822
+ radicΔ
3
120591(119895)
12119896
=
1
1205962119896
times arccos ((11990211205964
2119896+ [11990121199020minus (1199031+ 1199011) 1199021] 1205962
2119896
minus 1199020(1199030+ 1199010) )
times (1199022
0+ 1199022
11205962
2119896)
minus1
) + 2119895120587
(39)
where 119896 = 1 2 3 119895 = 0 1 2 then plusmn1198941205962119896is a pair of purely
imaginary roots of (38) corresponding to 120591(119895)
12119896
Define
120591120
= 120591(0)
121198960
= min119896=123
120591(0)
12119896
12059610
= 12059611198960
(40)
6 Abstract and Applied Analysis
Case d Consider1205911gt 0 1205912gt 0 1205911
= 1205912
The associated characteristic equation of system (3) is
1205823+11990121205822+1199011120582+1199010+ (1199021120582 + 1199020) 119890minus1205821205911+ (1199031120582+1199030) 119890minus1205821205912=0
(41)
We consider (41) with 1205912= 120591lowast
2in its stable interval [0 120591
210
)Regard 120591
1as a parameter
Let 120582 = 119894120596 (120596 gt 0) be a root of (41) then we have
minus 1198941205963minus 11990121205962+ 1198941199011120596 + 1199010+ (1198941199021120596 + 1199020) (cos120596120591
1minus 119894 sin120596120591
1)
+ (1199030+ 1198941199031120596) (cos120596120591lowast
2minus 119894 sin120596120591
lowast
2) = 0
(42)
Separating the real and imaginary parts we have
1205963minus 1199011120596 minus 1199031120596 cos120596120591lowast
2+ 1199030sin120596120591
lowast
2
= 1199021120596 cos120596120591
1minus 1199020sin120596120591
1
11990121205962minus 1199010minus 1199030cos120596120591lowast
2minus 1199031120596 sin120596120591
lowast
2
= 1199020cos120596120591
1+ 1199021120596 sin120596120591
1
(43)
It follows that
1205966+ 119898331205964+ 119898321205963+ 119898311205962+ 11989830
= 0 (44)
where
11989833
= 1199012
2minus 21199011minus 21199031cos120596120591lowast
2
11989832
= 2 (1199030minus 11990121199031) sin120596120591
lowast
2
11989831
= 1199012
1minus 211990101199012minus 2 (119901
21199030minus 11990111199031) cos120596120591lowast
2+ 1199032
1minus 1199022
1
11989830
= 1199012
0+ 211990101199030cos120596120591lowast
2+ 1199032
0minus 1199022
0
(45)
Denote 119865(120596) = 1205966+ 119898331205964+ 119898321205963+ 119898311205962+ 11989830 If
11989830
lt 0 then
119865 (0) lt 0 lim120596rarr+infin
119865 (120596) = +infin (46)
We can obtain that (44) has at most six positive roots1205961 1205962 120596
6 For every fixed 120596
119896 119896 = 1 2 6 there exists
a sequence 120591(119895)1119896
| 119895 = 0 1 2 3 such that (43) holdsLet
12059110
= min 120591(119895)
1119896| 119896 = 1 2 6 119895 = 0 1 2 3 (47)
When 1205911
= 120591(119895)
1119896 (41) has a pair of purely imaginary roots
plusmn119894120596(119895)
1119896for 120591lowast2isin [0 120591
210
)In the following we assume that
(H6) ((119889Re(120582))119889120591
1)|120582=plusmn119894120596
(119895)
1119896
= 0
Thus by the general Hopf bifurcation theorem for FDEsin Hale [22] we have the following result on the stability andHopf bifurcation in system (3)
Theorem 8 For 1205911
gt 0 1205912
gt 0 1205911
= 1205912 suppose that
(1198671)ndash(1198676) is satisfied If 119898
30lt 0 and 120591
lowast
2isin [0 120591
210
] thenthe positive equilibrium 119864
lowastis locally asymptotically stable for
1205911
isin [0 12059110) System (3) undergoes Hopf bifurcations at the
positive equilibrium 119864lowastfor 1205911= 120591(119895)
1119896
3 Direction and Stability ofthe Hopf Bifurcation
In Section 2 we obtain the conditions underwhich system (3)undergoes the Hopf bifurcation at the positive equilibrium119864lowast In this section we consider with 120591
2= 120591lowast
2isin [0 120591
210
)
and regard 1205911as a parameter We will derive the explicit
formulas determining the direction stability and period ofthese periodic solutions bifurcating from equilibrium 119864
lowastat
the critical values 1205911by using the normal form and the center
manifold theory developed by Hassard et al [23] Withoutloss of generality denote any one of these critical values 120591
1=
120591(119895)
1119896(119896 = 1 2 6 119895 = 0 1 2 ) by 120591
1 at which (43) has a
pair of purely imaginary roots plusmn119894120596 and system (3) undergoesHopf bifurcation from 119864
lowast
Throughout this section we always assume that 120591lowast2lt 12059110
Let 1199061= 1199091minus 119909lowast
1 1199062= 1199091minus 119909lowast
2 1199063= 1199092minus 119909lowast
3 119905 = 120591
1119905
and 120583 = 1205911minus 1205911 120583 isin R Then 120583 = 0 is the Hopf bifurcation
value of system (3) System (3) may be written as a functionaldifferential equation inC([minus1 0]R3)
(119905) = 119871120583(119906119905) + 119891 (120583 119906
119905) (48)
where 119906 = (1199061 1199062 1199063)119879isin R3 and
119871120583(120601) = (120591
1+ 120583) 119861[
[
1206011(0)
1206012(0)
1206013(0)
]
]
+ (1205911+ 120583)119862[
[
1206011(minus1)
1206012(minus1)
1206013(minus1)
]
]
+ (1205911+ 120583)119863
[[[[[[[[[[
[
1206011(minus
120591lowast
2
1205911
)
1206012(minus
120591lowast
2
1205911
)
1206013(minus
120591lowast
2
1205911
)
]]]]]]]]]]
]
(49)
119891 (120583 120601) = (1205911+ 120583)[
[
1198911
1198912
1198913
]
]
(50)
Abstract and Applied Analysis 7
where 120601 = (1206011 1206012 1206013)119879isin C([minus1 0]R3) and
119861 = [
[
11988711
11988712
0
0 11988722
11988723
0 0 11988733
]
]
119862 = [
[
0 0 0
11988821
0 0
0 0 0
]
]
119863 = [
[
0 0 0
0 0 0
0 11988932
0
]
]
1198911= minus (119886
11+ 1198973) 1206012
1(0) minus 119897
11206011(0) 1206012(0) minus 119897
21206012
1(0) 1206012(0)
minus 11989751206013
1(0) minus 119897
41206013
1(0) 1206012(0) + sdot sdot sdot
1198912= minus11989761206012(0) 1206013(0) minus (119897
7+ 11988622) 1206012
2(0) + 119897
11206011(minus1) 120601
2(0)
+ 11989731206012
1(minus1) minus 119897
81206013
2(0) minus 119897
91206012
2(0) 1206013(0)
+ 11989721206012
1(minus1) 120601
2(0) + 119897
31206013
1(minus1) + 119897
41206013
1(minus1) 120601
2(0)
minus 119897101206013
2(0) 1206013(0) + sdot sdot sdot
1198913= 11989761206012(minus
120591lowast
2
1205911
)1206013(0) + 119897
71206012
2(minus
120591lowast
2
1205911
) minus 119886331206012
3(0)
+ 11989791206012
2(minus
120591lowast
2
1205911
)1206013(0) + 119897
81206013
2(minus
120591lowast
2
1205911
)
+ 119897101206013
2(minus
120591lowast
2
1205911
)1206013(0) + sdot sdot sdot
1199011(119909) =
1198861119909
1 + 1198871119909
1199012(119909) =
1198862119909
1 + 1198872119909
1198971= 1199011015840
1(119909lowast)
1198972=
1
2
11990110158401015840
1(119909lowast) 1198973=
1
2
11990110158401015840
1(119909lowast) 1199101lowast
1198974=
1
3
119901101584010158401015840
1(119909lowast) 1198975=
1
3
119901101584010158401015840
1(119909lowast) 1199101lowast
1198976= 1199011015840
2(1199101lowast) 1198977=
1
2
11990110158401015840
2(1199101lowast) 1199102lowast
1198978=
1
3
119901101584010158401015840
2(1199101lowast) 1199102lowast 1198979=
1
2
11990110158401015840
2(1199101lowast)
11989710
=
1
3
119901101584010158401015840
2(1199101lowast)
(51)
Obviously 119871120583(120601) is a continuous linear function mapping
C([minus1 0]R3) intoR3 By the Riesz representation theoremthere exists a 3 times 3 matrix function 120578(120579 120583) (minus1 ⩽ 120579 ⩽ 0)whose elements are of bounded variation such that
119871120583120601 = int
0
minus1
119889120578 (120579 120583) 120601 (120579) for 120601 isin C ([minus1 0] R3) (52)
In fact we can choose
119889120578 (120579 120583) = (1205911+ 120583) [119861120575 (120579) + 119862120575 (120579 + 1) + 119863120575(120579 +
120591lowast
2
1205911
)]
(53)
where 120575 is Dirac-delta function For 120601 isin C([minus1 0]R3)define
119860 (120583) 120601 =
119889120601 (120579)
119889120579
120579 isin [minus1 0)
int
0
minus1
119889120578 (119904 120583) 120601 (119904) 120579 = 0
119877 (120583) 120601 =
0 120579 isin [minus1 0)
119891 (120583 120601) 120579 = 0
(54)
Then when 120579 = 0 the system is equivalent to
119905= 119860 (120583) 119909
119905+ 119877 (120583) 119909
119905 (55)
where 119909119905(120579) = 119909(119905+120579) 120579 isin [minus1 0] For120595 isin C1([0 1] (R3)
lowast)
define
119860lowast120595 (119904) =
minus
119889120595 (119904)
119889119904
119904 isin (0 1]
int
0
minus1
119889120578119879(119905 0) 120595 (minus119905) 119904 = 0
(56)
and a bilinear inner product
⟨120595 (119904) 120601 (120579)⟩=120595 (0) 120601 (0)minusint
0
minus1
int
120579
120585=0
120595 (120585minus120579) 119889120578 (120579) 120601 (120585) 119889120585
(57)
where 120578(120579) = 120578(120579 0) Let 119860 = 119860(0) then 119860 and 119860lowast
are adjoint operators By the discussion in Section 2 weknow that plusmn119894120596120591
1are eigenvalues of 119860 Thus they are also
eigenvalues of 119860lowast We first need to compute the eigenvectorof 119860 and 119860
lowast corresponding to 1198941205961205911and minus119894120596120591
1 respectively
Suppose that 119902(120579) = (1 120572 120573)1198791198901198941205791205961205911 is the eigenvector of 119860
corresponding to 1198941205961205911 Then 119860119902(120579) = 119894120596120591
1119902(120579) From the
definition of 119860119871120583(120601) and 120578(120579 120583) we can easily obtain 119902(120579) =
(1 120572 120573)1198791198901198941205791205961205911 where
120572 =
119894120596 minus 11988711
11988712
120573 =
11988932(119894120596 minus 119887
11)
11988712(119894120596 minus 119887
33) 119890119894120596120591lowast
2
(58)
and 119902(0) = (1 120572 120573)119879 Similarly let 119902lowast(119904) = 119863(1 120572
lowast 120573lowast)1198901198941199041205961205911
be the eigenvector of 119860lowast corresponding to minus119894120596120591
1 By the
definition of 119860lowast we can compute
120572lowast=
minus119894120596 minus 11988711
119888211198901198941205961205911
120573lowast=
11988723(minus119894120596 minus 119887
11)
11988821(119894120596 minus 119887
33) 1198901198941205961205911
(59)
From (57) we have⟨119902lowast(119904) 119902 (120579)⟩
= 119863(1 120572lowast 120573
lowast
) (1 120572 120573)119879
minus int
0
minus1
int
120579
120585=0
119863(1 120572lowast 120573
lowast
) 119890minus1198941205961205911(120585minus120579)
119889120578 (120579)
times (1 120572 120573)119879
1198901198941205961205911120585119889120585
= 1198631 + 120572120572lowast+ 120573120573
lowast
+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573
lowast
120591lowast
2119890minus119894120596120591lowast
2
(60)
8 Abstract and Applied Analysis
Thus we can choose
119863 = 1 + 120572120572lowast+ 120573120573
lowast
+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573
lowast
120591lowast
2119890minus119894120596120591lowast
2
minus1
(61)
such that ⟨119902lowast(119904) 119902(120579)⟩ = 1 ⟨119902lowast(119904) 119902(120579)⟩ = 0
In the remainder of this section we follow the ideas inHassard et al [23] and use the same notations as there tocompute the coordinates describing the center manifold 119862
0
at 120583 = 0 Let 119909119905be the solution of (48) when 120583 = 0 Define
119911 (119905) = ⟨119902lowast 119909119905⟩ 119882 (119905 120579) = 119909
119905(120579) minus 2Re 119911 (119905) 119902 (120579)
(62)
On the center manifold 1198620 we have
119882(119905 120579) = 119882 (119911 (119905) 119911 (119905) 120579)
= 11988220(120579)
1199112
2
+11988211(120579) 119911119911 +119882
02(120579)
1199112
2
+11988230(120579)
1199113
6
+ sdot sdot sdot
(63)
where 119911 and 119911 are local coordinates for center manifold 1198620in
the direction of 119902 and 119902 Note that 119882 is real if 119909119905is real We
consider only real solutions For the solution 119909119905isin 1198620of (48)
since 120583 = 0 we have
= 1198941205961205911119911 + ⟨119902
lowast(120579) 119891 (0119882 (119911 (119905) 119911 (119905) 120579)
+ 2Re 119911 (119905) 119902 (120579))⟩
= 1198941205961205911119911+119902lowast(0) 119891 (0119882 (119911 (119905) 119911 (119905) 0) +2Re 119911 (119905) 119902 (0))
= 1198941205961205911119911 + 119902lowast(0) 1198910(119911 119911) ≜ 119894120596120591
1119911 + 119892 (119911 119911)
(64)
where
119892 (119911 119911) = 119902lowast(0) 1198910(119911 119911) = 119892
20
1199112
2
+ 11989211119911119911
+ 11989202
1199112
2
+ 11989221
1199112119911
2
+ sdot sdot sdot
(65)
By (62) we have 119909119905(120579) = (119909
1119905(120579) 1199092119905(120579) 1199093119905(120579))119879= 119882(119905 120579) +
119911119902(120579) + 119911 119902(120579) and then
1199091119905(0) = 119911 + 119911 +119882
(1)
20(0)
1199112
2
+119882(1)
11(0) 119911119911
+119882(1)
02(0)
1199112
2
+ 119900 (|(119911 119911)|3)
1199092119905(0) = 119911120572 + 119911 120572 +119882
(2)
20(0)
1199112
2
+119882(2)
11(0) 119911119911 +119882
(2)
02(0)
1199112
2
+ 119900 (|(119911 119911)|3)
1199093119905(0) = 119911120573 + 119911120573 +119882
(3)
20(0)
1199112
2
+119882(3)
11(0) 119911119911
+119882(3)
02(0)
1199112
2
+ 119900 (|(119911 119911)|3)
1199091119905(minus1) = 119911119890
minus1198941205961205911+ 1199111198901198941205961205911+119882(1)
20(minus1)
1199112
2
+119882(1)
11(minus1) 119911119911 +119882
(1)
02(minus1)
1199112
2
+ 119900 (|(119911 119911)|3)
1199092119905(minus1) = 119911120572119890
minus1198941205961205911+ 119911 120572119890
1198941205961205911+119882(2)
20(minus1)
1199112
2
+119882(2)
11(minus1) 119911119911 +119882
(2)
02(minus1)
1199112
2
+ 119900 (|(119911 119911)|3)
1199093119905(minus1) = 119911120573119890
minus1198941205961205911+ 119911120573119890
1205961205911+119882(3)
20(minus1)
1199112
2
+119882(3)
11(minus1) 119911119911 +119882
(3)
02(minus1)
1199112
2
+ 119900 (|(119911 119911)|3)
1199091119905(minus
120591lowast
2
12059110
) = 119911119890minus119894120596120591lowast
2+ 119911119890119894120596120591lowast
2+119882(1)
20(minus
120591lowast
2
1205911
)
1199112
2
+119882(1)
11(minus
120591lowast
2
1205911
)119911119911 +119882(1)
02(minus
120591lowast
2
1205911
)
1199112
2
+ 119900 (|(119911 119911)|3)
1199092119905(minus
120591lowast
2
1205911
) = 119911120572119890minus119894120596120591lowast
2+ 119911 120572119890
119894120596120591lowast
2+119882(2)
20(minus
120591lowast
2
1205911
)
1199112
2
+119882(2)
11(minus
120591lowast
2
1205911
)119911119911 +119882(2)
02(minus
120591lowast
2
1205911
)
1199112
2
+ 119900 (|(119911 119911)|3)
1199093119905(minus
120591lowast
2
1205911
) = 119911120573119890minus119894120596120591lowast
2+ 119911120573119890
120596120591lowast
2+119882(3)
20(minus
120591lowast
2
1205911
)
1199112
2
+119882(3)
11(minus
120591lowast
2
1205911
)119911119911 +119882(3)
02(minus
120591lowast
2
1205911
)
1199112
2
+ 119900 (|(119911 119911)|3)
(66)
Abstract and Applied Analysis 9
It follows together with (50) that
119892 (119911 119911)=119902lowast(0) 1198910(119911 119911) =119863120591
1(1 120572lowast 120573
lowast
) (119891(0)
1119891(0)
2119891(0)
3)
119879
= 1198631205911[minus (119886
11+ 1198973) 1206012
1(0) minus 119897
11206011(0) 1206012(0)
minus 11989721206012
1(0) 1206012(0) minus 119897
51206013
1(0) minus 119897
41206013
1(0) 1206012(0)
+ sdot sdot sdot ]
+ 120572lowast[11989761206012(0) 1206013(0) (1198977+ 11988622) 1206012
2(0)
+ 11989711206011(minus1) 120601
2(0)
+ 11989731206012
1(minus1) minus 119897
81206013
2(0) minus 119897
91206012
2(0) 1206013(0)
+ 11989721206012
1(minus1) 120601
2(0) + 119897
31206013
1(minus1) + sdot sdot sdot ]
+ 120573
lowast
[11989761206012(minus
120591lowast
2
1205911
)1206013(0) + 119897
71206012
2(minus
120591lowast
2
1205911
)
minus 119886331206012
3(0) + 119897
81206013
2(minus
120591lowast
2
1205911
) + sdot sdot sdot ]
(67)
Comparing the coefficients with (65) we have
11989220
= 1198631205911[minus2 (119886
11+ 1198973) minus 2120572119897
1]
+ 120572lowast[21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591
1
minus21198976120572120573 minus 2 (119897
7+ 11988622) 1205722]
+ 120573
lowast
[21198976120572120573119890minus119894120596120591lowast
2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591
lowast
2
minus2119886331205732]
11989211
= 1198631205911[minus2 (119886
11+ 1198973) minus 1198971(120572 + 120572)]
+ 120572lowast[1198971(1205721198901198941205961205911+ 120572119890minus1198941205961205911) minus 1198976(120572120573 + 120572120573)
minus2 (1198977+ 11988622) 120572120572 + 2119897
3]
+ 120573
lowast
[1198976(120573120572119890119894120596120591lowast
2+ 120572120573119890
minus119894120596120591lowast
2)
+ 21198977120572120572 minus 119886
33120573120573]
11989202
= 21198631205911[minus2 (119886
11+ 1198973) minus 21198971120572]
+ 120572lowast[minus21198976120572120573 minus 2 (119897
7+ 11988622) 1205722+ 211989711205721198901198941205961205911
+2119897311989021198941205961205911]
+ 120573
lowast
[21198976120572120573119890119894120596120591lowast
2+ 2119897712057221198902119894120596120591lowast
2minus 11988633120573
2
]
11989221
= 1198631205911[minus (119886
11+ 1198973) (2119882
(1)
20(0) + 4119882
(1)
11(0))
minus 1198971(2120572119882
(1)
11(0) + 120572119882
(1)
20(0) + 119882
(2)
20(0)
+ 2119882(2)
11(0))]
+ 120572lowast[minus1198976(2120573119882
(2)
11(0) + 120572119882
(3)
20(0) + 120573119882
(2)
20(0)
+ 2120572119882(3)
11(0)) minus (119897
7+ 11988622)
times (4120572119882(2)
11(0) + 2120572119882
(2)
20(0))
+ 1198971(2120572119882
(1)
11(minus1) + 120572119882
(1)
20(minus1)
+ 119882(2)
20(0) 1198901198941205961205911+ 2119882
(2)
11(0) 119890minus1198941205961205911)
+ 1198973(4119882(1)
11(minus1) 119890
minus1198941205961205911+2119882(1)
20(minus1) 119890
1198941205961205911)]
+ 120573
lowast
[1198976(2120573119882
(2)
11(minus
120591lowast
2
1205911
) + 120573119882(2)
20(minus
120591lowast
2
1205911
)
+ 120572119882(3)
20(0) 119890119894120596120591lowast
2+ 2120572119882
(3)
11(0) 119890minus119894120596120591lowast
2)
+ 1198977(4120572119882
(2)
11(minus
120591lowast
2
1205911
) 119890minus119894120596120591lowast
2
+2120572119882(2)
20(minus
120591lowast
2
1205911
) 119890119894120596120591lowast
2)
minus 11988633(4120573119882
(3)
11(0) + 2120573119882
(3)
20(0))]
(68)
where
11988220(120579) =
11989411989220
1205961205911
119902 (0) 1198901198941205961205911120579+
11989411989202
31205961205911
119902 (0) 119890minus1198941205961205911120579+ 119864111989021198941205961205911120579
11988211(120579) = minus
11989411989211
1205961205911
119902 (0) 1198901198941205961205911120579+
11989411989211
1205961205911
119902 (0) 119890minus1198941205961205911120579+ 1198642
1198641= 2
[[[[[
[
2119894120596 minus 11988711
minus11988712
0
minus11988821119890minus2119894120596120591
12119894120596 minus 119887
22minus11988723
0 minus11988932119890minus2119894120596120591
lowast
22119894120596 minus 119887
33
]]]]]
]
minus1
times
[[[[[
[
minus2 (11988611
+ 1198973) minus 2120572119897
1
21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591
1minus 21198976120572120573 minus 2 (119897
7+ 11988622) 1205722
21198976120572120573119890minus119894120596120591lowast
2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591
lowast
2minus 2119886331205732
]]]]]
]
10 Abstract and Applied Analysis
1198642= 2
[[[[
[
minus11988711
minus11988712
0
minus11988821
minus11988722
minus11988723
0 minus11988932
minus11988733
]]]]
]
minus1
times
[[[[
[
minus2 (11988611
+ 1198973) minus 1198971(120572 + 120572)
1198971(1205721198901198941205961205911+ 120572119890minus1198941205961205911) minus 1198976(120572120573 + 120572120573) minus 2 (119897
7+ 11988622) 120572120572 + 2119897
3
1198976(120573120572119890119894120596120591lowast
2+ 120572120573119890
minus119894120596120591lowast
2) + 21198977120572120572 minus 119886
33120573120573
]]]]
]
(69)
Thus we can determine11988220(120579) and119882
11(120579) Furthermore
we can determine each 119892119894119895by the parameters and delay in (3)
Thus we can compute the following values
1198881(0) =
119894
21205961205911
(1198922011989211
minus 2100381610038161003816100381611989211
1003816100381610038161003816
2
minus
1
3
100381610038161003816100381611989202
1003816100381610038161003816
2
) +
1
2
11989221
1205832= minus
Re 1198881(0)
Re 1205821015840 (1205911)
1198792= minus
Im 1198881(0) + 120583
2Im 120582
1015840(1205911)
1205961205911
1205732= 2Re 119888
1(0)
(70)
which determine the quantities of bifurcating periodic solu-tions in the center manifold at the critical value 120591
1 Suppose
Re1205821015840(1205911) gt 0 120583
2determines the directions of the Hopf
bifurcation if 1205832
gt 0(lt 0) then the Hopf bifurcationis supercritical (subcritical) and the bifurcation exists for120591 gt 120591
1(lt 1205911) 1205732determines the stability of the bifurcation
periodic solutions the bifurcating periodic solutions arestable (unstable) if 120573
2lt 0(gt 0) and119879
2determines the period
of the bifurcating periodic solutions the period increases(decreases) if 119879
2gt 0(lt 0)
4 Numerical Simulation
We consider system (3) by taking the following coefficients1198861
= 03 11988611
= 58889 11988612
= 1 1198981
= 1 1198862
=
01 11988621
= 27 11988622
= 12 11988623
= 12 1198982
= 1 1198863
=
02 11988632
= 25 11988633
= 12 We have the unique positiveequilibrium 119864
lowast= (00451 00357 00551)
By computation we get 11989810
= minus00104 12059611
= 0516411991111
= 02666 ℎ10158401(11991111) = 03402 120591
20
= 32348 FromTheorem 6 we know that when 120591
1= 0 the positive
equilibrium 119864lowastis locally asymptotically stable for 120591
2isin
[0 32348) When 1205912crosses 120591
20
the equilibrium 119864lowastloses its
stability and Hopf bifurcation occurs From the algorithm inSection 3 we have 120583
2= 56646 120573
2= minus31583 119879
2= 5445
whichmeans that the bifurcation is supercritical and periodicsolution is stable The trajectories and the phase graphs areshown in Figures 1 and 2
Regarding 1205911as a parameter and let 120591
2= 29 isin
[0 32348) we can observe that with 1205911increasing the pos-
itive equilibrium 119864lowastloses its stability and Hopf bifurcation
occurs (see Figures 3 and 4)
5 Global Continuationof Local Hopf Bifurcations
In this section we study the global continuation of peri-odic solutions bifurcating from the positive equilibrium(119864lowast 120591(119895)
1119896) (119896 = 1 2 6 119895 = 0 1 ) Throughout this
section we follow closely the notations in [24] and assumethat 120591
2= 120591lowast
2isin [0 120591
210
) regarding 1205911as a parameter For
simplification of notations setting 119911119905(119905) = (119909
1119905 1199092119905 1199093119905)119879 we
may rewrite system (3) as the following functional differentialequation
(119905) = 119865 (119911119905 1205911 119901) (71)
where 119911119905(120579) = (119909
1119905(120579) 1199092119905(120579) 1199093119905(120579))119879
= (1199091(119905 + 120579) 119909
2(119905 +
120579) 1199093(119905 + 120579))
119879 for 119905 ge 0 and 120579 isin [minus1205911 0] Since 119909
1(119905) 1199092(119905)
and 1199093(119905) denote the densities of the prey the predator and
the top predator respectively the positive solution of system(3) is of interest and its periodic solutions only arise in the firstquadrant Thus we consider system (3) only in the domain1198773
+= (1199091 1199092 1199093) isin 1198773 1199091gt 0 119909
2gt 0 119909
3gt 0 It is obvious
that (71) has a unique positive equilibrium 119864lowast(119909lowast
1 119909lowast
2 119909lowast
3) in
1198773
+under the assumption (119867
1)ndash(1198674) Following the work of
[24] we need to define
119883 = 119862([minus1205911 0] 119877
3
+)
Γ = 119862119897 (119911 1205911 119901) isin X times R times R+
119911 is a 119901-periodic solution of system (71)
N = (119911 1205911 119901) 119865 (119911 120591
1 119901) = 0
(72)
Let ℓ(119864lowast120591(119895)
11198962120587120596
(119895)
1119896)denote the connected component pass-
ing through (119864lowast 120591(119895)
1119896 2120587120596
(119895)
1119896) in Γ where 120591
(119895)
1119896is defined by
(43) We know that ℓ(119864lowast120591(119895)
11198962120587120596
(119895)
1119896)through (119864
lowast 120591(119895)
1119896 2120587120596
(119895)
1119896)
is nonemptyFor the benefit of readers we first state the global Hopf
bifurcation theory due to Wu [24] for functional differentialequations
Lemma 9 Assume that (119911lowast 120591 119901) is an isolated center satis-
fying the hypotheses (A1)ndash(A4) in [24] Denote by ℓ(119911lowast120591119901)
theconnected component of (119911
lowast 120591 119901) in Γ Then either
(i) ℓ(119911lowast120591119901)
is unbounded or(ii) ℓ(119911lowast120591119901)
is bounded ℓ(119911lowast120591119901)
cap Γ is finite and
sum
(119911120591119901)isinℓ(119911lowast120591119901)capN
120574119898(119911lowast 120591 119901) = 0 (73)
for all 119898 = 1 2 where 120574119898(119911lowast 120591 119901) is the 119898119905ℎ crossing
number of (119911lowast 120591 119901) if119898 isin 119869(119911
lowast 120591 119901) or it is zero if otherwise
Clearly if (ii) in Lemma 9 is not true then ℓ(119911lowast120591119901)
isunbounded Thus if the projections of ℓ
(119911lowast120591119901)
onto 119911-spaceand onto 119901-space are bounded then the projection of ℓ
(119911lowast120591119901)
onto 120591-space is unbounded Further if we can show that the
Abstract and Applied Analysis 11
0 200 400 60000451
00451
00452
t
x1(t)
t
0 200 400 60000356
00357
00358
x2(t)
t
0 200 400 6000055
00551
00552
x3(t)
00451
00451
00452
00356
00357
00357
003570055
0055
00551
00551
00552
x 1(t)
x2 (t)
x3(t)
Figure 1 The trajectories and the phase graph with 1205911= 0 120591
2= 29 lt 120591
20= 32348 119864
lowastis locally asymptotically stable
0 500 1000004
0045
005
0 500 1000002
004
006
0 500 10000
005
01
t
t
t
x1(t)
x2(t)
x3(t)
0044
0046
0048
002
003
004
005003
004
005
006
007
008
x 1(t)
x2 (t)
x3(t)
Figure 2 The trajectories and the phase graph with 1205911= 0 120591
2= 33 gt 120591
20= 32348 a periodic orbit bifurcate from 119864
lowast
12 Abstract and Applied Analysis
0 500 1000 1500
00451
00452
0 500 1000 15000035
0036
0037
0 500 1000 15000054
0055
0056
t
t
t
x1(t)
x2(t)
x3(t)
0045100451
0045200452
0035
00355
003600546
00548
0055
00552
00554
00556
00558
x1(t)
x2 (t)
x3(t)
Figure 3 The trajectories and the phase graph with 1205911= 09 120591
2= 29 119864
lowastis locally asymptotically stable
0 500 1000 15000044
0045
0046
0 500 1000 1500002
003
004
0 500 1000 1500004
006
008
t
t
t
x1(t)
x2(t)
x3(t)
00445
0045
00455
0032
0034
0036
00380052
0054
0056
0058
006
0062
0064
x 1(t)
x2 (t)
x3(t)
Figure 4 The trajectories and the phase graph with 1205911= 12 120591
2= 29 a periodic orbit bifurcate from 119864
lowast
Abstract and Applied Analysis 13
projection of ℓ(119911lowast120591119901)
onto 120591-space is away from zero thenthe projection of ℓ
(119911lowast120591119901)
onto 120591-space must include interval[120591 infin) Following this ideal we can prove our results on theglobal continuation of local Hopf bifurcation
Lemma 10 If the conditions (1198671)ndash(1198674) hold then all nontriv-
ial periodic solutions of system (71) with initial conditions
1199091(120579) = 120593 (120579) ge 0 119909
2(120579) = 120595 (120579) ge 0
1199093(120579) = 120601 (120579) ge 0 120579 isin [minus120591
1 0)
120593 (0) gt 0 120595 (0) gt 0 120601 (0) gt 0
(74)
are uniformly bounded
Proof Suppose that (1199091(119905) 1199092(119905) 1199093(119905)) are nonconstant peri-
odic solutions of system (3) and define
1199091(1205851) = min 119909
1(119905) 119909
1(1205781) = max 119909
1(119905)
1199092(1205852) = min 119909
2(119905) 119909
2(1205782) = max 119909
2(119905)
1199093(1205853) = min 119909
3(119905) 119909
3(1205783) = max 119909
3(119905)
(75)
It follows from system (3) that
1199091(119905) = 119909
1(0) expint
119905
0
[1198861minus 119886111199091(119904) minus
119886121199092(119904)
1198981+ 1199091(119904)
] 119889119904
1199092(119905) = 119909
2(0) expint
119905
0
[minus1198862+
119886211199091(119904 minus 1205911)
1198981+ 1199091(119904 minus 1205911)
minus 119886221199092(119904) minus
119886231199093(119904)
1198982+ 1199092(119904)
] 119889119904
1199093(119905) = 119909
3(0) expint
119905
0
[minus1198863+
119886321199092(119904 minus 120591
lowast
2)
1198982+ 1199092(119904 minus 120591
lowast
2)
minus 119886331199093(119904)] 119889119904
(76)
which implies that the solutions of system (3) cannot crossthe 119909119894-axis (119894 = 1 2 3) Thus the nonconstant periodic orbits
must be located in the interior of first quadrant It followsfrom initial data of system (3) that 119909
1(119905) gt 0 119909
2(119905) gt
0 1199093(119905) gt 0 for 119905 ge 0
From the first equation of system (3) we can get
0 = 1198861minus 119886111199091(1205781) minus
119886121199092(1205781)
1198981+ 1199091(1205781)
le 1198861minus 119886111199091(1205781) (77)
thus we have
1199091(1205781) le
1198861
11988611
(78)
From the second equation of (3) we obtain
0 = minus 1198862+
119886211199091(1205782minus 1205911)
1198981+ 1199091(1205782minus 1205911)
minus 119886221199092(1205782)
minus
119886231199093(1205782)
1198982+ 1199092(1205782)
le minus1198862+
11988621(119886111988611)
1198981+ (119886111988611)
minus 119886221199092(1205782)
(79)
therefore one can get
1199092(1205782) le
minus1198862(119886111198981+ 1198861) + 119886111988621
11988622(119886111198981+ 1198861)
≜ 1198721 (80)
Applying the third equation of system (3) we know
0 = minus1198863+
119886321199092(1205783minus 120591lowast
2)
1198982+ 1199092(1205783minus 120591lowast
2)
minus 119886331199093(1205783)
le minus1198863+
119886321198721
1198982+1198721
minus 119886331199093(1205783)
(81)
It follows that
1199093(1205783) le
minus1198863(1198982+1198721) + 119886321198721
11988633(1198982+1198721)
≜ 1198722 (82)
This shows that the nontrivial periodic solution of system (3)is uniformly bounded and the proof is complete
Lemma 11 If the conditions (1198671)ndash(1198674) and
(H7) 11988622
minus (119886211198982) gt 0 [(119886
11minus 11988611198981)11988622
minus
(11988612119886211198982
1)]11988633
minus (11988611
minus (11988611198981))[(119886211198982)11988633
+
(11988623119886321198982
2)] gt 0
hold then system (3) has no nontrivial 1205911-periodic solution
Proof Suppose for a contradiction that system (3) has non-trivial periodic solution with period 120591
1 Then the following
system (83) of ordinary differential equations has nontrivialperiodic solution
1198891199091
119889119905
= 1199091(119905) [1198861minus 119886111199091(119905) minus
119886121199092(119905)
1198981+ 1199091(119905)
]
1198891199092
119889119905
= 1199092(119905) [minus119886
2+
119886211199091(119905)
1198981+ 1199091(119905)
minus 119886221199092(119905) minus
119886231199093(119905)
1198982+ 1199092(119905)
]
1198891199093
119889119905
= 1199093(119905) [minus119886
3+
119886321199092(119905 minus 120591
lowast
2)
1198982+ 1199092(119905 minus 120591
lowast
2)
minus 119886331199093(119905)]
(83)
which has the same equilibria to system (3) that is
1198641= (0 0 0) 119864
2= (
1198861
11988611
0 0)
1198643= (1199091 1199092 0) 119864
lowast= (119909lowast
1 119909lowast
2 119909lowast
3)
(84)
Note that 119909119894-axis (119894 = 1 2 3) are the invariable manifold of
system (83) and the orbits of system (83) do not intersect each
14 Abstract and Applied Analysis
other Thus there are no solutions crossing the coordinateaxes On the other hand note the fact that if system (83) hasa periodic solution then there must be the equilibrium in itsinterior and that 119864
1 1198642 1198643are located on the coordinate
axis Thus we conclude that the periodic orbit of system (83)must lie in the first quadrant If (119867
7) holds it is well known
that the positive equilibrium 119864lowastis globally asymptotically
stable in the first quadrant (see [1])Thus there is no periodicorbit in the first quadrant too The above discussion meansthat (83) does not have any nontrivial periodic solution It isa contradiction Therefore the lemma is confirmed
Theorem 12 Suppose the conditions of Theorem 8 and (1198677)
hold let120596119896and 120591(119895)1119896
be defined in Section 2 then when 1205911gt 120591(119895)
1119896
system (3) has at least 119895 minus 1 periodic solutions
Proof It is sufficient to prove that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is [120591
1 +infin) for each 119895 ge 1 where
1205911le 120591(119895)
1119896
In following we prove that the hypotheses (A1)ndash(A4) in[24] hold
(1) From system (3) we know easily that the followingconditions hold
(A1) 119865 isin 1198622(1198773
+times119877+times119877+) where 119865 = 119865|
1198773
+times119877+times119877+
rarr
1198773
+
(A3) 119865(120601 1205911 119901) is differential with respect to 120601
(2) It follows from system (3) that
119863119911119865 (119911 120591
1 119901) =
[[[[[[[[[[[
[
1198861minus 2119886111199091minus
1198861211989811199092
(1198981+ 1199091)2
minus
119886121199091
1198981+ 1199091
0
1198862111989811199092
(1198981+ 1199091)2
minus1198862+
119886211199091
1198981+ 1199091
minus 2119886221199092minus
1198862311989821199093
(1198982+ 1199092)2
minus
119886231199092
1198982+ 1199092
0
1198863211989821199093
(1198982+ 1199092)2
minus1198863+
119886321199092
1198982+ 1199092
minus 2119886331199093
]]]]]]]]]]]
]
(85)
Then under the assumption (1198671)ndash(1198674) we have
det119863119911119865 (119911lowast 1205911 119901)
= det
[[[[[[[[
[
minus11988611119909lowast
1+
11988612119909lowast
1119909lowast
2
(1198981+ 119909lowast
1)2
11988612119909lowast
1
1198981+ 119909lowast
1
0
119886211198981119909lowast
2
(1198981+ 119909lowast
1)2
minus11988622119909lowast
2+
11988623119909lowast
2119909lowast
3
(1198982+ 119909lowast
2)2
minus
11988623119909lowast
2
1198982+ 119909lowast
2
0
119886321198982119909lowast
3
(1198982+ 119909lowast
2)2
minus11988633119909lowast
3
]]]]]]]]
]
= minus
1198862
21199101lowast1199102lowast
(1 + 11988721199101lowast)3[minus119909lowast+
11988611198871119909lowast1199101lowast
(1 + 1198871119909lowast)2] = 0
(86)
From (86) we know that the hypothesis (A2) in [24]is satisfied
(3) The characteristic matrix of (71) at a stationarysolution (119911 120591
0 1199010) where 119911 = (119911
(1) 119911(2) 119911(3)) isin 119877
3
takes the following form
Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863
120601119865 (119911 120591
1 119901) (119890
120582119868) (87)
that is
Δ (119911 1205911 119901) (120582)
=
[[[[[[[[[[[[[[[[[
[
120582 minus 1198861+ 211988611119911(1)
+
119886121198981119911(2)
(1198981+ 119911(1))
2
11988612119911(1)
1198981+ 119911(1)
0
minus
119886211198981119911(2)
(1198981+ 119911(1))
2119890minus1205821205911
120582 + 1198862minus
11988621119911(1)
1198981+ 119911(1)
+ 211988622119911(2)
+
119886231198982119911(3)
(1198982+ 119911(2))
2
11988623119911(2)
1198982+ 119911(2)
0 minus
119886321198982119911(3)
(1198982+ 119911(2))
2119890minus120582120591lowast
2120582 + 1198863minus
11988632119911(2)
1198982+ 119911(2)
+ 211988633119911(3)
]]]]]]]]]]]]]]]]]
]
(88)
Abstract and Applied Analysis 15
From (88) we have
det (Δ (119864lowast 1205911 119901) (120582))
= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902
0+ 1199030)] 119890minus1205821205911
(89)
Note that (89) is the same as (20) from the discussion inSection 2 about the local Hopf bifurcation it is easy to verifythat (119864
lowast 120591(119895)
1119896 2120587120596
119896) is an isolated center and there exist 120598 gt
0 120575 gt 0 and a smooth curve 120582 (120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575) rarr C
such that det(Δ(120582(1205911))) = 0 |120582(120591
1) minus 120596119896| lt 120598 for all 120591
1isin
[120591(119895)
1119896minus 120575 120591(119895)
1119896+ 120575] and
120582 (120591(119895)
1119896) = 120596119896119894
119889Re 120582 (1205911)
1198891205911
1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)
1119896
gt 0 (90)
Let
Ω1205982120587120596
119896
= (120578 119901) 0 lt 120578 lt 120598
10038161003816100381610038161003816100381610038161003816
119901 minus
2120587
120596119896
10038161003816100381610038161003816100381610038161003816
lt 120598 (91)
It is easy to see that on [120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575] times 120597Ω
1205982120587120596119896
det(Δ(119864
lowast 1205911 119901)(120578 + (2120587119901)119894)) = 0 if and only if 120578 = 0
1205911= 120591(119895)
1119896 119901 = 2120587120596
119896 119896 = 1 2 3 119895 = 0 1 2
Therefore the hypothesis (A4) in [24] is satisfiedIf we define
119867plusmn(119864lowast 120591(119895)
1119896
2120587
120596119896
) (120578 119901)
= det(Δ (119864lowast 120591(119895)
1119896plusmn 120575 119901) (120578 +
2120587
119901
119894))
(92)
then we have the crossing number of isolated center(119864lowast 120591(119895)
1119896 2120587120596
119896) as follows
120574(119864lowast 120591(119895)
1119896
2120587
120596119896
) = deg119861(119867minus(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
minus deg119861(119867+(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
= minus1
(93)
Thus we have
sum
(1199111205911119901)isinC
(119864lowast120591(119895)
11198962120587120596119896)
120574 (119911 1205911 119901) lt 0
(94)
where (119911 1205911 119901) has all or parts of the form
(119864lowast 120591(119896)
1119895 2120587120596
119896) (119895 = 0 1 ) It follows from
Lemma 9 that the connected component ℓ(119864lowast120591(119895)
11198962120587120596
119896)
through (119864lowast 120591(119895)
1119896 2120587120596
119896) is unbounded for each center
(119911lowast 1205911 119901) (119895 = 0 1 ) From the discussion in Section 2
we have
120591(119895)
1119896=
1
120596119896
times arccos(((11990121205962
119896minus1199010) (1199030+1199020)
+ 1205962(1205962minus1199011) (1199031+1199021))
times ((1199030+1199020)2
+ (1199031+1199021)2
1205962
119896)
minus1
) +2119895120587
(95)
where 119896 = 1 2 3 119895 = 0 1 Thus one can get 2120587120596119896le 120591(119895)
1119896
for 119895 ge 1Now we prove that the projection of ℓ
(119864lowast120591(119895)
11198962120587120596
119896)onto
1205911-space is [120591
1 +infin) where 120591
1le 120591(119895)
1119896 Clearly it follows
from the proof of Lemma 11 that system (3) with 1205911
= 0
has no nontrivial periodic solution Hence the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is away from zero
For a contradiction we suppose that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is bounded this means that the
projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is included in a
interval (0 120591lowast) Noticing 2120587120596119896lt 120591119895
1119896and applying Lemma 11
we have 119901 lt 120591lowast for (119911(119905) 120591
1 119901) belonging to ℓ
(119864lowast120591(119895)
11198962120587120596
119896)
This implies that the projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 119901-
space is bounded Then applying Lemma 10 we get thatthe connected component ℓ
(119864lowast120591(119895)
11198962120587120596
119896)is bounded This
contradiction completes the proof
6 Conclusion
In this paper we take our attention to the stability and Hopfbifurcation analysis of a predator-prey systemwithMichaelis-Menten type functional response and two unequal delaysWe obtained some conditions for local stability and Hopfbifurcation occurring When 120591
1= 1205912 we derived the explicit
formulas to determine the properties of periodic solutionsby the normal form method and center manifold theoremSpecially the global existence results of periodic solutionsbifurcating from Hopf bifurcations are also established byusing a global Hopf bifurcation result due to Wu [24]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to thank the anonymous referees fortheir careful reading and constructive suggestions which leadto truly significant improvement of the paper This researchis supported by the National Natural Science Foundation ofChina (no 11061016)
16 Abstract and Applied Analysis
References
[1] R Xu and M A J Chaplain ldquoPersistence and global stabilityin a delayed predator-prey system with Michaelis-Menten typefunctional responserdquo Applied Mathematics and Computationvol 130 no 2-3 pp 441ndash455 2002
[2] Y L Song Y H Peng and J JWei ldquoBifurcations for a predator-prey system with two delaysrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 466ndash479 2008
[3] G-P Hu W-T Li and X-P Yan ldquoHopf bifurcations in apredator-prey system with multiple delaysrdquo Chaos Solitons ampFractals vol 42 no 2 pp 1273ndash1285 2009
[4] M Liao X Tang and C Xu ldquoBifurcation analysis for a three-species predator-prey systemwith two delaysrdquoCommunicationsin Nonlinear Science and Numerical Simulation vol 17 no 1 pp183ndash194 2012
[5] C Xu and P Li ldquoDynamical analysis in a delayed predator-prey model with two delaysrdquo Discrete Dynamics in Nature andSociety vol 2012 Article ID 652947 22 pages 2012
[6] J-F Zhang ldquoStability and bifurcation periodic solutions ina Lotka-Volterra competition system with multiple delaysrdquoNonlinear Dynamics vol 70 no 1 pp 849ndash860 2012
[7] J Xia Z Yu and R Yuan ldquoStability and Hopf bifurcation ina symmetric Lotka-Volterra predator-prey system with delaysrdquoElectronic Journal of Differential Equations vol 2013 no 9 pp1ndash16 2013
[8] H Wan and J Cui ldquoA malaria model with two delaysrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 601265 8pages 2013
[9] M Zhao ldquoHopf bifurcation analysis for a semiratio-dependentpredator-prey system with two delaysrdquo Abstract and AppliedAnalysis vol 2013 Article ID 495072 13 pages 2013
[10] S Feyissa and S Banerjee ldquoDelay-induced oscillatory dynamicsin humoral mediated immune response with two time delaysrdquoNonlinear Analysis Real World Applications vol 14 no 1 pp35ndash52 2013
[11] G Zhang Y Shen andB Chen ldquoHopf bifurcation of a predator-prey systemwith predator harvesting and twodelaysrdquoNonlinearDynamics vol 73 no 4 pp 2119ndash2131 2013
[12] X-Y Meng H-F Huo and H Xiang ldquoHopf bifurcation in athree-species systemwith delaysrdquo Journal of AppliedMathemat-ics and Computing vol 35 no 1-2 pp 635ndash661 2011
[13] J-F Zhang ldquoGlobal existence of bifurcated periodic solutions ina commensalism model with delaysrdquo Applied Mathematics andComputation vol 218 no 23 pp 11688ndash11699 2012
[14] Y Wang W Jiang and H Wang ldquoStability and global Hopfbifurcation in toxic phytoplankton-zooplankton model withdelay and selective harvestingrdquoNonlinear Dynamics vol 73 no1-2 pp 881ndash896 2013
[15] Y Ma ldquoGlobal Hopf bifurcation in the Leslie-Gower predator-prey model with two delaysrdquo Nonlinear Analysis Real WorldApplications vol 13 no 1 pp 370ndash375 2012
[16] X-Y Meng H-F Huo and X-B Zhang ldquoStability and globalHopf bifurcation in a delayed food web consisting of a preyand two predatorsrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4335ndash4348 2011
[17] Y Song J Wei andM Han ldquoLocal and global Hopf bifurcationin a delayed hematopoiesis modelrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol14 no 11 pp 3909ndash3919 2004
[18] Y Song and J Wei ldquoLocal Hopf bifurcation and global periodicsolutions in a delayed predator-prey systemrdquo Journal of Mathe-matical Analysis and Applications vol 301 no 1 pp 1ndash21 2005
[19] J Wei and M Y Li ldquoGlobal existence of periodic solutions in atri-neuron network model with delaysrdquo Physica D vol 198 no1-2 pp 106ndash119 2004
[20] T Zhao Y Kuang and H L Smith ldquoGlobal existence of peri-odic solutions in a class of delayed Gause-type predator-preysystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 28 no 8 pp 1373ndash1394 1997
[21] S J Gao L S Chen and Z D Teng ldquoHopf bifurcation andglobal stability for a delayed predator-prey system with stagestructure for predatorrdquo Applied Mathematics and Computationvol 202 no 2 pp 721ndash729 2008
[22] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977
[23] B Hassard D Kazarinoff and Y WanTheory and Applicationsof Hopf Bifurcation Cambridge University Press CambridgeUK 1981
[24] J H Wu ldquoSymmetric functional-differential equations andneural networks with memoryrdquo Transactions of the AmericanMathematical Society vol 350 no 12 pp 4799ndash4838 1998
[25] V Hutson ldquoThe existence of an equilibrium for permanentsystemsrdquo The Rocky Mountain Journal of Mathematics vol 20no 4 pp 1033ndash1040 1990
[26] S G Ruan and J J Wei ldquoOn the zeros of transcendentalfunctions with applications to stability of delay differentialequations with two delaysrdquo Dynamics of Continuous Discreteamp Impulsive Systems A vol 10 no 6 pp 863ndash874 2003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 3
where
1199012= minus (119887
11+ 11988722
+ 11988733)
1199011= 1198871111988722
+ 1198872211988733
+ 1198871111988733
1199010= minus119887111198872211988733
1199021= minus1198871211988821 1199020= 119887121198882111988733
1199031= minus1198872311988932 1199030= 119887111198872311988932
(11)
We consider the following cases
(1) 119864 = 1198641 The characteristic equation reduces to
(120582 minus 1198861) (120582 + 119886
2) (120582 + 119886
3) = 0 (12)
There are always a positive root 1198861and two negative roots
1198862 1198863of (12) hence 119864
1is a saddle point
(2) 119864 = 1198642 Equation (10) takes the form
(120582 + 1198861) (120582 + 119886
2minus
119886111988612
119898111988611
+ 1198861
) (120582 + 1198863) = 0 (13)
There is a positive root 120582 = (119886111988612(119898111988611
+ 1198861)) minus 119886
2if
119886111988612(119898111988611
+ 1198861) gt 119886
2 hence 119864
2is a saddle point If
119886111988612(119898111988611
+ 1198861) lt 119886
2 1198642is locally asymptotically stable
(3) 119864 = 1198643 The characteristic equation is
(120582 minus 11988733) [1205822minus (11988711
+ 11988722) 120582 + 119887
1111988722
minus 1198871211988821119890minus1205821205911] = 0 (14)
We will analyse the distribution of the characteristic root of(14) from Ruan and Wei [26] which is stated as follows
Lemma 1 Consider the exponential polynomial
119875 (120582 119890minus1205821205911 119890
minus120582120591119898)
= 120582119899+ 119901(0)
1120582119899minus1
+ sdot sdot sdot + 119901(0)
119899minus1120582 + 119901(0)
119899
+ [119901(1)
1120582119899minus1
+ sdot sdot sdot + 119901(1)
119899minus1120582 + 119901(1)
119899] 119890minus1205821205911
+ sdot sdot sdot + [119901(119898)
1120582119899minus1
+ sdot sdot sdot + 119901(119898)
119899minus1120582 + 119901(119898)
119899] 119890minus120582120591119898
(15)
where 120591119894⩾ 0 (119894 = 1 2 119898) and 119901
(119894)
119895(119894 = 0 1 119898 119895 =
1 2 119899) are constants As (1205911 1205912 120591
119898) vary the sum of
the order of the zeros of 119875(120582 119890minus1205821205911 119890minus120582120591119898) on the open righthalf plane can change only if a zero appears on or crosses theimaginary axis
By using Lemma 1 we can easily obtain the followingresults
Lemma 2 If 1198643is a nonnegative equilibrium point then
(1) 1198643is unstable if 119887
33gt 0
(2) 1198643is locally asymptotically stable if 119887
33lt 0 11988711
+ 11988722
lt
0 1198871111988722
minus 1198871211988821
gt 0 and 1198871111988722
+ 1198871211988821
gt 0
Proof (1) 120582 = 11988733
is a root of (14) if 11988733
gt 0 then 1198643is
unstable(2) Clearly 120582 = 0 is not a root of (14) we should discuss
the following equation instead of (14)
1205822minus (11988711
+ 11988722) 120582 + 119887
1111988722
minus 1198871211988821119890minus1205821205911= 0 (16)
Assume that 119894120596 with 120596 gt 0 is a solution of (16) Substituting120582 = 119894120596 into (16) and separating the real and imaginary partsyield
minus1205962+ 1198871111988722
= 1198871211988821cos120596120591
1
120596 (11988711
+ 11988722) = 1198871211988821sin120596120591
1
(17)
which implies
1205964+ (1198872
11+ 1198872
22) 1205962+ 1198872
111198872
22minus 1198872
121198882
21= 0 (18)
If 1198872111198872
22minus 1198872
121198882
21gt 0 that is (119887
1111988722
+ 1198871211988821)(1198871111988722
minus 1198871211988821) gt
0 there is no real root of (16) Hence there is no purelyimaginary root of (18) When 120591
1= 0 (16) reduces to
1205822minus (11988711
+ 11988722) 120582 + 119887
1111988722
minus 1198871211988821
= 0 (19)
If 11988711+ 11988722
lt 0 and 1198871111988722minus 1198871211988821
gt 0 both roots of (19) havenegative real parts Thus by using Lemma 1 when 119887
33lt 0
11988711
+ 11988722
lt 0 1198871111988722
minus 1198871211988821
gt 0 and 1198871111988722
+ 1198871211988821
gt 0 1198643is
locally asymptotically stable
(4)119864 = 119864lowastThe characteristic equation about119864
lowastis (10) In the
following we will analyse the distribution of roots of (10)Weconsider four cases
Case a Consider1205911= 1205912= 0
The associated characteristic equation of system (3) is
1205823+ 11990121205822+ (1199011+ 1199021+ 1199031) 120582 + (119901
0+ 1199020+ 1199030) = 0 (20)
Let
(H5) 1199012gt 0 119901
2(1199011+1199021+1199031)minus(1199010+1199020+1199030) gt 0 119901
0+1199020+1199030gt
0
By Routh-Hurwitz criterion we have the following
Theorem 3 For 1205911= 1205912= 0 assume that (119867
1)ndash(1198675) hold
Thenwhen 1205911= 1205912= 0 the positive equilibrium119864
lowast(119909lowast
1 119909lowast
2 119909lowast
3)
of system (3) is locally asymptotically stable
Case b Consider1205911= 0 1205912gt 0
The associated characteristic equation of system (3) is
1205823+ 11990121205822+ (1199011+ 1199021) 120582 + (119901
0+ 1199020) + (1199031120582 + 1199030) 119890minus1205821205912= 0
(21)
4 Abstract and Applied Analysis
We want to determine if the real part of some rootincreases to reach zero and eventually becomes positive as 120591varies Let 120582 = 119894120596 (120596 gt 0) be a root of (21) then we have
minus 1198941205963minus 11990121205962+ 119894 (1199011+ 1199021) 120596 + (119901
0+ 1199020)
+ (1199031120596119894 + 119903
0) (cos120596120591
2minus 119894 sin120596120591
2) = 0
(22)
Separating the real and imaginary parts we have
minus1205963+ (1199011+ 1199021) 120596 = 119903
0sin120596120591
2minus 1199031120596 cos120596120591
2
minus11990121205962+ (1199010+ 1199020) = minus119903
1120596 sin120596120591
2minus 1199030cos120596120591
2
(23)
It follows that
1205966+ 119898121205964+ 119898111205962+ 11989810
= 0 (24)
where 11989812
= 1199012
2minus 2(119901
1+ 1199021) 11989811
= (1199011+ 1199021)2minus 21199012(1199010+
1199020) minus 1199032
1 11989810
= (1199010+ 1199020)2minus 1199032
0
Denoting 119911 = 1205962 (24) becomes
1199113+ 119898121199112+ 11989811119911 + 119898
10= 0 (25)
Let
ℎ1(119911) = 119911
3+ 119898121199112+ 11989811119911 + 119898
10 (26)
we have
119889ℎ1(119911)
119889119911
= 31199112+ 211989812119911 + 119898
11 (27)
If 11989810
= (1199010+ 1199020)2minus 1199032
0lt 0 then ℎ
1(0) lt
0 lim119911rarr+infin
ℎ1(119911) = +infin We can know that (25) has at least
one positive rootIf 11989810
= (1199010+ 1199020)2minus 1199032
0ge 0 we obtain that when Δ =
1198982
12minus 311989811
le 0 (25) has no positive roots for 119911 isin [0 +infin)On the other hand when Δ = 119898
2
12minus 311989811
gt 0 the followingequation
31199112+ 211989812119911 + 119898
11= 0 (28)
has two real roots 119911lowast11
= (minus11989812
+ radicΔ)3 119911lowast
12= (minus119898
12minus
radicΔ)3 Because of ℎ101584010158401(119911lowast
11) = 2radicΔ gt 0 ℎ
10158401015840
1(119911lowast
12) = minus2radicΔ lt
0 119911lowast
11and 119911lowast12are the local minimum and the localmaximum
of ℎ1(119911) respectively By the above analysis we immediately
obtain the following
Lemma 4 (1) If 11989810
ge 0 and Δ = 1198982
12minus 311989811
le 0 (25) hasno positive root for 119911 isin [0 +infin)
(2) If 11989810
ge 0 and Δ = 1198982
12minus 311989811
gt 0 (25) has at leastone positive root if and only if 119911lowast
11= (minus119898
12+ radicΔ)3 gt 0 and
ℎ1(119911lowast
11) le 0
(3) If 11989810
lt 0 (25) has at least one positive root
Without loss of generality we assume that (25) has threepositive roots defined by 119911
11 11991112 11991113 respectively Then
(24) has three positive roots
12059611
= radic11991111 120596
12= radic11991112 120596
13= radic11991113 (29)
From (23) we have
cos120596111989612059121119896
=
11990311205964
1119896+ [11990121199030minus (1199021+ 1199011) 1199031] 1205962
1119896minus 1199030(1199020+ 1199010)
1199032
0+ 1199032
11205962
1119896
(30)
Thus if we denote
120591(119895)
21119896
=
1
1205961119896
times arccos ((11990311205964
1119896+ [11990121199030minus (1199021+ 1199011) 1199031] 1205962
1119896
minus 1199030(1199020+ 1199010))
times (1199032
0+ 1199032
11205962
1119896)
minus1
) + 2119895120587
(31)
where 119896 = 1 2 3 119895 = 0 1 2 then plusmn1198941205961119896is a pair of purely
imaginary roots of (21) corresponding to 120591(119895)
21119896
Define
120591210
= 120591(0)
211198960
= min119896=123
120591(0)
21119896
12059610
= 12059611198960
(32)
Let 120582(1205912) = 120572(120591
2)+119894120596(120591
2) be the root of (21) near 120591
2= 120591(119895)
21119896
satisfying
120572 (120591(119895)
21119896
) = 0 120596 (120591(119895)
21119896
) = 1205961119896 (33)
Substituting 120582(1205912) into (21) and taking the derivative with
respect to 1205912 we have
31205822+ 21199012120582 + (119901
1+ 1199021) + 1199031119890minus1205821205912minus 1205912(1199031120582 + 1199030) 119890minus1205821205912
119889120582
1198891205912
= 120582 (1199031120582 + 1199030) 119890minus1205821205912
(34)
Therefore
[
119889120582
1198891205912
]
minus1
=
[31205822+ 21199012120582 + (119901
1+ 1199021)] 1198901205821205912
120582 (1199031120582 + 1199030)
+
1199031
120582 (1199031120582 + 1199030)
minus
1205912
120582
(35)
When 1205912
= 120591(119895)
21119896
120582(120591(119895)
21119896
) = 1198941205961119896
(119896 = 1 2 3) 120582(1199031120582 +
1199030)|1205912=120591(119895)
21119896
= minus11990311205962
1119896+ 11989411990301205961119896 [31205822
+ 21199012120582 + (119901
1+
1199021)]1198901205821205912|1205912=120591(119895)
21119896
= [minus31205962
1119896+ (1199011+ 1199021)] cos(120596
1119896120591(119895)
21119896
) minus
211990121205961119896sin(1205961119896120591(119895)
21119896
) + 119894211990121205961119896cos(120596
1119896120591(119895)
21119896
) + [minus31205962
1119896+ (1199011+
1199021)] sin(120596
1119896120591(119895)
21119896
)
Abstract and Applied Analysis 5
According to (35) we have
[
Re 119889 (120582 (1205912))
1198891205912
]
minus1
1205912=120591(119895)
21119896
= Re[[31205822+ 21199012120582 + (119901
1+ 1199021)] 1198901205821205912
120582 (1199031120582 + 1199030)
]
1205912=120591(119895)
21119896
+ Re[ 1199031
120582 (1199031120582 + 1199030)
]
1205912=120591(119895)
21119896
=
1
Λ1
minus 11990311205962
1119896[minus31205962
1119896+ (1199011+ 1199021)] cos (120596
1119896120591(119895)
21119896
)
+ 2119903111990121205963
1119896sin (120596
1119896120591(119895)
21119896
) minus 1199032
11205962
1119896
+ 2119903011990121205962
1119896cos (120596
1119896120591(119895)
21119896
)
+ 1199030[minus31205962
1119896+ (1199011+ 1199021)] 1205961119896sin (120596
1119896120591(119895)
21119896
)
=
1
Λ1
31205966
1119896+ 2 [119901
2
2minus 2 (119901
1+ 1199021)] 1205964
1119896
+ [(1199011+ 1199021)2
minus 21199012(1199010+ 1199020) minus 1199032
1] 1205962
1119896
=
1
Λ1
1199111119896(31199112
1119896+ 2119898121199111119896
+ 11989811)
=
1
Λ1
1199111119896ℎ1015840
1(1199111119896)
(36)
where Λ1= 1199032
11205964
1119896+ 1199032
01205962
1119896gt 0 Notice that Λ
1gt 0 1199111119896
gt 0
sign
[
Re 119889 (120582 (1205912))
1198891205912
]
1205912=120591(119895)
21119896
= sign
[
Re 119889 (120582 (1205912))
1198891205912
]
minus1
1205912=120591(119895)
21119896
(37)
then we have the following lemma
Lemma 5 Suppose that 1199111119896
= 1205962
1119896and ℎ
1015840
1(1199111119896) = 0 where
ℎ1(119911) is defined by (26) then 119889(Re 120582(120591(119895)
21119896
))1198891205912has the same
sign with ℎ1015840
1(1199111119896)
From Lemmas 1 4 and 5 and Theorem 3 we can easilyobtain the following theorem
Theorem6 For 1205911= 0 120591
2gt 0 suppose that (119867
1)ndash(1198675) hold
(i) If 11989810
ge 0 and Δ = 1198982
12minus 311989811
le 0 then all rootsof (10) have negative real parts for all 120591
2ge 0 and the
positive equilibrium 119864lowastis locally asymptotically stable
for all 1205912ge 0
(ii) If either 11989810
lt 0 or 11989810
ge 0 Δ = 1198982
12minus 311989811
gt
0 119911lowast
11gt 0 and ℎ
1(119911lowast
11) le 0 then ℎ
1(119911) has at least one
positive roots and all roots of (23) have negative realparts for 120591
2isin [0 120591
210
) and the positive equilibrium 119864lowast
is locally asymptotically stable for 1205912isin [0 120591
210
)
(iii) If (ii) holds and ℎ1015840
1(1199111119896) = 0 then system (3)undergoes
Hopf bifurcations at the positive equilibrium119864lowastfor 1205912=
120591(119895)
21119896
(119896 = 1 2 3 119895 = 0 1 2 )
Case c Consider1205911gt 0 1205912= 0
The associated characteristic equation of system (3) is
1205823+ 11990121205822+ (1199011+ 1199031) 120582 + (119901
0+ 1199030) + (119902
1120582 + 1199020) 119890minus1205821205911= 0
(38)
Similar to the analysis of Case 119887 we get the followingtheorem
Theorem7 For 1205911gt 0 120591
2= 0 suppose that (119867
1)ndash(1198675) hold
(i) If 11989820
ge 0 and Δ = 1198982
22minus 311989821
le 0 then all rootsof (38) have negative real parts for all 120591
1ge 0 and the
positive equilibrium 119864lowastis locally asymptotically stable
for all 1205911ge 0
(ii) If either 11989820
lt 0 or 11989820
ge 0 Δ = 1198982
22minus 311989821
gt 0119911lowast
21gt 0 and ℎ
2(119911lowast
21) le 0 then ℎ
2(119911) has at least one
positive root 1199112119896 and all roots of (38) have negative real
parts for 1205911isin [0 120591
120
) and the positive equilibrium 119864lowast
is locally asymptotically stable for 1205911isin [0 120591
120
)
(iii) If (ii) holds and ℎ1015840
2(1199112119896) = 0 then system (3) undergoes
Hopf bifurcations at the positive equilibrium119864lowastfor 1205911=
120591(119895)
12119896
(119896 = 1 2 3 119895 = 0 1 2 )
where
11989822
= 1199012
2minus 2 (119901
1+ 1199031)
11989821
= (1199011+ 1199031)2
minus 21199012(1199010+ 1199030) minus 1199022
1
11989820
= (1199010+ 1199030)2
minus 1199022
0
ℎ2(119911) = 119911
3+ 119898221199112+ 11989821119911 + 119898
20 119911lowast
21=
minus11989822
+ radicΔ
3
120591(119895)
12119896
=
1
1205962119896
times arccos ((11990211205964
2119896+ [11990121199020minus (1199031+ 1199011) 1199021] 1205962
2119896
minus 1199020(1199030+ 1199010) )
times (1199022
0+ 1199022
11205962
2119896)
minus1
) + 2119895120587
(39)
where 119896 = 1 2 3 119895 = 0 1 2 then plusmn1198941205962119896is a pair of purely
imaginary roots of (38) corresponding to 120591(119895)
12119896
Define
120591120
= 120591(0)
121198960
= min119896=123
120591(0)
12119896
12059610
= 12059611198960
(40)
6 Abstract and Applied Analysis
Case d Consider1205911gt 0 1205912gt 0 1205911
= 1205912
The associated characteristic equation of system (3) is
1205823+11990121205822+1199011120582+1199010+ (1199021120582 + 1199020) 119890minus1205821205911+ (1199031120582+1199030) 119890minus1205821205912=0
(41)
We consider (41) with 1205912= 120591lowast
2in its stable interval [0 120591
210
)Regard 120591
1as a parameter
Let 120582 = 119894120596 (120596 gt 0) be a root of (41) then we have
minus 1198941205963minus 11990121205962+ 1198941199011120596 + 1199010+ (1198941199021120596 + 1199020) (cos120596120591
1minus 119894 sin120596120591
1)
+ (1199030+ 1198941199031120596) (cos120596120591lowast
2minus 119894 sin120596120591
lowast
2) = 0
(42)
Separating the real and imaginary parts we have
1205963minus 1199011120596 minus 1199031120596 cos120596120591lowast
2+ 1199030sin120596120591
lowast
2
= 1199021120596 cos120596120591
1minus 1199020sin120596120591
1
11990121205962minus 1199010minus 1199030cos120596120591lowast
2minus 1199031120596 sin120596120591
lowast
2
= 1199020cos120596120591
1+ 1199021120596 sin120596120591
1
(43)
It follows that
1205966+ 119898331205964+ 119898321205963+ 119898311205962+ 11989830
= 0 (44)
where
11989833
= 1199012
2minus 21199011minus 21199031cos120596120591lowast
2
11989832
= 2 (1199030minus 11990121199031) sin120596120591
lowast
2
11989831
= 1199012
1minus 211990101199012minus 2 (119901
21199030minus 11990111199031) cos120596120591lowast
2+ 1199032
1minus 1199022
1
11989830
= 1199012
0+ 211990101199030cos120596120591lowast
2+ 1199032
0minus 1199022
0
(45)
Denote 119865(120596) = 1205966+ 119898331205964+ 119898321205963+ 119898311205962+ 11989830 If
11989830
lt 0 then
119865 (0) lt 0 lim120596rarr+infin
119865 (120596) = +infin (46)
We can obtain that (44) has at most six positive roots1205961 1205962 120596
6 For every fixed 120596
119896 119896 = 1 2 6 there exists
a sequence 120591(119895)1119896
| 119895 = 0 1 2 3 such that (43) holdsLet
12059110
= min 120591(119895)
1119896| 119896 = 1 2 6 119895 = 0 1 2 3 (47)
When 1205911
= 120591(119895)
1119896 (41) has a pair of purely imaginary roots
plusmn119894120596(119895)
1119896for 120591lowast2isin [0 120591
210
)In the following we assume that
(H6) ((119889Re(120582))119889120591
1)|120582=plusmn119894120596
(119895)
1119896
= 0
Thus by the general Hopf bifurcation theorem for FDEsin Hale [22] we have the following result on the stability andHopf bifurcation in system (3)
Theorem 8 For 1205911
gt 0 1205912
gt 0 1205911
= 1205912 suppose that
(1198671)ndash(1198676) is satisfied If 119898
30lt 0 and 120591
lowast
2isin [0 120591
210
] thenthe positive equilibrium 119864
lowastis locally asymptotically stable for
1205911
isin [0 12059110) System (3) undergoes Hopf bifurcations at the
positive equilibrium 119864lowastfor 1205911= 120591(119895)
1119896
3 Direction and Stability ofthe Hopf Bifurcation
In Section 2 we obtain the conditions underwhich system (3)undergoes the Hopf bifurcation at the positive equilibrium119864lowast In this section we consider with 120591
2= 120591lowast
2isin [0 120591
210
)
and regard 1205911as a parameter We will derive the explicit
formulas determining the direction stability and period ofthese periodic solutions bifurcating from equilibrium 119864
lowastat
the critical values 1205911by using the normal form and the center
manifold theory developed by Hassard et al [23] Withoutloss of generality denote any one of these critical values 120591
1=
120591(119895)
1119896(119896 = 1 2 6 119895 = 0 1 2 ) by 120591
1 at which (43) has a
pair of purely imaginary roots plusmn119894120596 and system (3) undergoesHopf bifurcation from 119864
lowast
Throughout this section we always assume that 120591lowast2lt 12059110
Let 1199061= 1199091minus 119909lowast
1 1199062= 1199091minus 119909lowast
2 1199063= 1199092minus 119909lowast
3 119905 = 120591
1119905
and 120583 = 1205911minus 1205911 120583 isin R Then 120583 = 0 is the Hopf bifurcation
value of system (3) System (3) may be written as a functionaldifferential equation inC([minus1 0]R3)
(119905) = 119871120583(119906119905) + 119891 (120583 119906
119905) (48)
where 119906 = (1199061 1199062 1199063)119879isin R3 and
119871120583(120601) = (120591
1+ 120583) 119861[
[
1206011(0)
1206012(0)
1206013(0)
]
]
+ (1205911+ 120583)119862[
[
1206011(minus1)
1206012(minus1)
1206013(minus1)
]
]
+ (1205911+ 120583)119863
[[[[[[[[[[
[
1206011(minus
120591lowast
2
1205911
)
1206012(minus
120591lowast
2
1205911
)
1206013(minus
120591lowast
2
1205911
)
]]]]]]]]]]
]
(49)
119891 (120583 120601) = (1205911+ 120583)[
[
1198911
1198912
1198913
]
]
(50)
Abstract and Applied Analysis 7
where 120601 = (1206011 1206012 1206013)119879isin C([minus1 0]R3) and
119861 = [
[
11988711
11988712
0
0 11988722
11988723
0 0 11988733
]
]
119862 = [
[
0 0 0
11988821
0 0
0 0 0
]
]
119863 = [
[
0 0 0
0 0 0
0 11988932
0
]
]
1198911= minus (119886
11+ 1198973) 1206012
1(0) minus 119897
11206011(0) 1206012(0) minus 119897
21206012
1(0) 1206012(0)
minus 11989751206013
1(0) minus 119897
41206013
1(0) 1206012(0) + sdot sdot sdot
1198912= minus11989761206012(0) 1206013(0) minus (119897
7+ 11988622) 1206012
2(0) + 119897
11206011(minus1) 120601
2(0)
+ 11989731206012
1(minus1) minus 119897
81206013
2(0) minus 119897
91206012
2(0) 1206013(0)
+ 11989721206012
1(minus1) 120601
2(0) + 119897
31206013
1(minus1) + 119897
41206013
1(minus1) 120601
2(0)
minus 119897101206013
2(0) 1206013(0) + sdot sdot sdot
1198913= 11989761206012(minus
120591lowast
2
1205911
)1206013(0) + 119897
71206012
2(minus
120591lowast
2
1205911
) minus 119886331206012
3(0)
+ 11989791206012
2(minus
120591lowast
2
1205911
)1206013(0) + 119897
81206013
2(minus
120591lowast
2
1205911
)
+ 119897101206013
2(minus
120591lowast
2
1205911
)1206013(0) + sdot sdot sdot
1199011(119909) =
1198861119909
1 + 1198871119909
1199012(119909) =
1198862119909
1 + 1198872119909
1198971= 1199011015840
1(119909lowast)
1198972=
1
2
11990110158401015840
1(119909lowast) 1198973=
1
2
11990110158401015840
1(119909lowast) 1199101lowast
1198974=
1
3
119901101584010158401015840
1(119909lowast) 1198975=
1
3
119901101584010158401015840
1(119909lowast) 1199101lowast
1198976= 1199011015840
2(1199101lowast) 1198977=
1
2
11990110158401015840
2(1199101lowast) 1199102lowast
1198978=
1
3
119901101584010158401015840
2(1199101lowast) 1199102lowast 1198979=
1
2
11990110158401015840
2(1199101lowast)
11989710
=
1
3
119901101584010158401015840
2(1199101lowast)
(51)
Obviously 119871120583(120601) is a continuous linear function mapping
C([minus1 0]R3) intoR3 By the Riesz representation theoremthere exists a 3 times 3 matrix function 120578(120579 120583) (minus1 ⩽ 120579 ⩽ 0)whose elements are of bounded variation such that
119871120583120601 = int
0
minus1
119889120578 (120579 120583) 120601 (120579) for 120601 isin C ([minus1 0] R3) (52)
In fact we can choose
119889120578 (120579 120583) = (1205911+ 120583) [119861120575 (120579) + 119862120575 (120579 + 1) + 119863120575(120579 +
120591lowast
2
1205911
)]
(53)
where 120575 is Dirac-delta function For 120601 isin C([minus1 0]R3)define
119860 (120583) 120601 =
119889120601 (120579)
119889120579
120579 isin [minus1 0)
int
0
minus1
119889120578 (119904 120583) 120601 (119904) 120579 = 0
119877 (120583) 120601 =
0 120579 isin [minus1 0)
119891 (120583 120601) 120579 = 0
(54)
Then when 120579 = 0 the system is equivalent to
119905= 119860 (120583) 119909
119905+ 119877 (120583) 119909
119905 (55)
where 119909119905(120579) = 119909(119905+120579) 120579 isin [minus1 0] For120595 isin C1([0 1] (R3)
lowast)
define
119860lowast120595 (119904) =
minus
119889120595 (119904)
119889119904
119904 isin (0 1]
int
0
minus1
119889120578119879(119905 0) 120595 (minus119905) 119904 = 0
(56)
and a bilinear inner product
⟨120595 (119904) 120601 (120579)⟩=120595 (0) 120601 (0)minusint
0
minus1
int
120579
120585=0
120595 (120585minus120579) 119889120578 (120579) 120601 (120585) 119889120585
(57)
where 120578(120579) = 120578(120579 0) Let 119860 = 119860(0) then 119860 and 119860lowast
are adjoint operators By the discussion in Section 2 weknow that plusmn119894120596120591
1are eigenvalues of 119860 Thus they are also
eigenvalues of 119860lowast We first need to compute the eigenvectorof 119860 and 119860
lowast corresponding to 1198941205961205911and minus119894120596120591
1 respectively
Suppose that 119902(120579) = (1 120572 120573)1198791198901198941205791205961205911 is the eigenvector of 119860
corresponding to 1198941205961205911 Then 119860119902(120579) = 119894120596120591
1119902(120579) From the
definition of 119860119871120583(120601) and 120578(120579 120583) we can easily obtain 119902(120579) =
(1 120572 120573)1198791198901198941205791205961205911 where
120572 =
119894120596 minus 11988711
11988712
120573 =
11988932(119894120596 minus 119887
11)
11988712(119894120596 minus 119887
33) 119890119894120596120591lowast
2
(58)
and 119902(0) = (1 120572 120573)119879 Similarly let 119902lowast(119904) = 119863(1 120572
lowast 120573lowast)1198901198941199041205961205911
be the eigenvector of 119860lowast corresponding to minus119894120596120591
1 By the
definition of 119860lowast we can compute
120572lowast=
minus119894120596 minus 11988711
119888211198901198941205961205911
120573lowast=
11988723(minus119894120596 minus 119887
11)
11988821(119894120596 minus 119887
33) 1198901198941205961205911
(59)
From (57) we have⟨119902lowast(119904) 119902 (120579)⟩
= 119863(1 120572lowast 120573
lowast
) (1 120572 120573)119879
minus int
0
minus1
int
120579
120585=0
119863(1 120572lowast 120573
lowast
) 119890minus1198941205961205911(120585minus120579)
119889120578 (120579)
times (1 120572 120573)119879
1198901198941205961205911120585119889120585
= 1198631 + 120572120572lowast+ 120573120573
lowast
+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573
lowast
120591lowast
2119890minus119894120596120591lowast
2
(60)
8 Abstract and Applied Analysis
Thus we can choose
119863 = 1 + 120572120572lowast+ 120573120573
lowast
+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573
lowast
120591lowast
2119890minus119894120596120591lowast
2
minus1
(61)
such that ⟨119902lowast(119904) 119902(120579)⟩ = 1 ⟨119902lowast(119904) 119902(120579)⟩ = 0
In the remainder of this section we follow the ideas inHassard et al [23] and use the same notations as there tocompute the coordinates describing the center manifold 119862
0
at 120583 = 0 Let 119909119905be the solution of (48) when 120583 = 0 Define
119911 (119905) = ⟨119902lowast 119909119905⟩ 119882 (119905 120579) = 119909
119905(120579) minus 2Re 119911 (119905) 119902 (120579)
(62)
On the center manifold 1198620 we have
119882(119905 120579) = 119882 (119911 (119905) 119911 (119905) 120579)
= 11988220(120579)
1199112
2
+11988211(120579) 119911119911 +119882
02(120579)
1199112
2
+11988230(120579)
1199113
6
+ sdot sdot sdot
(63)
where 119911 and 119911 are local coordinates for center manifold 1198620in
the direction of 119902 and 119902 Note that 119882 is real if 119909119905is real We
consider only real solutions For the solution 119909119905isin 1198620of (48)
since 120583 = 0 we have
= 1198941205961205911119911 + ⟨119902
lowast(120579) 119891 (0119882 (119911 (119905) 119911 (119905) 120579)
+ 2Re 119911 (119905) 119902 (120579))⟩
= 1198941205961205911119911+119902lowast(0) 119891 (0119882 (119911 (119905) 119911 (119905) 0) +2Re 119911 (119905) 119902 (0))
= 1198941205961205911119911 + 119902lowast(0) 1198910(119911 119911) ≜ 119894120596120591
1119911 + 119892 (119911 119911)
(64)
where
119892 (119911 119911) = 119902lowast(0) 1198910(119911 119911) = 119892
20
1199112
2
+ 11989211119911119911
+ 11989202
1199112
2
+ 11989221
1199112119911
2
+ sdot sdot sdot
(65)
By (62) we have 119909119905(120579) = (119909
1119905(120579) 1199092119905(120579) 1199093119905(120579))119879= 119882(119905 120579) +
119911119902(120579) + 119911 119902(120579) and then
1199091119905(0) = 119911 + 119911 +119882
(1)
20(0)
1199112
2
+119882(1)
11(0) 119911119911
+119882(1)
02(0)
1199112
2
+ 119900 (|(119911 119911)|3)
1199092119905(0) = 119911120572 + 119911 120572 +119882
(2)
20(0)
1199112
2
+119882(2)
11(0) 119911119911 +119882
(2)
02(0)
1199112
2
+ 119900 (|(119911 119911)|3)
1199093119905(0) = 119911120573 + 119911120573 +119882
(3)
20(0)
1199112
2
+119882(3)
11(0) 119911119911
+119882(3)
02(0)
1199112
2
+ 119900 (|(119911 119911)|3)
1199091119905(minus1) = 119911119890
minus1198941205961205911+ 1199111198901198941205961205911+119882(1)
20(minus1)
1199112
2
+119882(1)
11(minus1) 119911119911 +119882
(1)
02(minus1)
1199112
2
+ 119900 (|(119911 119911)|3)
1199092119905(minus1) = 119911120572119890
minus1198941205961205911+ 119911 120572119890
1198941205961205911+119882(2)
20(minus1)
1199112
2
+119882(2)
11(minus1) 119911119911 +119882
(2)
02(minus1)
1199112
2
+ 119900 (|(119911 119911)|3)
1199093119905(minus1) = 119911120573119890
minus1198941205961205911+ 119911120573119890
1205961205911+119882(3)
20(minus1)
1199112
2
+119882(3)
11(minus1) 119911119911 +119882
(3)
02(minus1)
1199112
2
+ 119900 (|(119911 119911)|3)
1199091119905(minus
120591lowast
2
12059110
) = 119911119890minus119894120596120591lowast
2+ 119911119890119894120596120591lowast
2+119882(1)
20(minus
120591lowast
2
1205911
)
1199112
2
+119882(1)
11(minus
120591lowast
2
1205911
)119911119911 +119882(1)
02(minus
120591lowast
2
1205911
)
1199112
2
+ 119900 (|(119911 119911)|3)
1199092119905(minus
120591lowast
2
1205911
) = 119911120572119890minus119894120596120591lowast
2+ 119911 120572119890
119894120596120591lowast
2+119882(2)
20(minus
120591lowast
2
1205911
)
1199112
2
+119882(2)
11(minus
120591lowast
2
1205911
)119911119911 +119882(2)
02(minus
120591lowast
2
1205911
)
1199112
2
+ 119900 (|(119911 119911)|3)
1199093119905(minus
120591lowast
2
1205911
) = 119911120573119890minus119894120596120591lowast
2+ 119911120573119890
120596120591lowast
2+119882(3)
20(minus
120591lowast
2
1205911
)
1199112
2
+119882(3)
11(minus
120591lowast
2
1205911
)119911119911 +119882(3)
02(minus
120591lowast
2
1205911
)
1199112
2
+ 119900 (|(119911 119911)|3)
(66)
Abstract and Applied Analysis 9
It follows together with (50) that
119892 (119911 119911)=119902lowast(0) 1198910(119911 119911) =119863120591
1(1 120572lowast 120573
lowast
) (119891(0)
1119891(0)
2119891(0)
3)
119879
= 1198631205911[minus (119886
11+ 1198973) 1206012
1(0) minus 119897
11206011(0) 1206012(0)
minus 11989721206012
1(0) 1206012(0) minus 119897
51206013
1(0) minus 119897
41206013
1(0) 1206012(0)
+ sdot sdot sdot ]
+ 120572lowast[11989761206012(0) 1206013(0) (1198977+ 11988622) 1206012
2(0)
+ 11989711206011(minus1) 120601
2(0)
+ 11989731206012
1(minus1) minus 119897
81206013
2(0) minus 119897
91206012
2(0) 1206013(0)
+ 11989721206012
1(minus1) 120601
2(0) + 119897
31206013
1(minus1) + sdot sdot sdot ]
+ 120573
lowast
[11989761206012(minus
120591lowast
2
1205911
)1206013(0) + 119897
71206012
2(minus
120591lowast
2
1205911
)
minus 119886331206012
3(0) + 119897
81206013
2(minus
120591lowast
2
1205911
) + sdot sdot sdot ]
(67)
Comparing the coefficients with (65) we have
11989220
= 1198631205911[minus2 (119886
11+ 1198973) minus 2120572119897
1]
+ 120572lowast[21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591
1
minus21198976120572120573 minus 2 (119897
7+ 11988622) 1205722]
+ 120573
lowast
[21198976120572120573119890minus119894120596120591lowast
2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591
lowast
2
minus2119886331205732]
11989211
= 1198631205911[minus2 (119886
11+ 1198973) minus 1198971(120572 + 120572)]
+ 120572lowast[1198971(1205721198901198941205961205911+ 120572119890minus1198941205961205911) minus 1198976(120572120573 + 120572120573)
minus2 (1198977+ 11988622) 120572120572 + 2119897
3]
+ 120573
lowast
[1198976(120573120572119890119894120596120591lowast
2+ 120572120573119890
minus119894120596120591lowast
2)
+ 21198977120572120572 minus 119886
33120573120573]
11989202
= 21198631205911[minus2 (119886
11+ 1198973) minus 21198971120572]
+ 120572lowast[minus21198976120572120573 minus 2 (119897
7+ 11988622) 1205722+ 211989711205721198901198941205961205911
+2119897311989021198941205961205911]
+ 120573
lowast
[21198976120572120573119890119894120596120591lowast
2+ 2119897712057221198902119894120596120591lowast
2minus 11988633120573
2
]
11989221
= 1198631205911[minus (119886
11+ 1198973) (2119882
(1)
20(0) + 4119882
(1)
11(0))
minus 1198971(2120572119882
(1)
11(0) + 120572119882
(1)
20(0) + 119882
(2)
20(0)
+ 2119882(2)
11(0))]
+ 120572lowast[minus1198976(2120573119882
(2)
11(0) + 120572119882
(3)
20(0) + 120573119882
(2)
20(0)
+ 2120572119882(3)
11(0)) minus (119897
7+ 11988622)
times (4120572119882(2)
11(0) + 2120572119882
(2)
20(0))
+ 1198971(2120572119882
(1)
11(minus1) + 120572119882
(1)
20(minus1)
+ 119882(2)
20(0) 1198901198941205961205911+ 2119882
(2)
11(0) 119890minus1198941205961205911)
+ 1198973(4119882(1)
11(minus1) 119890
minus1198941205961205911+2119882(1)
20(minus1) 119890
1198941205961205911)]
+ 120573
lowast
[1198976(2120573119882
(2)
11(minus
120591lowast
2
1205911
) + 120573119882(2)
20(minus
120591lowast
2
1205911
)
+ 120572119882(3)
20(0) 119890119894120596120591lowast
2+ 2120572119882
(3)
11(0) 119890minus119894120596120591lowast
2)
+ 1198977(4120572119882
(2)
11(minus
120591lowast
2
1205911
) 119890minus119894120596120591lowast
2
+2120572119882(2)
20(minus
120591lowast
2
1205911
) 119890119894120596120591lowast
2)
minus 11988633(4120573119882
(3)
11(0) + 2120573119882
(3)
20(0))]
(68)
where
11988220(120579) =
11989411989220
1205961205911
119902 (0) 1198901198941205961205911120579+
11989411989202
31205961205911
119902 (0) 119890minus1198941205961205911120579+ 119864111989021198941205961205911120579
11988211(120579) = minus
11989411989211
1205961205911
119902 (0) 1198901198941205961205911120579+
11989411989211
1205961205911
119902 (0) 119890minus1198941205961205911120579+ 1198642
1198641= 2
[[[[[
[
2119894120596 minus 11988711
minus11988712
0
minus11988821119890minus2119894120596120591
12119894120596 minus 119887
22minus11988723
0 minus11988932119890minus2119894120596120591
lowast
22119894120596 minus 119887
33
]]]]]
]
minus1
times
[[[[[
[
minus2 (11988611
+ 1198973) minus 2120572119897
1
21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591
1minus 21198976120572120573 minus 2 (119897
7+ 11988622) 1205722
21198976120572120573119890minus119894120596120591lowast
2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591
lowast
2minus 2119886331205732
]]]]]
]
10 Abstract and Applied Analysis
1198642= 2
[[[[
[
minus11988711
minus11988712
0
minus11988821
minus11988722
minus11988723
0 minus11988932
minus11988733
]]]]
]
minus1
times
[[[[
[
minus2 (11988611
+ 1198973) minus 1198971(120572 + 120572)
1198971(1205721198901198941205961205911+ 120572119890minus1198941205961205911) minus 1198976(120572120573 + 120572120573) minus 2 (119897
7+ 11988622) 120572120572 + 2119897
3
1198976(120573120572119890119894120596120591lowast
2+ 120572120573119890
minus119894120596120591lowast
2) + 21198977120572120572 minus 119886
33120573120573
]]]]
]
(69)
Thus we can determine11988220(120579) and119882
11(120579) Furthermore
we can determine each 119892119894119895by the parameters and delay in (3)
Thus we can compute the following values
1198881(0) =
119894
21205961205911
(1198922011989211
minus 2100381610038161003816100381611989211
1003816100381610038161003816
2
minus
1
3
100381610038161003816100381611989202
1003816100381610038161003816
2
) +
1
2
11989221
1205832= minus
Re 1198881(0)
Re 1205821015840 (1205911)
1198792= minus
Im 1198881(0) + 120583
2Im 120582
1015840(1205911)
1205961205911
1205732= 2Re 119888
1(0)
(70)
which determine the quantities of bifurcating periodic solu-tions in the center manifold at the critical value 120591
1 Suppose
Re1205821015840(1205911) gt 0 120583
2determines the directions of the Hopf
bifurcation if 1205832
gt 0(lt 0) then the Hopf bifurcationis supercritical (subcritical) and the bifurcation exists for120591 gt 120591
1(lt 1205911) 1205732determines the stability of the bifurcation
periodic solutions the bifurcating periodic solutions arestable (unstable) if 120573
2lt 0(gt 0) and119879
2determines the period
of the bifurcating periodic solutions the period increases(decreases) if 119879
2gt 0(lt 0)
4 Numerical Simulation
We consider system (3) by taking the following coefficients1198861
= 03 11988611
= 58889 11988612
= 1 1198981
= 1 1198862
=
01 11988621
= 27 11988622
= 12 11988623
= 12 1198982
= 1 1198863
=
02 11988632
= 25 11988633
= 12 We have the unique positiveequilibrium 119864
lowast= (00451 00357 00551)
By computation we get 11989810
= minus00104 12059611
= 0516411991111
= 02666 ℎ10158401(11991111) = 03402 120591
20
= 32348 FromTheorem 6 we know that when 120591
1= 0 the positive
equilibrium 119864lowastis locally asymptotically stable for 120591
2isin
[0 32348) When 1205912crosses 120591
20
the equilibrium 119864lowastloses its
stability and Hopf bifurcation occurs From the algorithm inSection 3 we have 120583
2= 56646 120573
2= minus31583 119879
2= 5445
whichmeans that the bifurcation is supercritical and periodicsolution is stable The trajectories and the phase graphs areshown in Figures 1 and 2
Regarding 1205911as a parameter and let 120591
2= 29 isin
[0 32348) we can observe that with 1205911increasing the pos-
itive equilibrium 119864lowastloses its stability and Hopf bifurcation
occurs (see Figures 3 and 4)
5 Global Continuationof Local Hopf Bifurcations
In this section we study the global continuation of peri-odic solutions bifurcating from the positive equilibrium(119864lowast 120591(119895)
1119896) (119896 = 1 2 6 119895 = 0 1 ) Throughout this
section we follow closely the notations in [24] and assumethat 120591
2= 120591lowast
2isin [0 120591
210
) regarding 1205911as a parameter For
simplification of notations setting 119911119905(119905) = (119909
1119905 1199092119905 1199093119905)119879 we
may rewrite system (3) as the following functional differentialequation
(119905) = 119865 (119911119905 1205911 119901) (71)
where 119911119905(120579) = (119909
1119905(120579) 1199092119905(120579) 1199093119905(120579))119879
= (1199091(119905 + 120579) 119909
2(119905 +
120579) 1199093(119905 + 120579))
119879 for 119905 ge 0 and 120579 isin [minus1205911 0] Since 119909
1(119905) 1199092(119905)
and 1199093(119905) denote the densities of the prey the predator and
the top predator respectively the positive solution of system(3) is of interest and its periodic solutions only arise in the firstquadrant Thus we consider system (3) only in the domain1198773
+= (1199091 1199092 1199093) isin 1198773 1199091gt 0 119909
2gt 0 119909
3gt 0 It is obvious
that (71) has a unique positive equilibrium 119864lowast(119909lowast
1 119909lowast
2 119909lowast
3) in
1198773
+under the assumption (119867
1)ndash(1198674) Following the work of
[24] we need to define
119883 = 119862([minus1205911 0] 119877
3
+)
Γ = 119862119897 (119911 1205911 119901) isin X times R times R+
119911 is a 119901-periodic solution of system (71)
N = (119911 1205911 119901) 119865 (119911 120591
1 119901) = 0
(72)
Let ℓ(119864lowast120591(119895)
11198962120587120596
(119895)
1119896)denote the connected component pass-
ing through (119864lowast 120591(119895)
1119896 2120587120596
(119895)
1119896) in Γ where 120591
(119895)
1119896is defined by
(43) We know that ℓ(119864lowast120591(119895)
11198962120587120596
(119895)
1119896)through (119864
lowast 120591(119895)
1119896 2120587120596
(119895)
1119896)
is nonemptyFor the benefit of readers we first state the global Hopf
bifurcation theory due to Wu [24] for functional differentialequations
Lemma 9 Assume that (119911lowast 120591 119901) is an isolated center satis-
fying the hypotheses (A1)ndash(A4) in [24] Denote by ℓ(119911lowast120591119901)
theconnected component of (119911
lowast 120591 119901) in Γ Then either
(i) ℓ(119911lowast120591119901)
is unbounded or(ii) ℓ(119911lowast120591119901)
is bounded ℓ(119911lowast120591119901)
cap Γ is finite and
sum
(119911120591119901)isinℓ(119911lowast120591119901)capN
120574119898(119911lowast 120591 119901) = 0 (73)
for all 119898 = 1 2 where 120574119898(119911lowast 120591 119901) is the 119898119905ℎ crossing
number of (119911lowast 120591 119901) if119898 isin 119869(119911
lowast 120591 119901) or it is zero if otherwise
Clearly if (ii) in Lemma 9 is not true then ℓ(119911lowast120591119901)
isunbounded Thus if the projections of ℓ
(119911lowast120591119901)
onto 119911-spaceand onto 119901-space are bounded then the projection of ℓ
(119911lowast120591119901)
onto 120591-space is unbounded Further if we can show that the
Abstract and Applied Analysis 11
0 200 400 60000451
00451
00452
t
x1(t)
t
0 200 400 60000356
00357
00358
x2(t)
t
0 200 400 6000055
00551
00552
x3(t)
00451
00451
00452
00356
00357
00357
003570055
0055
00551
00551
00552
x 1(t)
x2 (t)
x3(t)
Figure 1 The trajectories and the phase graph with 1205911= 0 120591
2= 29 lt 120591
20= 32348 119864
lowastis locally asymptotically stable
0 500 1000004
0045
005
0 500 1000002
004
006
0 500 10000
005
01
t
t
t
x1(t)
x2(t)
x3(t)
0044
0046
0048
002
003
004
005003
004
005
006
007
008
x 1(t)
x2 (t)
x3(t)
Figure 2 The trajectories and the phase graph with 1205911= 0 120591
2= 33 gt 120591
20= 32348 a periodic orbit bifurcate from 119864
lowast
12 Abstract and Applied Analysis
0 500 1000 1500
00451
00452
0 500 1000 15000035
0036
0037
0 500 1000 15000054
0055
0056
t
t
t
x1(t)
x2(t)
x3(t)
0045100451
0045200452
0035
00355
003600546
00548
0055
00552
00554
00556
00558
x1(t)
x2 (t)
x3(t)
Figure 3 The trajectories and the phase graph with 1205911= 09 120591
2= 29 119864
lowastis locally asymptotically stable
0 500 1000 15000044
0045
0046
0 500 1000 1500002
003
004
0 500 1000 1500004
006
008
t
t
t
x1(t)
x2(t)
x3(t)
00445
0045
00455
0032
0034
0036
00380052
0054
0056
0058
006
0062
0064
x 1(t)
x2 (t)
x3(t)
Figure 4 The trajectories and the phase graph with 1205911= 12 120591
2= 29 a periodic orbit bifurcate from 119864
lowast
Abstract and Applied Analysis 13
projection of ℓ(119911lowast120591119901)
onto 120591-space is away from zero thenthe projection of ℓ
(119911lowast120591119901)
onto 120591-space must include interval[120591 infin) Following this ideal we can prove our results on theglobal continuation of local Hopf bifurcation
Lemma 10 If the conditions (1198671)ndash(1198674) hold then all nontriv-
ial periodic solutions of system (71) with initial conditions
1199091(120579) = 120593 (120579) ge 0 119909
2(120579) = 120595 (120579) ge 0
1199093(120579) = 120601 (120579) ge 0 120579 isin [minus120591
1 0)
120593 (0) gt 0 120595 (0) gt 0 120601 (0) gt 0
(74)
are uniformly bounded
Proof Suppose that (1199091(119905) 1199092(119905) 1199093(119905)) are nonconstant peri-
odic solutions of system (3) and define
1199091(1205851) = min 119909
1(119905) 119909
1(1205781) = max 119909
1(119905)
1199092(1205852) = min 119909
2(119905) 119909
2(1205782) = max 119909
2(119905)
1199093(1205853) = min 119909
3(119905) 119909
3(1205783) = max 119909
3(119905)
(75)
It follows from system (3) that
1199091(119905) = 119909
1(0) expint
119905
0
[1198861minus 119886111199091(119904) minus
119886121199092(119904)
1198981+ 1199091(119904)
] 119889119904
1199092(119905) = 119909
2(0) expint
119905
0
[minus1198862+
119886211199091(119904 minus 1205911)
1198981+ 1199091(119904 minus 1205911)
minus 119886221199092(119904) minus
119886231199093(119904)
1198982+ 1199092(119904)
] 119889119904
1199093(119905) = 119909
3(0) expint
119905
0
[minus1198863+
119886321199092(119904 minus 120591
lowast
2)
1198982+ 1199092(119904 minus 120591
lowast
2)
minus 119886331199093(119904)] 119889119904
(76)
which implies that the solutions of system (3) cannot crossthe 119909119894-axis (119894 = 1 2 3) Thus the nonconstant periodic orbits
must be located in the interior of first quadrant It followsfrom initial data of system (3) that 119909
1(119905) gt 0 119909
2(119905) gt
0 1199093(119905) gt 0 for 119905 ge 0
From the first equation of system (3) we can get
0 = 1198861minus 119886111199091(1205781) minus
119886121199092(1205781)
1198981+ 1199091(1205781)
le 1198861minus 119886111199091(1205781) (77)
thus we have
1199091(1205781) le
1198861
11988611
(78)
From the second equation of (3) we obtain
0 = minus 1198862+
119886211199091(1205782minus 1205911)
1198981+ 1199091(1205782minus 1205911)
minus 119886221199092(1205782)
minus
119886231199093(1205782)
1198982+ 1199092(1205782)
le minus1198862+
11988621(119886111988611)
1198981+ (119886111988611)
minus 119886221199092(1205782)
(79)
therefore one can get
1199092(1205782) le
minus1198862(119886111198981+ 1198861) + 119886111988621
11988622(119886111198981+ 1198861)
≜ 1198721 (80)
Applying the third equation of system (3) we know
0 = minus1198863+
119886321199092(1205783minus 120591lowast
2)
1198982+ 1199092(1205783minus 120591lowast
2)
minus 119886331199093(1205783)
le minus1198863+
119886321198721
1198982+1198721
minus 119886331199093(1205783)
(81)
It follows that
1199093(1205783) le
minus1198863(1198982+1198721) + 119886321198721
11988633(1198982+1198721)
≜ 1198722 (82)
This shows that the nontrivial periodic solution of system (3)is uniformly bounded and the proof is complete
Lemma 11 If the conditions (1198671)ndash(1198674) and
(H7) 11988622
minus (119886211198982) gt 0 [(119886
11minus 11988611198981)11988622
minus
(11988612119886211198982
1)]11988633
minus (11988611
minus (11988611198981))[(119886211198982)11988633
+
(11988623119886321198982
2)] gt 0
hold then system (3) has no nontrivial 1205911-periodic solution
Proof Suppose for a contradiction that system (3) has non-trivial periodic solution with period 120591
1 Then the following
system (83) of ordinary differential equations has nontrivialperiodic solution
1198891199091
119889119905
= 1199091(119905) [1198861minus 119886111199091(119905) minus
119886121199092(119905)
1198981+ 1199091(119905)
]
1198891199092
119889119905
= 1199092(119905) [minus119886
2+
119886211199091(119905)
1198981+ 1199091(119905)
minus 119886221199092(119905) minus
119886231199093(119905)
1198982+ 1199092(119905)
]
1198891199093
119889119905
= 1199093(119905) [minus119886
3+
119886321199092(119905 minus 120591
lowast
2)
1198982+ 1199092(119905 minus 120591
lowast
2)
minus 119886331199093(119905)]
(83)
which has the same equilibria to system (3) that is
1198641= (0 0 0) 119864
2= (
1198861
11988611
0 0)
1198643= (1199091 1199092 0) 119864
lowast= (119909lowast
1 119909lowast
2 119909lowast
3)
(84)
Note that 119909119894-axis (119894 = 1 2 3) are the invariable manifold of
system (83) and the orbits of system (83) do not intersect each
14 Abstract and Applied Analysis
other Thus there are no solutions crossing the coordinateaxes On the other hand note the fact that if system (83) hasa periodic solution then there must be the equilibrium in itsinterior and that 119864
1 1198642 1198643are located on the coordinate
axis Thus we conclude that the periodic orbit of system (83)must lie in the first quadrant If (119867
7) holds it is well known
that the positive equilibrium 119864lowastis globally asymptotically
stable in the first quadrant (see [1])Thus there is no periodicorbit in the first quadrant too The above discussion meansthat (83) does not have any nontrivial periodic solution It isa contradiction Therefore the lemma is confirmed
Theorem 12 Suppose the conditions of Theorem 8 and (1198677)
hold let120596119896and 120591(119895)1119896
be defined in Section 2 then when 1205911gt 120591(119895)
1119896
system (3) has at least 119895 minus 1 periodic solutions
Proof It is sufficient to prove that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is [120591
1 +infin) for each 119895 ge 1 where
1205911le 120591(119895)
1119896
In following we prove that the hypotheses (A1)ndash(A4) in[24] hold
(1) From system (3) we know easily that the followingconditions hold
(A1) 119865 isin 1198622(1198773
+times119877+times119877+) where 119865 = 119865|
1198773
+times119877+times119877+
rarr
1198773
+
(A3) 119865(120601 1205911 119901) is differential with respect to 120601
(2) It follows from system (3) that
119863119911119865 (119911 120591
1 119901) =
[[[[[[[[[[[
[
1198861minus 2119886111199091minus
1198861211989811199092
(1198981+ 1199091)2
minus
119886121199091
1198981+ 1199091
0
1198862111989811199092
(1198981+ 1199091)2
minus1198862+
119886211199091
1198981+ 1199091
minus 2119886221199092minus
1198862311989821199093
(1198982+ 1199092)2
minus
119886231199092
1198982+ 1199092
0
1198863211989821199093
(1198982+ 1199092)2
minus1198863+
119886321199092
1198982+ 1199092
minus 2119886331199093
]]]]]]]]]]]
]
(85)
Then under the assumption (1198671)ndash(1198674) we have
det119863119911119865 (119911lowast 1205911 119901)
= det
[[[[[[[[
[
minus11988611119909lowast
1+
11988612119909lowast
1119909lowast
2
(1198981+ 119909lowast
1)2
11988612119909lowast
1
1198981+ 119909lowast
1
0
119886211198981119909lowast
2
(1198981+ 119909lowast
1)2
minus11988622119909lowast
2+
11988623119909lowast
2119909lowast
3
(1198982+ 119909lowast
2)2
minus
11988623119909lowast
2
1198982+ 119909lowast
2
0
119886321198982119909lowast
3
(1198982+ 119909lowast
2)2
minus11988633119909lowast
3
]]]]]]]]
]
= minus
1198862
21199101lowast1199102lowast
(1 + 11988721199101lowast)3[minus119909lowast+
11988611198871119909lowast1199101lowast
(1 + 1198871119909lowast)2] = 0
(86)
From (86) we know that the hypothesis (A2) in [24]is satisfied
(3) The characteristic matrix of (71) at a stationarysolution (119911 120591
0 1199010) where 119911 = (119911
(1) 119911(2) 119911(3)) isin 119877
3
takes the following form
Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863
120601119865 (119911 120591
1 119901) (119890
120582119868) (87)
that is
Δ (119911 1205911 119901) (120582)
=
[[[[[[[[[[[[[[[[[
[
120582 minus 1198861+ 211988611119911(1)
+
119886121198981119911(2)
(1198981+ 119911(1))
2
11988612119911(1)
1198981+ 119911(1)
0
minus
119886211198981119911(2)
(1198981+ 119911(1))
2119890minus1205821205911
120582 + 1198862minus
11988621119911(1)
1198981+ 119911(1)
+ 211988622119911(2)
+
119886231198982119911(3)
(1198982+ 119911(2))
2
11988623119911(2)
1198982+ 119911(2)
0 minus
119886321198982119911(3)
(1198982+ 119911(2))
2119890minus120582120591lowast
2120582 + 1198863minus
11988632119911(2)
1198982+ 119911(2)
+ 211988633119911(3)
]]]]]]]]]]]]]]]]]
]
(88)
Abstract and Applied Analysis 15
From (88) we have
det (Δ (119864lowast 1205911 119901) (120582))
= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902
0+ 1199030)] 119890minus1205821205911
(89)
Note that (89) is the same as (20) from the discussion inSection 2 about the local Hopf bifurcation it is easy to verifythat (119864
lowast 120591(119895)
1119896 2120587120596
119896) is an isolated center and there exist 120598 gt
0 120575 gt 0 and a smooth curve 120582 (120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575) rarr C
such that det(Δ(120582(1205911))) = 0 |120582(120591
1) minus 120596119896| lt 120598 for all 120591
1isin
[120591(119895)
1119896minus 120575 120591(119895)
1119896+ 120575] and
120582 (120591(119895)
1119896) = 120596119896119894
119889Re 120582 (1205911)
1198891205911
1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)
1119896
gt 0 (90)
Let
Ω1205982120587120596
119896
= (120578 119901) 0 lt 120578 lt 120598
10038161003816100381610038161003816100381610038161003816
119901 minus
2120587
120596119896
10038161003816100381610038161003816100381610038161003816
lt 120598 (91)
It is easy to see that on [120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575] times 120597Ω
1205982120587120596119896
det(Δ(119864
lowast 1205911 119901)(120578 + (2120587119901)119894)) = 0 if and only if 120578 = 0
1205911= 120591(119895)
1119896 119901 = 2120587120596
119896 119896 = 1 2 3 119895 = 0 1 2
Therefore the hypothesis (A4) in [24] is satisfiedIf we define
119867plusmn(119864lowast 120591(119895)
1119896
2120587
120596119896
) (120578 119901)
= det(Δ (119864lowast 120591(119895)
1119896plusmn 120575 119901) (120578 +
2120587
119901
119894))
(92)
then we have the crossing number of isolated center(119864lowast 120591(119895)
1119896 2120587120596
119896) as follows
120574(119864lowast 120591(119895)
1119896
2120587
120596119896
) = deg119861(119867minus(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
minus deg119861(119867+(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
= minus1
(93)
Thus we have
sum
(1199111205911119901)isinC
(119864lowast120591(119895)
11198962120587120596119896)
120574 (119911 1205911 119901) lt 0
(94)
where (119911 1205911 119901) has all or parts of the form
(119864lowast 120591(119896)
1119895 2120587120596
119896) (119895 = 0 1 ) It follows from
Lemma 9 that the connected component ℓ(119864lowast120591(119895)
11198962120587120596
119896)
through (119864lowast 120591(119895)
1119896 2120587120596
119896) is unbounded for each center
(119911lowast 1205911 119901) (119895 = 0 1 ) From the discussion in Section 2
we have
120591(119895)
1119896=
1
120596119896
times arccos(((11990121205962
119896minus1199010) (1199030+1199020)
+ 1205962(1205962minus1199011) (1199031+1199021))
times ((1199030+1199020)2
+ (1199031+1199021)2
1205962
119896)
minus1
) +2119895120587
(95)
where 119896 = 1 2 3 119895 = 0 1 Thus one can get 2120587120596119896le 120591(119895)
1119896
for 119895 ge 1Now we prove that the projection of ℓ
(119864lowast120591(119895)
11198962120587120596
119896)onto
1205911-space is [120591
1 +infin) where 120591
1le 120591(119895)
1119896 Clearly it follows
from the proof of Lemma 11 that system (3) with 1205911
= 0
has no nontrivial periodic solution Hence the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is away from zero
For a contradiction we suppose that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is bounded this means that the
projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is included in a
interval (0 120591lowast) Noticing 2120587120596119896lt 120591119895
1119896and applying Lemma 11
we have 119901 lt 120591lowast for (119911(119905) 120591
1 119901) belonging to ℓ
(119864lowast120591(119895)
11198962120587120596
119896)
This implies that the projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 119901-
space is bounded Then applying Lemma 10 we get thatthe connected component ℓ
(119864lowast120591(119895)
11198962120587120596
119896)is bounded This
contradiction completes the proof
6 Conclusion
In this paper we take our attention to the stability and Hopfbifurcation analysis of a predator-prey systemwithMichaelis-Menten type functional response and two unequal delaysWe obtained some conditions for local stability and Hopfbifurcation occurring When 120591
1= 1205912 we derived the explicit
formulas to determine the properties of periodic solutionsby the normal form method and center manifold theoremSpecially the global existence results of periodic solutionsbifurcating from Hopf bifurcations are also established byusing a global Hopf bifurcation result due to Wu [24]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to thank the anonymous referees fortheir careful reading and constructive suggestions which leadto truly significant improvement of the paper This researchis supported by the National Natural Science Foundation ofChina (no 11061016)
16 Abstract and Applied Analysis
References
[1] R Xu and M A J Chaplain ldquoPersistence and global stabilityin a delayed predator-prey system with Michaelis-Menten typefunctional responserdquo Applied Mathematics and Computationvol 130 no 2-3 pp 441ndash455 2002
[2] Y L Song Y H Peng and J JWei ldquoBifurcations for a predator-prey system with two delaysrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 466ndash479 2008
[3] G-P Hu W-T Li and X-P Yan ldquoHopf bifurcations in apredator-prey system with multiple delaysrdquo Chaos Solitons ampFractals vol 42 no 2 pp 1273ndash1285 2009
[4] M Liao X Tang and C Xu ldquoBifurcation analysis for a three-species predator-prey systemwith two delaysrdquoCommunicationsin Nonlinear Science and Numerical Simulation vol 17 no 1 pp183ndash194 2012
[5] C Xu and P Li ldquoDynamical analysis in a delayed predator-prey model with two delaysrdquo Discrete Dynamics in Nature andSociety vol 2012 Article ID 652947 22 pages 2012
[6] J-F Zhang ldquoStability and bifurcation periodic solutions ina Lotka-Volterra competition system with multiple delaysrdquoNonlinear Dynamics vol 70 no 1 pp 849ndash860 2012
[7] J Xia Z Yu and R Yuan ldquoStability and Hopf bifurcation ina symmetric Lotka-Volterra predator-prey system with delaysrdquoElectronic Journal of Differential Equations vol 2013 no 9 pp1ndash16 2013
[8] H Wan and J Cui ldquoA malaria model with two delaysrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 601265 8pages 2013
[9] M Zhao ldquoHopf bifurcation analysis for a semiratio-dependentpredator-prey system with two delaysrdquo Abstract and AppliedAnalysis vol 2013 Article ID 495072 13 pages 2013
[10] S Feyissa and S Banerjee ldquoDelay-induced oscillatory dynamicsin humoral mediated immune response with two time delaysrdquoNonlinear Analysis Real World Applications vol 14 no 1 pp35ndash52 2013
[11] G Zhang Y Shen andB Chen ldquoHopf bifurcation of a predator-prey systemwith predator harvesting and twodelaysrdquoNonlinearDynamics vol 73 no 4 pp 2119ndash2131 2013
[12] X-Y Meng H-F Huo and H Xiang ldquoHopf bifurcation in athree-species systemwith delaysrdquo Journal of AppliedMathemat-ics and Computing vol 35 no 1-2 pp 635ndash661 2011
[13] J-F Zhang ldquoGlobal existence of bifurcated periodic solutions ina commensalism model with delaysrdquo Applied Mathematics andComputation vol 218 no 23 pp 11688ndash11699 2012
[14] Y Wang W Jiang and H Wang ldquoStability and global Hopfbifurcation in toxic phytoplankton-zooplankton model withdelay and selective harvestingrdquoNonlinear Dynamics vol 73 no1-2 pp 881ndash896 2013
[15] Y Ma ldquoGlobal Hopf bifurcation in the Leslie-Gower predator-prey model with two delaysrdquo Nonlinear Analysis Real WorldApplications vol 13 no 1 pp 370ndash375 2012
[16] X-Y Meng H-F Huo and X-B Zhang ldquoStability and globalHopf bifurcation in a delayed food web consisting of a preyand two predatorsrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4335ndash4348 2011
[17] Y Song J Wei andM Han ldquoLocal and global Hopf bifurcationin a delayed hematopoiesis modelrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol14 no 11 pp 3909ndash3919 2004
[18] Y Song and J Wei ldquoLocal Hopf bifurcation and global periodicsolutions in a delayed predator-prey systemrdquo Journal of Mathe-matical Analysis and Applications vol 301 no 1 pp 1ndash21 2005
[19] J Wei and M Y Li ldquoGlobal existence of periodic solutions in atri-neuron network model with delaysrdquo Physica D vol 198 no1-2 pp 106ndash119 2004
[20] T Zhao Y Kuang and H L Smith ldquoGlobal existence of peri-odic solutions in a class of delayed Gause-type predator-preysystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 28 no 8 pp 1373ndash1394 1997
[21] S J Gao L S Chen and Z D Teng ldquoHopf bifurcation andglobal stability for a delayed predator-prey system with stagestructure for predatorrdquo Applied Mathematics and Computationvol 202 no 2 pp 721ndash729 2008
[22] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977
[23] B Hassard D Kazarinoff and Y WanTheory and Applicationsof Hopf Bifurcation Cambridge University Press CambridgeUK 1981
[24] J H Wu ldquoSymmetric functional-differential equations andneural networks with memoryrdquo Transactions of the AmericanMathematical Society vol 350 no 12 pp 4799ndash4838 1998
[25] V Hutson ldquoThe existence of an equilibrium for permanentsystemsrdquo The Rocky Mountain Journal of Mathematics vol 20no 4 pp 1033ndash1040 1990
[26] S G Ruan and J J Wei ldquoOn the zeros of transcendentalfunctions with applications to stability of delay differentialequations with two delaysrdquo Dynamics of Continuous Discreteamp Impulsive Systems A vol 10 no 6 pp 863ndash874 2003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Abstract and Applied Analysis
We want to determine if the real part of some rootincreases to reach zero and eventually becomes positive as 120591varies Let 120582 = 119894120596 (120596 gt 0) be a root of (21) then we have
minus 1198941205963minus 11990121205962+ 119894 (1199011+ 1199021) 120596 + (119901
0+ 1199020)
+ (1199031120596119894 + 119903
0) (cos120596120591
2minus 119894 sin120596120591
2) = 0
(22)
Separating the real and imaginary parts we have
minus1205963+ (1199011+ 1199021) 120596 = 119903
0sin120596120591
2minus 1199031120596 cos120596120591
2
minus11990121205962+ (1199010+ 1199020) = minus119903
1120596 sin120596120591
2minus 1199030cos120596120591
2
(23)
It follows that
1205966+ 119898121205964+ 119898111205962+ 11989810
= 0 (24)
where 11989812
= 1199012
2minus 2(119901
1+ 1199021) 11989811
= (1199011+ 1199021)2minus 21199012(1199010+
1199020) minus 1199032
1 11989810
= (1199010+ 1199020)2minus 1199032
0
Denoting 119911 = 1205962 (24) becomes
1199113+ 119898121199112+ 11989811119911 + 119898
10= 0 (25)
Let
ℎ1(119911) = 119911
3+ 119898121199112+ 11989811119911 + 119898
10 (26)
we have
119889ℎ1(119911)
119889119911
= 31199112+ 211989812119911 + 119898
11 (27)
If 11989810
= (1199010+ 1199020)2minus 1199032
0lt 0 then ℎ
1(0) lt
0 lim119911rarr+infin
ℎ1(119911) = +infin We can know that (25) has at least
one positive rootIf 11989810
= (1199010+ 1199020)2minus 1199032
0ge 0 we obtain that when Δ =
1198982
12minus 311989811
le 0 (25) has no positive roots for 119911 isin [0 +infin)On the other hand when Δ = 119898
2
12minus 311989811
gt 0 the followingequation
31199112+ 211989812119911 + 119898
11= 0 (28)
has two real roots 119911lowast11
= (minus11989812
+ radicΔ)3 119911lowast
12= (minus119898
12minus
radicΔ)3 Because of ℎ101584010158401(119911lowast
11) = 2radicΔ gt 0 ℎ
10158401015840
1(119911lowast
12) = minus2radicΔ lt
0 119911lowast
11and 119911lowast12are the local minimum and the localmaximum
of ℎ1(119911) respectively By the above analysis we immediately
obtain the following
Lemma 4 (1) If 11989810
ge 0 and Δ = 1198982
12minus 311989811
le 0 (25) hasno positive root for 119911 isin [0 +infin)
(2) If 11989810
ge 0 and Δ = 1198982
12minus 311989811
gt 0 (25) has at leastone positive root if and only if 119911lowast
11= (minus119898
12+ radicΔ)3 gt 0 and
ℎ1(119911lowast
11) le 0
(3) If 11989810
lt 0 (25) has at least one positive root
Without loss of generality we assume that (25) has threepositive roots defined by 119911
11 11991112 11991113 respectively Then
(24) has three positive roots
12059611
= radic11991111 120596
12= radic11991112 120596
13= radic11991113 (29)
From (23) we have
cos120596111989612059121119896
=
11990311205964
1119896+ [11990121199030minus (1199021+ 1199011) 1199031] 1205962
1119896minus 1199030(1199020+ 1199010)
1199032
0+ 1199032
11205962
1119896
(30)
Thus if we denote
120591(119895)
21119896
=
1
1205961119896
times arccos ((11990311205964
1119896+ [11990121199030minus (1199021+ 1199011) 1199031] 1205962
1119896
minus 1199030(1199020+ 1199010))
times (1199032
0+ 1199032
11205962
1119896)
minus1
) + 2119895120587
(31)
where 119896 = 1 2 3 119895 = 0 1 2 then plusmn1198941205961119896is a pair of purely
imaginary roots of (21) corresponding to 120591(119895)
21119896
Define
120591210
= 120591(0)
211198960
= min119896=123
120591(0)
21119896
12059610
= 12059611198960
(32)
Let 120582(1205912) = 120572(120591
2)+119894120596(120591
2) be the root of (21) near 120591
2= 120591(119895)
21119896
satisfying
120572 (120591(119895)
21119896
) = 0 120596 (120591(119895)
21119896
) = 1205961119896 (33)
Substituting 120582(1205912) into (21) and taking the derivative with
respect to 1205912 we have
31205822+ 21199012120582 + (119901
1+ 1199021) + 1199031119890minus1205821205912minus 1205912(1199031120582 + 1199030) 119890minus1205821205912
119889120582
1198891205912
= 120582 (1199031120582 + 1199030) 119890minus1205821205912
(34)
Therefore
[
119889120582
1198891205912
]
minus1
=
[31205822+ 21199012120582 + (119901
1+ 1199021)] 1198901205821205912
120582 (1199031120582 + 1199030)
+
1199031
120582 (1199031120582 + 1199030)
minus
1205912
120582
(35)
When 1205912
= 120591(119895)
21119896
120582(120591(119895)
21119896
) = 1198941205961119896
(119896 = 1 2 3) 120582(1199031120582 +
1199030)|1205912=120591(119895)
21119896
= minus11990311205962
1119896+ 11989411990301205961119896 [31205822
+ 21199012120582 + (119901
1+
1199021)]1198901205821205912|1205912=120591(119895)
21119896
= [minus31205962
1119896+ (1199011+ 1199021)] cos(120596
1119896120591(119895)
21119896
) minus
211990121205961119896sin(1205961119896120591(119895)
21119896
) + 119894211990121205961119896cos(120596
1119896120591(119895)
21119896
) + [minus31205962
1119896+ (1199011+
1199021)] sin(120596
1119896120591(119895)
21119896
)
Abstract and Applied Analysis 5
According to (35) we have
[
Re 119889 (120582 (1205912))
1198891205912
]
minus1
1205912=120591(119895)
21119896
= Re[[31205822+ 21199012120582 + (119901
1+ 1199021)] 1198901205821205912
120582 (1199031120582 + 1199030)
]
1205912=120591(119895)
21119896
+ Re[ 1199031
120582 (1199031120582 + 1199030)
]
1205912=120591(119895)
21119896
=
1
Λ1
minus 11990311205962
1119896[minus31205962
1119896+ (1199011+ 1199021)] cos (120596
1119896120591(119895)
21119896
)
+ 2119903111990121205963
1119896sin (120596
1119896120591(119895)
21119896
) minus 1199032
11205962
1119896
+ 2119903011990121205962
1119896cos (120596
1119896120591(119895)
21119896
)
+ 1199030[minus31205962
1119896+ (1199011+ 1199021)] 1205961119896sin (120596
1119896120591(119895)
21119896
)
=
1
Λ1
31205966
1119896+ 2 [119901
2
2minus 2 (119901
1+ 1199021)] 1205964
1119896
+ [(1199011+ 1199021)2
minus 21199012(1199010+ 1199020) minus 1199032
1] 1205962
1119896
=
1
Λ1
1199111119896(31199112
1119896+ 2119898121199111119896
+ 11989811)
=
1
Λ1
1199111119896ℎ1015840
1(1199111119896)
(36)
where Λ1= 1199032
11205964
1119896+ 1199032
01205962
1119896gt 0 Notice that Λ
1gt 0 1199111119896
gt 0
sign
[
Re 119889 (120582 (1205912))
1198891205912
]
1205912=120591(119895)
21119896
= sign
[
Re 119889 (120582 (1205912))
1198891205912
]
minus1
1205912=120591(119895)
21119896
(37)
then we have the following lemma
Lemma 5 Suppose that 1199111119896
= 1205962
1119896and ℎ
1015840
1(1199111119896) = 0 where
ℎ1(119911) is defined by (26) then 119889(Re 120582(120591(119895)
21119896
))1198891205912has the same
sign with ℎ1015840
1(1199111119896)
From Lemmas 1 4 and 5 and Theorem 3 we can easilyobtain the following theorem
Theorem6 For 1205911= 0 120591
2gt 0 suppose that (119867
1)ndash(1198675) hold
(i) If 11989810
ge 0 and Δ = 1198982
12minus 311989811
le 0 then all rootsof (10) have negative real parts for all 120591
2ge 0 and the
positive equilibrium 119864lowastis locally asymptotically stable
for all 1205912ge 0
(ii) If either 11989810
lt 0 or 11989810
ge 0 Δ = 1198982
12minus 311989811
gt
0 119911lowast
11gt 0 and ℎ
1(119911lowast
11) le 0 then ℎ
1(119911) has at least one
positive roots and all roots of (23) have negative realparts for 120591
2isin [0 120591
210
) and the positive equilibrium 119864lowast
is locally asymptotically stable for 1205912isin [0 120591
210
)
(iii) If (ii) holds and ℎ1015840
1(1199111119896) = 0 then system (3)undergoes
Hopf bifurcations at the positive equilibrium119864lowastfor 1205912=
120591(119895)
21119896
(119896 = 1 2 3 119895 = 0 1 2 )
Case c Consider1205911gt 0 1205912= 0
The associated characteristic equation of system (3) is
1205823+ 11990121205822+ (1199011+ 1199031) 120582 + (119901
0+ 1199030) + (119902
1120582 + 1199020) 119890minus1205821205911= 0
(38)
Similar to the analysis of Case 119887 we get the followingtheorem
Theorem7 For 1205911gt 0 120591
2= 0 suppose that (119867
1)ndash(1198675) hold
(i) If 11989820
ge 0 and Δ = 1198982
22minus 311989821
le 0 then all rootsof (38) have negative real parts for all 120591
1ge 0 and the
positive equilibrium 119864lowastis locally asymptotically stable
for all 1205911ge 0
(ii) If either 11989820
lt 0 or 11989820
ge 0 Δ = 1198982
22minus 311989821
gt 0119911lowast
21gt 0 and ℎ
2(119911lowast
21) le 0 then ℎ
2(119911) has at least one
positive root 1199112119896 and all roots of (38) have negative real
parts for 1205911isin [0 120591
120
) and the positive equilibrium 119864lowast
is locally asymptotically stable for 1205911isin [0 120591
120
)
(iii) If (ii) holds and ℎ1015840
2(1199112119896) = 0 then system (3) undergoes
Hopf bifurcations at the positive equilibrium119864lowastfor 1205911=
120591(119895)
12119896
(119896 = 1 2 3 119895 = 0 1 2 )
where
11989822
= 1199012
2minus 2 (119901
1+ 1199031)
11989821
= (1199011+ 1199031)2
minus 21199012(1199010+ 1199030) minus 1199022
1
11989820
= (1199010+ 1199030)2
minus 1199022
0
ℎ2(119911) = 119911
3+ 119898221199112+ 11989821119911 + 119898
20 119911lowast
21=
minus11989822
+ radicΔ
3
120591(119895)
12119896
=
1
1205962119896
times arccos ((11990211205964
2119896+ [11990121199020minus (1199031+ 1199011) 1199021] 1205962
2119896
minus 1199020(1199030+ 1199010) )
times (1199022
0+ 1199022
11205962
2119896)
minus1
) + 2119895120587
(39)
where 119896 = 1 2 3 119895 = 0 1 2 then plusmn1198941205962119896is a pair of purely
imaginary roots of (38) corresponding to 120591(119895)
12119896
Define
120591120
= 120591(0)
121198960
= min119896=123
120591(0)
12119896
12059610
= 12059611198960
(40)
6 Abstract and Applied Analysis
Case d Consider1205911gt 0 1205912gt 0 1205911
= 1205912
The associated characteristic equation of system (3) is
1205823+11990121205822+1199011120582+1199010+ (1199021120582 + 1199020) 119890minus1205821205911+ (1199031120582+1199030) 119890minus1205821205912=0
(41)
We consider (41) with 1205912= 120591lowast
2in its stable interval [0 120591
210
)Regard 120591
1as a parameter
Let 120582 = 119894120596 (120596 gt 0) be a root of (41) then we have
minus 1198941205963minus 11990121205962+ 1198941199011120596 + 1199010+ (1198941199021120596 + 1199020) (cos120596120591
1minus 119894 sin120596120591
1)
+ (1199030+ 1198941199031120596) (cos120596120591lowast
2minus 119894 sin120596120591
lowast
2) = 0
(42)
Separating the real and imaginary parts we have
1205963minus 1199011120596 minus 1199031120596 cos120596120591lowast
2+ 1199030sin120596120591
lowast
2
= 1199021120596 cos120596120591
1minus 1199020sin120596120591
1
11990121205962minus 1199010minus 1199030cos120596120591lowast
2minus 1199031120596 sin120596120591
lowast
2
= 1199020cos120596120591
1+ 1199021120596 sin120596120591
1
(43)
It follows that
1205966+ 119898331205964+ 119898321205963+ 119898311205962+ 11989830
= 0 (44)
where
11989833
= 1199012
2minus 21199011minus 21199031cos120596120591lowast
2
11989832
= 2 (1199030minus 11990121199031) sin120596120591
lowast
2
11989831
= 1199012
1minus 211990101199012minus 2 (119901
21199030minus 11990111199031) cos120596120591lowast
2+ 1199032
1minus 1199022
1
11989830
= 1199012
0+ 211990101199030cos120596120591lowast
2+ 1199032
0minus 1199022
0
(45)
Denote 119865(120596) = 1205966+ 119898331205964+ 119898321205963+ 119898311205962+ 11989830 If
11989830
lt 0 then
119865 (0) lt 0 lim120596rarr+infin
119865 (120596) = +infin (46)
We can obtain that (44) has at most six positive roots1205961 1205962 120596
6 For every fixed 120596
119896 119896 = 1 2 6 there exists
a sequence 120591(119895)1119896
| 119895 = 0 1 2 3 such that (43) holdsLet
12059110
= min 120591(119895)
1119896| 119896 = 1 2 6 119895 = 0 1 2 3 (47)
When 1205911
= 120591(119895)
1119896 (41) has a pair of purely imaginary roots
plusmn119894120596(119895)
1119896for 120591lowast2isin [0 120591
210
)In the following we assume that
(H6) ((119889Re(120582))119889120591
1)|120582=plusmn119894120596
(119895)
1119896
= 0
Thus by the general Hopf bifurcation theorem for FDEsin Hale [22] we have the following result on the stability andHopf bifurcation in system (3)
Theorem 8 For 1205911
gt 0 1205912
gt 0 1205911
= 1205912 suppose that
(1198671)ndash(1198676) is satisfied If 119898
30lt 0 and 120591
lowast
2isin [0 120591
210
] thenthe positive equilibrium 119864
lowastis locally asymptotically stable for
1205911
isin [0 12059110) System (3) undergoes Hopf bifurcations at the
positive equilibrium 119864lowastfor 1205911= 120591(119895)
1119896
3 Direction and Stability ofthe Hopf Bifurcation
In Section 2 we obtain the conditions underwhich system (3)undergoes the Hopf bifurcation at the positive equilibrium119864lowast In this section we consider with 120591
2= 120591lowast
2isin [0 120591
210
)
and regard 1205911as a parameter We will derive the explicit
formulas determining the direction stability and period ofthese periodic solutions bifurcating from equilibrium 119864
lowastat
the critical values 1205911by using the normal form and the center
manifold theory developed by Hassard et al [23] Withoutloss of generality denote any one of these critical values 120591
1=
120591(119895)
1119896(119896 = 1 2 6 119895 = 0 1 2 ) by 120591
1 at which (43) has a
pair of purely imaginary roots plusmn119894120596 and system (3) undergoesHopf bifurcation from 119864
lowast
Throughout this section we always assume that 120591lowast2lt 12059110
Let 1199061= 1199091minus 119909lowast
1 1199062= 1199091minus 119909lowast
2 1199063= 1199092minus 119909lowast
3 119905 = 120591
1119905
and 120583 = 1205911minus 1205911 120583 isin R Then 120583 = 0 is the Hopf bifurcation
value of system (3) System (3) may be written as a functionaldifferential equation inC([minus1 0]R3)
(119905) = 119871120583(119906119905) + 119891 (120583 119906
119905) (48)
where 119906 = (1199061 1199062 1199063)119879isin R3 and
119871120583(120601) = (120591
1+ 120583) 119861[
[
1206011(0)
1206012(0)
1206013(0)
]
]
+ (1205911+ 120583)119862[
[
1206011(minus1)
1206012(minus1)
1206013(minus1)
]
]
+ (1205911+ 120583)119863
[[[[[[[[[[
[
1206011(minus
120591lowast
2
1205911
)
1206012(minus
120591lowast
2
1205911
)
1206013(minus
120591lowast
2
1205911
)
]]]]]]]]]]
]
(49)
119891 (120583 120601) = (1205911+ 120583)[
[
1198911
1198912
1198913
]
]
(50)
Abstract and Applied Analysis 7
where 120601 = (1206011 1206012 1206013)119879isin C([minus1 0]R3) and
119861 = [
[
11988711
11988712
0
0 11988722
11988723
0 0 11988733
]
]
119862 = [
[
0 0 0
11988821
0 0
0 0 0
]
]
119863 = [
[
0 0 0
0 0 0
0 11988932
0
]
]
1198911= minus (119886
11+ 1198973) 1206012
1(0) minus 119897
11206011(0) 1206012(0) minus 119897
21206012
1(0) 1206012(0)
minus 11989751206013
1(0) minus 119897
41206013
1(0) 1206012(0) + sdot sdot sdot
1198912= minus11989761206012(0) 1206013(0) minus (119897
7+ 11988622) 1206012
2(0) + 119897
11206011(minus1) 120601
2(0)
+ 11989731206012
1(minus1) minus 119897
81206013
2(0) minus 119897
91206012
2(0) 1206013(0)
+ 11989721206012
1(minus1) 120601
2(0) + 119897
31206013
1(minus1) + 119897
41206013
1(minus1) 120601
2(0)
minus 119897101206013
2(0) 1206013(0) + sdot sdot sdot
1198913= 11989761206012(minus
120591lowast
2
1205911
)1206013(0) + 119897
71206012
2(minus
120591lowast
2
1205911
) minus 119886331206012
3(0)
+ 11989791206012
2(minus
120591lowast
2
1205911
)1206013(0) + 119897
81206013
2(minus
120591lowast
2
1205911
)
+ 119897101206013
2(minus
120591lowast
2
1205911
)1206013(0) + sdot sdot sdot
1199011(119909) =
1198861119909
1 + 1198871119909
1199012(119909) =
1198862119909
1 + 1198872119909
1198971= 1199011015840
1(119909lowast)
1198972=
1
2
11990110158401015840
1(119909lowast) 1198973=
1
2
11990110158401015840
1(119909lowast) 1199101lowast
1198974=
1
3
119901101584010158401015840
1(119909lowast) 1198975=
1
3
119901101584010158401015840
1(119909lowast) 1199101lowast
1198976= 1199011015840
2(1199101lowast) 1198977=
1
2
11990110158401015840
2(1199101lowast) 1199102lowast
1198978=
1
3
119901101584010158401015840
2(1199101lowast) 1199102lowast 1198979=
1
2
11990110158401015840
2(1199101lowast)
11989710
=
1
3
119901101584010158401015840
2(1199101lowast)
(51)
Obviously 119871120583(120601) is a continuous linear function mapping
C([minus1 0]R3) intoR3 By the Riesz representation theoremthere exists a 3 times 3 matrix function 120578(120579 120583) (minus1 ⩽ 120579 ⩽ 0)whose elements are of bounded variation such that
119871120583120601 = int
0
minus1
119889120578 (120579 120583) 120601 (120579) for 120601 isin C ([minus1 0] R3) (52)
In fact we can choose
119889120578 (120579 120583) = (1205911+ 120583) [119861120575 (120579) + 119862120575 (120579 + 1) + 119863120575(120579 +
120591lowast
2
1205911
)]
(53)
where 120575 is Dirac-delta function For 120601 isin C([minus1 0]R3)define
119860 (120583) 120601 =
119889120601 (120579)
119889120579
120579 isin [minus1 0)
int
0
minus1
119889120578 (119904 120583) 120601 (119904) 120579 = 0
119877 (120583) 120601 =
0 120579 isin [minus1 0)
119891 (120583 120601) 120579 = 0
(54)
Then when 120579 = 0 the system is equivalent to
119905= 119860 (120583) 119909
119905+ 119877 (120583) 119909
119905 (55)
where 119909119905(120579) = 119909(119905+120579) 120579 isin [minus1 0] For120595 isin C1([0 1] (R3)
lowast)
define
119860lowast120595 (119904) =
minus
119889120595 (119904)
119889119904
119904 isin (0 1]
int
0
minus1
119889120578119879(119905 0) 120595 (minus119905) 119904 = 0
(56)
and a bilinear inner product
⟨120595 (119904) 120601 (120579)⟩=120595 (0) 120601 (0)minusint
0
minus1
int
120579
120585=0
120595 (120585minus120579) 119889120578 (120579) 120601 (120585) 119889120585
(57)
where 120578(120579) = 120578(120579 0) Let 119860 = 119860(0) then 119860 and 119860lowast
are adjoint operators By the discussion in Section 2 weknow that plusmn119894120596120591
1are eigenvalues of 119860 Thus they are also
eigenvalues of 119860lowast We first need to compute the eigenvectorof 119860 and 119860
lowast corresponding to 1198941205961205911and minus119894120596120591
1 respectively
Suppose that 119902(120579) = (1 120572 120573)1198791198901198941205791205961205911 is the eigenvector of 119860
corresponding to 1198941205961205911 Then 119860119902(120579) = 119894120596120591
1119902(120579) From the
definition of 119860119871120583(120601) and 120578(120579 120583) we can easily obtain 119902(120579) =
(1 120572 120573)1198791198901198941205791205961205911 where
120572 =
119894120596 minus 11988711
11988712
120573 =
11988932(119894120596 minus 119887
11)
11988712(119894120596 minus 119887
33) 119890119894120596120591lowast
2
(58)
and 119902(0) = (1 120572 120573)119879 Similarly let 119902lowast(119904) = 119863(1 120572
lowast 120573lowast)1198901198941199041205961205911
be the eigenvector of 119860lowast corresponding to minus119894120596120591
1 By the
definition of 119860lowast we can compute
120572lowast=
minus119894120596 minus 11988711
119888211198901198941205961205911
120573lowast=
11988723(minus119894120596 minus 119887
11)
11988821(119894120596 minus 119887
33) 1198901198941205961205911
(59)
From (57) we have⟨119902lowast(119904) 119902 (120579)⟩
= 119863(1 120572lowast 120573
lowast
) (1 120572 120573)119879
minus int
0
minus1
int
120579
120585=0
119863(1 120572lowast 120573
lowast
) 119890minus1198941205961205911(120585minus120579)
119889120578 (120579)
times (1 120572 120573)119879
1198901198941205961205911120585119889120585
= 1198631 + 120572120572lowast+ 120573120573
lowast
+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573
lowast
120591lowast
2119890minus119894120596120591lowast
2
(60)
8 Abstract and Applied Analysis
Thus we can choose
119863 = 1 + 120572120572lowast+ 120573120573
lowast
+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573
lowast
120591lowast
2119890minus119894120596120591lowast
2
minus1
(61)
such that ⟨119902lowast(119904) 119902(120579)⟩ = 1 ⟨119902lowast(119904) 119902(120579)⟩ = 0
In the remainder of this section we follow the ideas inHassard et al [23] and use the same notations as there tocompute the coordinates describing the center manifold 119862
0
at 120583 = 0 Let 119909119905be the solution of (48) when 120583 = 0 Define
119911 (119905) = ⟨119902lowast 119909119905⟩ 119882 (119905 120579) = 119909
119905(120579) minus 2Re 119911 (119905) 119902 (120579)
(62)
On the center manifold 1198620 we have
119882(119905 120579) = 119882 (119911 (119905) 119911 (119905) 120579)
= 11988220(120579)
1199112
2
+11988211(120579) 119911119911 +119882
02(120579)
1199112
2
+11988230(120579)
1199113
6
+ sdot sdot sdot
(63)
where 119911 and 119911 are local coordinates for center manifold 1198620in
the direction of 119902 and 119902 Note that 119882 is real if 119909119905is real We
consider only real solutions For the solution 119909119905isin 1198620of (48)
since 120583 = 0 we have
= 1198941205961205911119911 + ⟨119902
lowast(120579) 119891 (0119882 (119911 (119905) 119911 (119905) 120579)
+ 2Re 119911 (119905) 119902 (120579))⟩
= 1198941205961205911119911+119902lowast(0) 119891 (0119882 (119911 (119905) 119911 (119905) 0) +2Re 119911 (119905) 119902 (0))
= 1198941205961205911119911 + 119902lowast(0) 1198910(119911 119911) ≜ 119894120596120591
1119911 + 119892 (119911 119911)
(64)
where
119892 (119911 119911) = 119902lowast(0) 1198910(119911 119911) = 119892
20
1199112
2
+ 11989211119911119911
+ 11989202
1199112
2
+ 11989221
1199112119911
2
+ sdot sdot sdot
(65)
By (62) we have 119909119905(120579) = (119909
1119905(120579) 1199092119905(120579) 1199093119905(120579))119879= 119882(119905 120579) +
119911119902(120579) + 119911 119902(120579) and then
1199091119905(0) = 119911 + 119911 +119882
(1)
20(0)
1199112
2
+119882(1)
11(0) 119911119911
+119882(1)
02(0)
1199112
2
+ 119900 (|(119911 119911)|3)
1199092119905(0) = 119911120572 + 119911 120572 +119882
(2)
20(0)
1199112
2
+119882(2)
11(0) 119911119911 +119882
(2)
02(0)
1199112
2
+ 119900 (|(119911 119911)|3)
1199093119905(0) = 119911120573 + 119911120573 +119882
(3)
20(0)
1199112
2
+119882(3)
11(0) 119911119911
+119882(3)
02(0)
1199112
2
+ 119900 (|(119911 119911)|3)
1199091119905(minus1) = 119911119890
minus1198941205961205911+ 1199111198901198941205961205911+119882(1)
20(minus1)
1199112
2
+119882(1)
11(minus1) 119911119911 +119882
(1)
02(minus1)
1199112
2
+ 119900 (|(119911 119911)|3)
1199092119905(minus1) = 119911120572119890
minus1198941205961205911+ 119911 120572119890
1198941205961205911+119882(2)
20(minus1)
1199112
2
+119882(2)
11(minus1) 119911119911 +119882
(2)
02(minus1)
1199112
2
+ 119900 (|(119911 119911)|3)
1199093119905(minus1) = 119911120573119890
minus1198941205961205911+ 119911120573119890
1205961205911+119882(3)
20(minus1)
1199112
2
+119882(3)
11(minus1) 119911119911 +119882
(3)
02(minus1)
1199112
2
+ 119900 (|(119911 119911)|3)
1199091119905(minus
120591lowast
2
12059110
) = 119911119890minus119894120596120591lowast
2+ 119911119890119894120596120591lowast
2+119882(1)
20(minus
120591lowast
2
1205911
)
1199112
2
+119882(1)
11(minus
120591lowast
2
1205911
)119911119911 +119882(1)
02(minus
120591lowast
2
1205911
)
1199112
2
+ 119900 (|(119911 119911)|3)
1199092119905(minus
120591lowast
2
1205911
) = 119911120572119890minus119894120596120591lowast
2+ 119911 120572119890
119894120596120591lowast
2+119882(2)
20(minus
120591lowast
2
1205911
)
1199112
2
+119882(2)
11(minus
120591lowast
2
1205911
)119911119911 +119882(2)
02(minus
120591lowast
2
1205911
)
1199112
2
+ 119900 (|(119911 119911)|3)
1199093119905(minus
120591lowast
2
1205911
) = 119911120573119890minus119894120596120591lowast
2+ 119911120573119890
120596120591lowast
2+119882(3)
20(minus
120591lowast
2
1205911
)
1199112
2
+119882(3)
11(minus
120591lowast
2
1205911
)119911119911 +119882(3)
02(minus
120591lowast
2
1205911
)
1199112
2
+ 119900 (|(119911 119911)|3)
(66)
Abstract and Applied Analysis 9
It follows together with (50) that
119892 (119911 119911)=119902lowast(0) 1198910(119911 119911) =119863120591
1(1 120572lowast 120573
lowast
) (119891(0)
1119891(0)
2119891(0)
3)
119879
= 1198631205911[minus (119886
11+ 1198973) 1206012
1(0) minus 119897
11206011(0) 1206012(0)
minus 11989721206012
1(0) 1206012(0) minus 119897
51206013
1(0) minus 119897
41206013
1(0) 1206012(0)
+ sdot sdot sdot ]
+ 120572lowast[11989761206012(0) 1206013(0) (1198977+ 11988622) 1206012
2(0)
+ 11989711206011(minus1) 120601
2(0)
+ 11989731206012
1(minus1) minus 119897
81206013
2(0) minus 119897
91206012
2(0) 1206013(0)
+ 11989721206012
1(minus1) 120601
2(0) + 119897
31206013
1(minus1) + sdot sdot sdot ]
+ 120573
lowast
[11989761206012(minus
120591lowast
2
1205911
)1206013(0) + 119897
71206012
2(minus
120591lowast
2
1205911
)
minus 119886331206012
3(0) + 119897
81206013
2(minus
120591lowast
2
1205911
) + sdot sdot sdot ]
(67)
Comparing the coefficients with (65) we have
11989220
= 1198631205911[minus2 (119886
11+ 1198973) minus 2120572119897
1]
+ 120572lowast[21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591
1
minus21198976120572120573 minus 2 (119897
7+ 11988622) 1205722]
+ 120573
lowast
[21198976120572120573119890minus119894120596120591lowast
2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591
lowast
2
minus2119886331205732]
11989211
= 1198631205911[minus2 (119886
11+ 1198973) minus 1198971(120572 + 120572)]
+ 120572lowast[1198971(1205721198901198941205961205911+ 120572119890minus1198941205961205911) minus 1198976(120572120573 + 120572120573)
minus2 (1198977+ 11988622) 120572120572 + 2119897
3]
+ 120573
lowast
[1198976(120573120572119890119894120596120591lowast
2+ 120572120573119890
minus119894120596120591lowast
2)
+ 21198977120572120572 minus 119886
33120573120573]
11989202
= 21198631205911[minus2 (119886
11+ 1198973) minus 21198971120572]
+ 120572lowast[minus21198976120572120573 minus 2 (119897
7+ 11988622) 1205722+ 211989711205721198901198941205961205911
+2119897311989021198941205961205911]
+ 120573
lowast
[21198976120572120573119890119894120596120591lowast
2+ 2119897712057221198902119894120596120591lowast
2minus 11988633120573
2
]
11989221
= 1198631205911[minus (119886
11+ 1198973) (2119882
(1)
20(0) + 4119882
(1)
11(0))
minus 1198971(2120572119882
(1)
11(0) + 120572119882
(1)
20(0) + 119882
(2)
20(0)
+ 2119882(2)
11(0))]
+ 120572lowast[minus1198976(2120573119882
(2)
11(0) + 120572119882
(3)
20(0) + 120573119882
(2)
20(0)
+ 2120572119882(3)
11(0)) minus (119897
7+ 11988622)
times (4120572119882(2)
11(0) + 2120572119882
(2)
20(0))
+ 1198971(2120572119882
(1)
11(minus1) + 120572119882
(1)
20(minus1)
+ 119882(2)
20(0) 1198901198941205961205911+ 2119882
(2)
11(0) 119890minus1198941205961205911)
+ 1198973(4119882(1)
11(minus1) 119890
minus1198941205961205911+2119882(1)
20(minus1) 119890
1198941205961205911)]
+ 120573
lowast
[1198976(2120573119882
(2)
11(minus
120591lowast
2
1205911
) + 120573119882(2)
20(minus
120591lowast
2
1205911
)
+ 120572119882(3)
20(0) 119890119894120596120591lowast
2+ 2120572119882
(3)
11(0) 119890minus119894120596120591lowast
2)
+ 1198977(4120572119882
(2)
11(minus
120591lowast
2
1205911
) 119890minus119894120596120591lowast
2
+2120572119882(2)
20(minus
120591lowast
2
1205911
) 119890119894120596120591lowast
2)
minus 11988633(4120573119882
(3)
11(0) + 2120573119882
(3)
20(0))]
(68)
where
11988220(120579) =
11989411989220
1205961205911
119902 (0) 1198901198941205961205911120579+
11989411989202
31205961205911
119902 (0) 119890minus1198941205961205911120579+ 119864111989021198941205961205911120579
11988211(120579) = minus
11989411989211
1205961205911
119902 (0) 1198901198941205961205911120579+
11989411989211
1205961205911
119902 (0) 119890minus1198941205961205911120579+ 1198642
1198641= 2
[[[[[
[
2119894120596 minus 11988711
minus11988712
0
minus11988821119890minus2119894120596120591
12119894120596 minus 119887
22minus11988723
0 minus11988932119890minus2119894120596120591
lowast
22119894120596 minus 119887
33
]]]]]
]
minus1
times
[[[[[
[
minus2 (11988611
+ 1198973) minus 2120572119897
1
21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591
1minus 21198976120572120573 minus 2 (119897
7+ 11988622) 1205722
21198976120572120573119890minus119894120596120591lowast
2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591
lowast
2minus 2119886331205732
]]]]]
]
10 Abstract and Applied Analysis
1198642= 2
[[[[
[
minus11988711
minus11988712
0
minus11988821
minus11988722
minus11988723
0 minus11988932
minus11988733
]]]]
]
minus1
times
[[[[
[
minus2 (11988611
+ 1198973) minus 1198971(120572 + 120572)
1198971(1205721198901198941205961205911+ 120572119890minus1198941205961205911) minus 1198976(120572120573 + 120572120573) minus 2 (119897
7+ 11988622) 120572120572 + 2119897
3
1198976(120573120572119890119894120596120591lowast
2+ 120572120573119890
minus119894120596120591lowast
2) + 21198977120572120572 minus 119886
33120573120573
]]]]
]
(69)
Thus we can determine11988220(120579) and119882
11(120579) Furthermore
we can determine each 119892119894119895by the parameters and delay in (3)
Thus we can compute the following values
1198881(0) =
119894
21205961205911
(1198922011989211
minus 2100381610038161003816100381611989211
1003816100381610038161003816
2
minus
1
3
100381610038161003816100381611989202
1003816100381610038161003816
2
) +
1
2
11989221
1205832= minus
Re 1198881(0)
Re 1205821015840 (1205911)
1198792= minus
Im 1198881(0) + 120583
2Im 120582
1015840(1205911)
1205961205911
1205732= 2Re 119888
1(0)
(70)
which determine the quantities of bifurcating periodic solu-tions in the center manifold at the critical value 120591
1 Suppose
Re1205821015840(1205911) gt 0 120583
2determines the directions of the Hopf
bifurcation if 1205832
gt 0(lt 0) then the Hopf bifurcationis supercritical (subcritical) and the bifurcation exists for120591 gt 120591
1(lt 1205911) 1205732determines the stability of the bifurcation
periodic solutions the bifurcating periodic solutions arestable (unstable) if 120573
2lt 0(gt 0) and119879
2determines the period
of the bifurcating periodic solutions the period increases(decreases) if 119879
2gt 0(lt 0)
4 Numerical Simulation
We consider system (3) by taking the following coefficients1198861
= 03 11988611
= 58889 11988612
= 1 1198981
= 1 1198862
=
01 11988621
= 27 11988622
= 12 11988623
= 12 1198982
= 1 1198863
=
02 11988632
= 25 11988633
= 12 We have the unique positiveequilibrium 119864
lowast= (00451 00357 00551)
By computation we get 11989810
= minus00104 12059611
= 0516411991111
= 02666 ℎ10158401(11991111) = 03402 120591
20
= 32348 FromTheorem 6 we know that when 120591
1= 0 the positive
equilibrium 119864lowastis locally asymptotically stable for 120591
2isin
[0 32348) When 1205912crosses 120591
20
the equilibrium 119864lowastloses its
stability and Hopf bifurcation occurs From the algorithm inSection 3 we have 120583
2= 56646 120573
2= minus31583 119879
2= 5445
whichmeans that the bifurcation is supercritical and periodicsolution is stable The trajectories and the phase graphs areshown in Figures 1 and 2
Regarding 1205911as a parameter and let 120591
2= 29 isin
[0 32348) we can observe that with 1205911increasing the pos-
itive equilibrium 119864lowastloses its stability and Hopf bifurcation
occurs (see Figures 3 and 4)
5 Global Continuationof Local Hopf Bifurcations
In this section we study the global continuation of peri-odic solutions bifurcating from the positive equilibrium(119864lowast 120591(119895)
1119896) (119896 = 1 2 6 119895 = 0 1 ) Throughout this
section we follow closely the notations in [24] and assumethat 120591
2= 120591lowast
2isin [0 120591
210
) regarding 1205911as a parameter For
simplification of notations setting 119911119905(119905) = (119909
1119905 1199092119905 1199093119905)119879 we
may rewrite system (3) as the following functional differentialequation
(119905) = 119865 (119911119905 1205911 119901) (71)
where 119911119905(120579) = (119909
1119905(120579) 1199092119905(120579) 1199093119905(120579))119879
= (1199091(119905 + 120579) 119909
2(119905 +
120579) 1199093(119905 + 120579))
119879 for 119905 ge 0 and 120579 isin [minus1205911 0] Since 119909
1(119905) 1199092(119905)
and 1199093(119905) denote the densities of the prey the predator and
the top predator respectively the positive solution of system(3) is of interest and its periodic solutions only arise in the firstquadrant Thus we consider system (3) only in the domain1198773
+= (1199091 1199092 1199093) isin 1198773 1199091gt 0 119909
2gt 0 119909
3gt 0 It is obvious
that (71) has a unique positive equilibrium 119864lowast(119909lowast
1 119909lowast
2 119909lowast
3) in
1198773
+under the assumption (119867
1)ndash(1198674) Following the work of
[24] we need to define
119883 = 119862([minus1205911 0] 119877
3
+)
Γ = 119862119897 (119911 1205911 119901) isin X times R times R+
119911 is a 119901-periodic solution of system (71)
N = (119911 1205911 119901) 119865 (119911 120591
1 119901) = 0
(72)
Let ℓ(119864lowast120591(119895)
11198962120587120596
(119895)
1119896)denote the connected component pass-
ing through (119864lowast 120591(119895)
1119896 2120587120596
(119895)
1119896) in Γ where 120591
(119895)
1119896is defined by
(43) We know that ℓ(119864lowast120591(119895)
11198962120587120596
(119895)
1119896)through (119864
lowast 120591(119895)
1119896 2120587120596
(119895)
1119896)
is nonemptyFor the benefit of readers we first state the global Hopf
bifurcation theory due to Wu [24] for functional differentialequations
Lemma 9 Assume that (119911lowast 120591 119901) is an isolated center satis-
fying the hypotheses (A1)ndash(A4) in [24] Denote by ℓ(119911lowast120591119901)
theconnected component of (119911
lowast 120591 119901) in Γ Then either
(i) ℓ(119911lowast120591119901)
is unbounded or(ii) ℓ(119911lowast120591119901)
is bounded ℓ(119911lowast120591119901)
cap Γ is finite and
sum
(119911120591119901)isinℓ(119911lowast120591119901)capN
120574119898(119911lowast 120591 119901) = 0 (73)
for all 119898 = 1 2 where 120574119898(119911lowast 120591 119901) is the 119898119905ℎ crossing
number of (119911lowast 120591 119901) if119898 isin 119869(119911
lowast 120591 119901) or it is zero if otherwise
Clearly if (ii) in Lemma 9 is not true then ℓ(119911lowast120591119901)
isunbounded Thus if the projections of ℓ
(119911lowast120591119901)
onto 119911-spaceand onto 119901-space are bounded then the projection of ℓ
(119911lowast120591119901)
onto 120591-space is unbounded Further if we can show that the
Abstract and Applied Analysis 11
0 200 400 60000451
00451
00452
t
x1(t)
t
0 200 400 60000356
00357
00358
x2(t)
t
0 200 400 6000055
00551
00552
x3(t)
00451
00451
00452
00356
00357
00357
003570055
0055
00551
00551
00552
x 1(t)
x2 (t)
x3(t)
Figure 1 The trajectories and the phase graph with 1205911= 0 120591
2= 29 lt 120591
20= 32348 119864
lowastis locally asymptotically stable
0 500 1000004
0045
005
0 500 1000002
004
006
0 500 10000
005
01
t
t
t
x1(t)
x2(t)
x3(t)
0044
0046
0048
002
003
004
005003
004
005
006
007
008
x 1(t)
x2 (t)
x3(t)
Figure 2 The trajectories and the phase graph with 1205911= 0 120591
2= 33 gt 120591
20= 32348 a periodic orbit bifurcate from 119864
lowast
12 Abstract and Applied Analysis
0 500 1000 1500
00451
00452
0 500 1000 15000035
0036
0037
0 500 1000 15000054
0055
0056
t
t
t
x1(t)
x2(t)
x3(t)
0045100451
0045200452
0035
00355
003600546
00548
0055
00552
00554
00556
00558
x1(t)
x2 (t)
x3(t)
Figure 3 The trajectories and the phase graph with 1205911= 09 120591
2= 29 119864
lowastis locally asymptotically stable
0 500 1000 15000044
0045
0046
0 500 1000 1500002
003
004
0 500 1000 1500004
006
008
t
t
t
x1(t)
x2(t)
x3(t)
00445
0045
00455
0032
0034
0036
00380052
0054
0056
0058
006
0062
0064
x 1(t)
x2 (t)
x3(t)
Figure 4 The trajectories and the phase graph with 1205911= 12 120591
2= 29 a periodic orbit bifurcate from 119864
lowast
Abstract and Applied Analysis 13
projection of ℓ(119911lowast120591119901)
onto 120591-space is away from zero thenthe projection of ℓ
(119911lowast120591119901)
onto 120591-space must include interval[120591 infin) Following this ideal we can prove our results on theglobal continuation of local Hopf bifurcation
Lemma 10 If the conditions (1198671)ndash(1198674) hold then all nontriv-
ial periodic solutions of system (71) with initial conditions
1199091(120579) = 120593 (120579) ge 0 119909
2(120579) = 120595 (120579) ge 0
1199093(120579) = 120601 (120579) ge 0 120579 isin [minus120591
1 0)
120593 (0) gt 0 120595 (0) gt 0 120601 (0) gt 0
(74)
are uniformly bounded
Proof Suppose that (1199091(119905) 1199092(119905) 1199093(119905)) are nonconstant peri-
odic solutions of system (3) and define
1199091(1205851) = min 119909
1(119905) 119909
1(1205781) = max 119909
1(119905)
1199092(1205852) = min 119909
2(119905) 119909
2(1205782) = max 119909
2(119905)
1199093(1205853) = min 119909
3(119905) 119909
3(1205783) = max 119909
3(119905)
(75)
It follows from system (3) that
1199091(119905) = 119909
1(0) expint
119905
0
[1198861minus 119886111199091(119904) minus
119886121199092(119904)
1198981+ 1199091(119904)
] 119889119904
1199092(119905) = 119909
2(0) expint
119905
0
[minus1198862+
119886211199091(119904 minus 1205911)
1198981+ 1199091(119904 minus 1205911)
minus 119886221199092(119904) minus
119886231199093(119904)
1198982+ 1199092(119904)
] 119889119904
1199093(119905) = 119909
3(0) expint
119905
0
[minus1198863+
119886321199092(119904 minus 120591
lowast
2)
1198982+ 1199092(119904 minus 120591
lowast
2)
minus 119886331199093(119904)] 119889119904
(76)
which implies that the solutions of system (3) cannot crossthe 119909119894-axis (119894 = 1 2 3) Thus the nonconstant periodic orbits
must be located in the interior of first quadrant It followsfrom initial data of system (3) that 119909
1(119905) gt 0 119909
2(119905) gt
0 1199093(119905) gt 0 for 119905 ge 0
From the first equation of system (3) we can get
0 = 1198861minus 119886111199091(1205781) minus
119886121199092(1205781)
1198981+ 1199091(1205781)
le 1198861minus 119886111199091(1205781) (77)
thus we have
1199091(1205781) le
1198861
11988611
(78)
From the second equation of (3) we obtain
0 = minus 1198862+
119886211199091(1205782minus 1205911)
1198981+ 1199091(1205782minus 1205911)
minus 119886221199092(1205782)
minus
119886231199093(1205782)
1198982+ 1199092(1205782)
le minus1198862+
11988621(119886111988611)
1198981+ (119886111988611)
minus 119886221199092(1205782)
(79)
therefore one can get
1199092(1205782) le
minus1198862(119886111198981+ 1198861) + 119886111988621
11988622(119886111198981+ 1198861)
≜ 1198721 (80)
Applying the third equation of system (3) we know
0 = minus1198863+
119886321199092(1205783minus 120591lowast
2)
1198982+ 1199092(1205783minus 120591lowast
2)
minus 119886331199093(1205783)
le minus1198863+
119886321198721
1198982+1198721
minus 119886331199093(1205783)
(81)
It follows that
1199093(1205783) le
minus1198863(1198982+1198721) + 119886321198721
11988633(1198982+1198721)
≜ 1198722 (82)
This shows that the nontrivial periodic solution of system (3)is uniformly bounded and the proof is complete
Lemma 11 If the conditions (1198671)ndash(1198674) and
(H7) 11988622
minus (119886211198982) gt 0 [(119886
11minus 11988611198981)11988622
minus
(11988612119886211198982
1)]11988633
minus (11988611
minus (11988611198981))[(119886211198982)11988633
+
(11988623119886321198982
2)] gt 0
hold then system (3) has no nontrivial 1205911-periodic solution
Proof Suppose for a contradiction that system (3) has non-trivial periodic solution with period 120591
1 Then the following
system (83) of ordinary differential equations has nontrivialperiodic solution
1198891199091
119889119905
= 1199091(119905) [1198861minus 119886111199091(119905) minus
119886121199092(119905)
1198981+ 1199091(119905)
]
1198891199092
119889119905
= 1199092(119905) [minus119886
2+
119886211199091(119905)
1198981+ 1199091(119905)
minus 119886221199092(119905) minus
119886231199093(119905)
1198982+ 1199092(119905)
]
1198891199093
119889119905
= 1199093(119905) [minus119886
3+
119886321199092(119905 minus 120591
lowast
2)
1198982+ 1199092(119905 minus 120591
lowast
2)
minus 119886331199093(119905)]
(83)
which has the same equilibria to system (3) that is
1198641= (0 0 0) 119864
2= (
1198861
11988611
0 0)
1198643= (1199091 1199092 0) 119864
lowast= (119909lowast
1 119909lowast
2 119909lowast
3)
(84)
Note that 119909119894-axis (119894 = 1 2 3) are the invariable manifold of
system (83) and the orbits of system (83) do not intersect each
14 Abstract and Applied Analysis
other Thus there are no solutions crossing the coordinateaxes On the other hand note the fact that if system (83) hasa periodic solution then there must be the equilibrium in itsinterior and that 119864
1 1198642 1198643are located on the coordinate
axis Thus we conclude that the periodic orbit of system (83)must lie in the first quadrant If (119867
7) holds it is well known
that the positive equilibrium 119864lowastis globally asymptotically
stable in the first quadrant (see [1])Thus there is no periodicorbit in the first quadrant too The above discussion meansthat (83) does not have any nontrivial periodic solution It isa contradiction Therefore the lemma is confirmed
Theorem 12 Suppose the conditions of Theorem 8 and (1198677)
hold let120596119896and 120591(119895)1119896
be defined in Section 2 then when 1205911gt 120591(119895)
1119896
system (3) has at least 119895 minus 1 periodic solutions
Proof It is sufficient to prove that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is [120591
1 +infin) for each 119895 ge 1 where
1205911le 120591(119895)
1119896
In following we prove that the hypotheses (A1)ndash(A4) in[24] hold
(1) From system (3) we know easily that the followingconditions hold
(A1) 119865 isin 1198622(1198773
+times119877+times119877+) where 119865 = 119865|
1198773
+times119877+times119877+
rarr
1198773
+
(A3) 119865(120601 1205911 119901) is differential with respect to 120601
(2) It follows from system (3) that
119863119911119865 (119911 120591
1 119901) =
[[[[[[[[[[[
[
1198861minus 2119886111199091minus
1198861211989811199092
(1198981+ 1199091)2
minus
119886121199091
1198981+ 1199091
0
1198862111989811199092
(1198981+ 1199091)2
minus1198862+
119886211199091
1198981+ 1199091
minus 2119886221199092minus
1198862311989821199093
(1198982+ 1199092)2
minus
119886231199092
1198982+ 1199092
0
1198863211989821199093
(1198982+ 1199092)2
minus1198863+
119886321199092
1198982+ 1199092
minus 2119886331199093
]]]]]]]]]]]
]
(85)
Then under the assumption (1198671)ndash(1198674) we have
det119863119911119865 (119911lowast 1205911 119901)
= det
[[[[[[[[
[
minus11988611119909lowast
1+
11988612119909lowast
1119909lowast
2
(1198981+ 119909lowast
1)2
11988612119909lowast
1
1198981+ 119909lowast
1
0
119886211198981119909lowast
2
(1198981+ 119909lowast
1)2
minus11988622119909lowast
2+
11988623119909lowast
2119909lowast
3
(1198982+ 119909lowast
2)2
minus
11988623119909lowast
2
1198982+ 119909lowast
2
0
119886321198982119909lowast
3
(1198982+ 119909lowast
2)2
minus11988633119909lowast
3
]]]]]]]]
]
= minus
1198862
21199101lowast1199102lowast
(1 + 11988721199101lowast)3[minus119909lowast+
11988611198871119909lowast1199101lowast
(1 + 1198871119909lowast)2] = 0
(86)
From (86) we know that the hypothesis (A2) in [24]is satisfied
(3) The characteristic matrix of (71) at a stationarysolution (119911 120591
0 1199010) where 119911 = (119911
(1) 119911(2) 119911(3)) isin 119877
3
takes the following form
Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863
120601119865 (119911 120591
1 119901) (119890
120582119868) (87)
that is
Δ (119911 1205911 119901) (120582)
=
[[[[[[[[[[[[[[[[[
[
120582 minus 1198861+ 211988611119911(1)
+
119886121198981119911(2)
(1198981+ 119911(1))
2
11988612119911(1)
1198981+ 119911(1)
0
minus
119886211198981119911(2)
(1198981+ 119911(1))
2119890minus1205821205911
120582 + 1198862minus
11988621119911(1)
1198981+ 119911(1)
+ 211988622119911(2)
+
119886231198982119911(3)
(1198982+ 119911(2))
2
11988623119911(2)
1198982+ 119911(2)
0 minus
119886321198982119911(3)
(1198982+ 119911(2))
2119890minus120582120591lowast
2120582 + 1198863minus
11988632119911(2)
1198982+ 119911(2)
+ 211988633119911(3)
]]]]]]]]]]]]]]]]]
]
(88)
Abstract and Applied Analysis 15
From (88) we have
det (Δ (119864lowast 1205911 119901) (120582))
= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902
0+ 1199030)] 119890minus1205821205911
(89)
Note that (89) is the same as (20) from the discussion inSection 2 about the local Hopf bifurcation it is easy to verifythat (119864
lowast 120591(119895)
1119896 2120587120596
119896) is an isolated center and there exist 120598 gt
0 120575 gt 0 and a smooth curve 120582 (120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575) rarr C
such that det(Δ(120582(1205911))) = 0 |120582(120591
1) minus 120596119896| lt 120598 for all 120591
1isin
[120591(119895)
1119896minus 120575 120591(119895)
1119896+ 120575] and
120582 (120591(119895)
1119896) = 120596119896119894
119889Re 120582 (1205911)
1198891205911
1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)
1119896
gt 0 (90)
Let
Ω1205982120587120596
119896
= (120578 119901) 0 lt 120578 lt 120598
10038161003816100381610038161003816100381610038161003816
119901 minus
2120587
120596119896
10038161003816100381610038161003816100381610038161003816
lt 120598 (91)
It is easy to see that on [120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575] times 120597Ω
1205982120587120596119896
det(Δ(119864
lowast 1205911 119901)(120578 + (2120587119901)119894)) = 0 if and only if 120578 = 0
1205911= 120591(119895)
1119896 119901 = 2120587120596
119896 119896 = 1 2 3 119895 = 0 1 2
Therefore the hypothesis (A4) in [24] is satisfiedIf we define
119867plusmn(119864lowast 120591(119895)
1119896
2120587
120596119896
) (120578 119901)
= det(Δ (119864lowast 120591(119895)
1119896plusmn 120575 119901) (120578 +
2120587
119901
119894))
(92)
then we have the crossing number of isolated center(119864lowast 120591(119895)
1119896 2120587120596
119896) as follows
120574(119864lowast 120591(119895)
1119896
2120587
120596119896
) = deg119861(119867minus(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
minus deg119861(119867+(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
= minus1
(93)
Thus we have
sum
(1199111205911119901)isinC
(119864lowast120591(119895)
11198962120587120596119896)
120574 (119911 1205911 119901) lt 0
(94)
where (119911 1205911 119901) has all or parts of the form
(119864lowast 120591(119896)
1119895 2120587120596
119896) (119895 = 0 1 ) It follows from
Lemma 9 that the connected component ℓ(119864lowast120591(119895)
11198962120587120596
119896)
through (119864lowast 120591(119895)
1119896 2120587120596
119896) is unbounded for each center
(119911lowast 1205911 119901) (119895 = 0 1 ) From the discussion in Section 2
we have
120591(119895)
1119896=
1
120596119896
times arccos(((11990121205962
119896minus1199010) (1199030+1199020)
+ 1205962(1205962minus1199011) (1199031+1199021))
times ((1199030+1199020)2
+ (1199031+1199021)2
1205962
119896)
minus1
) +2119895120587
(95)
where 119896 = 1 2 3 119895 = 0 1 Thus one can get 2120587120596119896le 120591(119895)
1119896
for 119895 ge 1Now we prove that the projection of ℓ
(119864lowast120591(119895)
11198962120587120596
119896)onto
1205911-space is [120591
1 +infin) where 120591
1le 120591(119895)
1119896 Clearly it follows
from the proof of Lemma 11 that system (3) with 1205911
= 0
has no nontrivial periodic solution Hence the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is away from zero
For a contradiction we suppose that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is bounded this means that the
projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is included in a
interval (0 120591lowast) Noticing 2120587120596119896lt 120591119895
1119896and applying Lemma 11
we have 119901 lt 120591lowast for (119911(119905) 120591
1 119901) belonging to ℓ
(119864lowast120591(119895)
11198962120587120596
119896)
This implies that the projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 119901-
space is bounded Then applying Lemma 10 we get thatthe connected component ℓ
(119864lowast120591(119895)
11198962120587120596
119896)is bounded This
contradiction completes the proof
6 Conclusion
In this paper we take our attention to the stability and Hopfbifurcation analysis of a predator-prey systemwithMichaelis-Menten type functional response and two unequal delaysWe obtained some conditions for local stability and Hopfbifurcation occurring When 120591
1= 1205912 we derived the explicit
formulas to determine the properties of periodic solutionsby the normal form method and center manifold theoremSpecially the global existence results of periodic solutionsbifurcating from Hopf bifurcations are also established byusing a global Hopf bifurcation result due to Wu [24]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to thank the anonymous referees fortheir careful reading and constructive suggestions which leadto truly significant improvement of the paper This researchis supported by the National Natural Science Foundation ofChina (no 11061016)
16 Abstract and Applied Analysis
References
[1] R Xu and M A J Chaplain ldquoPersistence and global stabilityin a delayed predator-prey system with Michaelis-Menten typefunctional responserdquo Applied Mathematics and Computationvol 130 no 2-3 pp 441ndash455 2002
[2] Y L Song Y H Peng and J JWei ldquoBifurcations for a predator-prey system with two delaysrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 466ndash479 2008
[3] G-P Hu W-T Li and X-P Yan ldquoHopf bifurcations in apredator-prey system with multiple delaysrdquo Chaos Solitons ampFractals vol 42 no 2 pp 1273ndash1285 2009
[4] M Liao X Tang and C Xu ldquoBifurcation analysis for a three-species predator-prey systemwith two delaysrdquoCommunicationsin Nonlinear Science and Numerical Simulation vol 17 no 1 pp183ndash194 2012
[5] C Xu and P Li ldquoDynamical analysis in a delayed predator-prey model with two delaysrdquo Discrete Dynamics in Nature andSociety vol 2012 Article ID 652947 22 pages 2012
[6] J-F Zhang ldquoStability and bifurcation periodic solutions ina Lotka-Volterra competition system with multiple delaysrdquoNonlinear Dynamics vol 70 no 1 pp 849ndash860 2012
[7] J Xia Z Yu and R Yuan ldquoStability and Hopf bifurcation ina symmetric Lotka-Volterra predator-prey system with delaysrdquoElectronic Journal of Differential Equations vol 2013 no 9 pp1ndash16 2013
[8] H Wan and J Cui ldquoA malaria model with two delaysrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 601265 8pages 2013
[9] M Zhao ldquoHopf bifurcation analysis for a semiratio-dependentpredator-prey system with two delaysrdquo Abstract and AppliedAnalysis vol 2013 Article ID 495072 13 pages 2013
[10] S Feyissa and S Banerjee ldquoDelay-induced oscillatory dynamicsin humoral mediated immune response with two time delaysrdquoNonlinear Analysis Real World Applications vol 14 no 1 pp35ndash52 2013
[11] G Zhang Y Shen andB Chen ldquoHopf bifurcation of a predator-prey systemwith predator harvesting and twodelaysrdquoNonlinearDynamics vol 73 no 4 pp 2119ndash2131 2013
[12] X-Y Meng H-F Huo and H Xiang ldquoHopf bifurcation in athree-species systemwith delaysrdquo Journal of AppliedMathemat-ics and Computing vol 35 no 1-2 pp 635ndash661 2011
[13] J-F Zhang ldquoGlobal existence of bifurcated periodic solutions ina commensalism model with delaysrdquo Applied Mathematics andComputation vol 218 no 23 pp 11688ndash11699 2012
[14] Y Wang W Jiang and H Wang ldquoStability and global Hopfbifurcation in toxic phytoplankton-zooplankton model withdelay and selective harvestingrdquoNonlinear Dynamics vol 73 no1-2 pp 881ndash896 2013
[15] Y Ma ldquoGlobal Hopf bifurcation in the Leslie-Gower predator-prey model with two delaysrdquo Nonlinear Analysis Real WorldApplications vol 13 no 1 pp 370ndash375 2012
[16] X-Y Meng H-F Huo and X-B Zhang ldquoStability and globalHopf bifurcation in a delayed food web consisting of a preyand two predatorsrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4335ndash4348 2011
[17] Y Song J Wei andM Han ldquoLocal and global Hopf bifurcationin a delayed hematopoiesis modelrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol14 no 11 pp 3909ndash3919 2004
[18] Y Song and J Wei ldquoLocal Hopf bifurcation and global periodicsolutions in a delayed predator-prey systemrdquo Journal of Mathe-matical Analysis and Applications vol 301 no 1 pp 1ndash21 2005
[19] J Wei and M Y Li ldquoGlobal existence of periodic solutions in atri-neuron network model with delaysrdquo Physica D vol 198 no1-2 pp 106ndash119 2004
[20] T Zhao Y Kuang and H L Smith ldquoGlobal existence of peri-odic solutions in a class of delayed Gause-type predator-preysystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 28 no 8 pp 1373ndash1394 1997
[21] S J Gao L S Chen and Z D Teng ldquoHopf bifurcation andglobal stability for a delayed predator-prey system with stagestructure for predatorrdquo Applied Mathematics and Computationvol 202 no 2 pp 721ndash729 2008
[22] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977
[23] B Hassard D Kazarinoff and Y WanTheory and Applicationsof Hopf Bifurcation Cambridge University Press CambridgeUK 1981
[24] J H Wu ldquoSymmetric functional-differential equations andneural networks with memoryrdquo Transactions of the AmericanMathematical Society vol 350 no 12 pp 4799ndash4838 1998
[25] V Hutson ldquoThe existence of an equilibrium for permanentsystemsrdquo The Rocky Mountain Journal of Mathematics vol 20no 4 pp 1033ndash1040 1990
[26] S G Ruan and J J Wei ldquoOn the zeros of transcendentalfunctions with applications to stability of delay differentialequations with two delaysrdquo Dynamics of Continuous Discreteamp Impulsive Systems A vol 10 no 6 pp 863ndash874 2003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 5
According to (35) we have
[
Re 119889 (120582 (1205912))
1198891205912
]
minus1
1205912=120591(119895)
21119896
= Re[[31205822+ 21199012120582 + (119901
1+ 1199021)] 1198901205821205912
120582 (1199031120582 + 1199030)
]
1205912=120591(119895)
21119896
+ Re[ 1199031
120582 (1199031120582 + 1199030)
]
1205912=120591(119895)
21119896
=
1
Λ1
minus 11990311205962
1119896[minus31205962
1119896+ (1199011+ 1199021)] cos (120596
1119896120591(119895)
21119896
)
+ 2119903111990121205963
1119896sin (120596
1119896120591(119895)
21119896
) minus 1199032
11205962
1119896
+ 2119903011990121205962
1119896cos (120596
1119896120591(119895)
21119896
)
+ 1199030[minus31205962
1119896+ (1199011+ 1199021)] 1205961119896sin (120596
1119896120591(119895)
21119896
)
=
1
Λ1
31205966
1119896+ 2 [119901
2
2minus 2 (119901
1+ 1199021)] 1205964
1119896
+ [(1199011+ 1199021)2
minus 21199012(1199010+ 1199020) minus 1199032
1] 1205962
1119896
=
1
Λ1
1199111119896(31199112
1119896+ 2119898121199111119896
+ 11989811)
=
1
Λ1
1199111119896ℎ1015840
1(1199111119896)
(36)
where Λ1= 1199032
11205964
1119896+ 1199032
01205962
1119896gt 0 Notice that Λ
1gt 0 1199111119896
gt 0
sign
[
Re 119889 (120582 (1205912))
1198891205912
]
1205912=120591(119895)
21119896
= sign
[
Re 119889 (120582 (1205912))
1198891205912
]
minus1
1205912=120591(119895)
21119896
(37)
then we have the following lemma
Lemma 5 Suppose that 1199111119896
= 1205962
1119896and ℎ
1015840
1(1199111119896) = 0 where
ℎ1(119911) is defined by (26) then 119889(Re 120582(120591(119895)
21119896
))1198891205912has the same
sign with ℎ1015840
1(1199111119896)
From Lemmas 1 4 and 5 and Theorem 3 we can easilyobtain the following theorem
Theorem6 For 1205911= 0 120591
2gt 0 suppose that (119867
1)ndash(1198675) hold
(i) If 11989810
ge 0 and Δ = 1198982
12minus 311989811
le 0 then all rootsof (10) have negative real parts for all 120591
2ge 0 and the
positive equilibrium 119864lowastis locally asymptotically stable
for all 1205912ge 0
(ii) If either 11989810
lt 0 or 11989810
ge 0 Δ = 1198982
12minus 311989811
gt
0 119911lowast
11gt 0 and ℎ
1(119911lowast
11) le 0 then ℎ
1(119911) has at least one
positive roots and all roots of (23) have negative realparts for 120591
2isin [0 120591
210
) and the positive equilibrium 119864lowast
is locally asymptotically stable for 1205912isin [0 120591
210
)
(iii) If (ii) holds and ℎ1015840
1(1199111119896) = 0 then system (3)undergoes
Hopf bifurcations at the positive equilibrium119864lowastfor 1205912=
120591(119895)
21119896
(119896 = 1 2 3 119895 = 0 1 2 )
Case c Consider1205911gt 0 1205912= 0
The associated characteristic equation of system (3) is
1205823+ 11990121205822+ (1199011+ 1199031) 120582 + (119901
0+ 1199030) + (119902
1120582 + 1199020) 119890minus1205821205911= 0
(38)
Similar to the analysis of Case 119887 we get the followingtheorem
Theorem7 For 1205911gt 0 120591
2= 0 suppose that (119867
1)ndash(1198675) hold
(i) If 11989820
ge 0 and Δ = 1198982
22minus 311989821
le 0 then all rootsof (38) have negative real parts for all 120591
1ge 0 and the
positive equilibrium 119864lowastis locally asymptotically stable
for all 1205911ge 0
(ii) If either 11989820
lt 0 or 11989820
ge 0 Δ = 1198982
22minus 311989821
gt 0119911lowast
21gt 0 and ℎ
2(119911lowast
21) le 0 then ℎ
2(119911) has at least one
positive root 1199112119896 and all roots of (38) have negative real
parts for 1205911isin [0 120591
120
) and the positive equilibrium 119864lowast
is locally asymptotically stable for 1205911isin [0 120591
120
)
(iii) If (ii) holds and ℎ1015840
2(1199112119896) = 0 then system (3) undergoes
Hopf bifurcations at the positive equilibrium119864lowastfor 1205911=
120591(119895)
12119896
(119896 = 1 2 3 119895 = 0 1 2 )
where
11989822
= 1199012
2minus 2 (119901
1+ 1199031)
11989821
= (1199011+ 1199031)2
minus 21199012(1199010+ 1199030) minus 1199022
1
11989820
= (1199010+ 1199030)2
minus 1199022
0
ℎ2(119911) = 119911
3+ 119898221199112+ 11989821119911 + 119898
20 119911lowast
21=
minus11989822
+ radicΔ
3
120591(119895)
12119896
=
1
1205962119896
times arccos ((11990211205964
2119896+ [11990121199020minus (1199031+ 1199011) 1199021] 1205962
2119896
minus 1199020(1199030+ 1199010) )
times (1199022
0+ 1199022
11205962
2119896)
minus1
) + 2119895120587
(39)
where 119896 = 1 2 3 119895 = 0 1 2 then plusmn1198941205962119896is a pair of purely
imaginary roots of (38) corresponding to 120591(119895)
12119896
Define
120591120
= 120591(0)
121198960
= min119896=123
120591(0)
12119896
12059610
= 12059611198960
(40)
6 Abstract and Applied Analysis
Case d Consider1205911gt 0 1205912gt 0 1205911
= 1205912
The associated characteristic equation of system (3) is
1205823+11990121205822+1199011120582+1199010+ (1199021120582 + 1199020) 119890minus1205821205911+ (1199031120582+1199030) 119890minus1205821205912=0
(41)
We consider (41) with 1205912= 120591lowast
2in its stable interval [0 120591
210
)Regard 120591
1as a parameter
Let 120582 = 119894120596 (120596 gt 0) be a root of (41) then we have
minus 1198941205963minus 11990121205962+ 1198941199011120596 + 1199010+ (1198941199021120596 + 1199020) (cos120596120591
1minus 119894 sin120596120591
1)
+ (1199030+ 1198941199031120596) (cos120596120591lowast
2minus 119894 sin120596120591
lowast
2) = 0
(42)
Separating the real and imaginary parts we have
1205963minus 1199011120596 minus 1199031120596 cos120596120591lowast
2+ 1199030sin120596120591
lowast
2
= 1199021120596 cos120596120591
1minus 1199020sin120596120591
1
11990121205962minus 1199010minus 1199030cos120596120591lowast
2minus 1199031120596 sin120596120591
lowast
2
= 1199020cos120596120591
1+ 1199021120596 sin120596120591
1
(43)
It follows that
1205966+ 119898331205964+ 119898321205963+ 119898311205962+ 11989830
= 0 (44)
where
11989833
= 1199012
2minus 21199011minus 21199031cos120596120591lowast
2
11989832
= 2 (1199030minus 11990121199031) sin120596120591
lowast
2
11989831
= 1199012
1minus 211990101199012minus 2 (119901
21199030minus 11990111199031) cos120596120591lowast
2+ 1199032
1minus 1199022
1
11989830
= 1199012
0+ 211990101199030cos120596120591lowast
2+ 1199032
0minus 1199022
0
(45)
Denote 119865(120596) = 1205966+ 119898331205964+ 119898321205963+ 119898311205962+ 11989830 If
11989830
lt 0 then
119865 (0) lt 0 lim120596rarr+infin
119865 (120596) = +infin (46)
We can obtain that (44) has at most six positive roots1205961 1205962 120596
6 For every fixed 120596
119896 119896 = 1 2 6 there exists
a sequence 120591(119895)1119896
| 119895 = 0 1 2 3 such that (43) holdsLet
12059110
= min 120591(119895)
1119896| 119896 = 1 2 6 119895 = 0 1 2 3 (47)
When 1205911
= 120591(119895)
1119896 (41) has a pair of purely imaginary roots
plusmn119894120596(119895)
1119896for 120591lowast2isin [0 120591
210
)In the following we assume that
(H6) ((119889Re(120582))119889120591
1)|120582=plusmn119894120596
(119895)
1119896
= 0
Thus by the general Hopf bifurcation theorem for FDEsin Hale [22] we have the following result on the stability andHopf bifurcation in system (3)
Theorem 8 For 1205911
gt 0 1205912
gt 0 1205911
= 1205912 suppose that
(1198671)ndash(1198676) is satisfied If 119898
30lt 0 and 120591
lowast
2isin [0 120591
210
] thenthe positive equilibrium 119864
lowastis locally asymptotically stable for
1205911
isin [0 12059110) System (3) undergoes Hopf bifurcations at the
positive equilibrium 119864lowastfor 1205911= 120591(119895)
1119896
3 Direction and Stability ofthe Hopf Bifurcation
In Section 2 we obtain the conditions underwhich system (3)undergoes the Hopf bifurcation at the positive equilibrium119864lowast In this section we consider with 120591
2= 120591lowast
2isin [0 120591
210
)
and regard 1205911as a parameter We will derive the explicit
formulas determining the direction stability and period ofthese periodic solutions bifurcating from equilibrium 119864
lowastat
the critical values 1205911by using the normal form and the center
manifold theory developed by Hassard et al [23] Withoutloss of generality denote any one of these critical values 120591
1=
120591(119895)
1119896(119896 = 1 2 6 119895 = 0 1 2 ) by 120591
1 at which (43) has a
pair of purely imaginary roots plusmn119894120596 and system (3) undergoesHopf bifurcation from 119864
lowast
Throughout this section we always assume that 120591lowast2lt 12059110
Let 1199061= 1199091minus 119909lowast
1 1199062= 1199091minus 119909lowast
2 1199063= 1199092minus 119909lowast
3 119905 = 120591
1119905
and 120583 = 1205911minus 1205911 120583 isin R Then 120583 = 0 is the Hopf bifurcation
value of system (3) System (3) may be written as a functionaldifferential equation inC([minus1 0]R3)
(119905) = 119871120583(119906119905) + 119891 (120583 119906
119905) (48)
where 119906 = (1199061 1199062 1199063)119879isin R3 and
119871120583(120601) = (120591
1+ 120583) 119861[
[
1206011(0)
1206012(0)
1206013(0)
]
]
+ (1205911+ 120583)119862[
[
1206011(minus1)
1206012(minus1)
1206013(minus1)
]
]
+ (1205911+ 120583)119863
[[[[[[[[[[
[
1206011(minus
120591lowast
2
1205911
)
1206012(minus
120591lowast
2
1205911
)
1206013(minus
120591lowast
2
1205911
)
]]]]]]]]]]
]
(49)
119891 (120583 120601) = (1205911+ 120583)[
[
1198911
1198912
1198913
]
]
(50)
Abstract and Applied Analysis 7
where 120601 = (1206011 1206012 1206013)119879isin C([minus1 0]R3) and
119861 = [
[
11988711
11988712
0
0 11988722
11988723
0 0 11988733
]
]
119862 = [
[
0 0 0
11988821
0 0
0 0 0
]
]
119863 = [
[
0 0 0
0 0 0
0 11988932
0
]
]
1198911= minus (119886
11+ 1198973) 1206012
1(0) minus 119897
11206011(0) 1206012(0) minus 119897
21206012
1(0) 1206012(0)
minus 11989751206013
1(0) minus 119897
41206013
1(0) 1206012(0) + sdot sdot sdot
1198912= minus11989761206012(0) 1206013(0) minus (119897
7+ 11988622) 1206012
2(0) + 119897
11206011(minus1) 120601
2(0)
+ 11989731206012
1(minus1) minus 119897
81206013
2(0) minus 119897
91206012
2(0) 1206013(0)
+ 11989721206012
1(minus1) 120601
2(0) + 119897
31206013
1(minus1) + 119897
41206013
1(minus1) 120601
2(0)
minus 119897101206013
2(0) 1206013(0) + sdot sdot sdot
1198913= 11989761206012(minus
120591lowast
2
1205911
)1206013(0) + 119897
71206012
2(minus
120591lowast
2
1205911
) minus 119886331206012
3(0)
+ 11989791206012
2(minus
120591lowast
2
1205911
)1206013(0) + 119897
81206013
2(minus
120591lowast
2
1205911
)
+ 119897101206013
2(minus
120591lowast
2
1205911
)1206013(0) + sdot sdot sdot
1199011(119909) =
1198861119909
1 + 1198871119909
1199012(119909) =
1198862119909
1 + 1198872119909
1198971= 1199011015840
1(119909lowast)
1198972=
1
2
11990110158401015840
1(119909lowast) 1198973=
1
2
11990110158401015840
1(119909lowast) 1199101lowast
1198974=
1
3
119901101584010158401015840
1(119909lowast) 1198975=
1
3
119901101584010158401015840
1(119909lowast) 1199101lowast
1198976= 1199011015840
2(1199101lowast) 1198977=
1
2
11990110158401015840
2(1199101lowast) 1199102lowast
1198978=
1
3
119901101584010158401015840
2(1199101lowast) 1199102lowast 1198979=
1
2
11990110158401015840
2(1199101lowast)
11989710
=
1
3
119901101584010158401015840
2(1199101lowast)
(51)
Obviously 119871120583(120601) is a continuous linear function mapping
C([minus1 0]R3) intoR3 By the Riesz representation theoremthere exists a 3 times 3 matrix function 120578(120579 120583) (minus1 ⩽ 120579 ⩽ 0)whose elements are of bounded variation such that
119871120583120601 = int
0
minus1
119889120578 (120579 120583) 120601 (120579) for 120601 isin C ([minus1 0] R3) (52)
In fact we can choose
119889120578 (120579 120583) = (1205911+ 120583) [119861120575 (120579) + 119862120575 (120579 + 1) + 119863120575(120579 +
120591lowast
2
1205911
)]
(53)
where 120575 is Dirac-delta function For 120601 isin C([minus1 0]R3)define
119860 (120583) 120601 =
119889120601 (120579)
119889120579
120579 isin [minus1 0)
int
0
minus1
119889120578 (119904 120583) 120601 (119904) 120579 = 0
119877 (120583) 120601 =
0 120579 isin [minus1 0)
119891 (120583 120601) 120579 = 0
(54)
Then when 120579 = 0 the system is equivalent to
119905= 119860 (120583) 119909
119905+ 119877 (120583) 119909
119905 (55)
where 119909119905(120579) = 119909(119905+120579) 120579 isin [minus1 0] For120595 isin C1([0 1] (R3)
lowast)
define
119860lowast120595 (119904) =
minus
119889120595 (119904)
119889119904
119904 isin (0 1]
int
0
minus1
119889120578119879(119905 0) 120595 (minus119905) 119904 = 0
(56)
and a bilinear inner product
⟨120595 (119904) 120601 (120579)⟩=120595 (0) 120601 (0)minusint
0
minus1
int
120579
120585=0
120595 (120585minus120579) 119889120578 (120579) 120601 (120585) 119889120585
(57)
where 120578(120579) = 120578(120579 0) Let 119860 = 119860(0) then 119860 and 119860lowast
are adjoint operators By the discussion in Section 2 weknow that plusmn119894120596120591
1are eigenvalues of 119860 Thus they are also
eigenvalues of 119860lowast We first need to compute the eigenvectorof 119860 and 119860
lowast corresponding to 1198941205961205911and minus119894120596120591
1 respectively
Suppose that 119902(120579) = (1 120572 120573)1198791198901198941205791205961205911 is the eigenvector of 119860
corresponding to 1198941205961205911 Then 119860119902(120579) = 119894120596120591
1119902(120579) From the
definition of 119860119871120583(120601) and 120578(120579 120583) we can easily obtain 119902(120579) =
(1 120572 120573)1198791198901198941205791205961205911 where
120572 =
119894120596 minus 11988711
11988712
120573 =
11988932(119894120596 minus 119887
11)
11988712(119894120596 minus 119887
33) 119890119894120596120591lowast
2
(58)
and 119902(0) = (1 120572 120573)119879 Similarly let 119902lowast(119904) = 119863(1 120572
lowast 120573lowast)1198901198941199041205961205911
be the eigenvector of 119860lowast corresponding to minus119894120596120591
1 By the
definition of 119860lowast we can compute
120572lowast=
minus119894120596 minus 11988711
119888211198901198941205961205911
120573lowast=
11988723(minus119894120596 minus 119887
11)
11988821(119894120596 minus 119887
33) 1198901198941205961205911
(59)
From (57) we have⟨119902lowast(119904) 119902 (120579)⟩
= 119863(1 120572lowast 120573
lowast
) (1 120572 120573)119879
minus int
0
minus1
int
120579
120585=0
119863(1 120572lowast 120573
lowast
) 119890minus1198941205961205911(120585minus120579)
119889120578 (120579)
times (1 120572 120573)119879
1198901198941205961205911120585119889120585
= 1198631 + 120572120572lowast+ 120573120573
lowast
+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573
lowast
120591lowast
2119890minus119894120596120591lowast
2
(60)
8 Abstract and Applied Analysis
Thus we can choose
119863 = 1 + 120572120572lowast+ 120573120573
lowast
+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573
lowast
120591lowast
2119890minus119894120596120591lowast
2
minus1
(61)
such that ⟨119902lowast(119904) 119902(120579)⟩ = 1 ⟨119902lowast(119904) 119902(120579)⟩ = 0
In the remainder of this section we follow the ideas inHassard et al [23] and use the same notations as there tocompute the coordinates describing the center manifold 119862
0
at 120583 = 0 Let 119909119905be the solution of (48) when 120583 = 0 Define
119911 (119905) = ⟨119902lowast 119909119905⟩ 119882 (119905 120579) = 119909
119905(120579) minus 2Re 119911 (119905) 119902 (120579)
(62)
On the center manifold 1198620 we have
119882(119905 120579) = 119882 (119911 (119905) 119911 (119905) 120579)
= 11988220(120579)
1199112
2
+11988211(120579) 119911119911 +119882
02(120579)
1199112
2
+11988230(120579)
1199113
6
+ sdot sdot sdot
(63)
where 119911 and 119911 are local coordinates for center manifold 1198620in
the direction of 119902 and 119902 Note that 119882 is real if 119909119905is real We
consider only real solutions For the solution 119909119905isin 1198620of (48)
since 120583 = 0 we have
= 1198941205961205911119911 + ⟨119902
lowast(120579) 119891 (0119882 (119911 (119905) 119911 (119905) 120579)
+ 2Re 119911 (119905) 119902 (120579))⟩
= 1198941205961205911119911+119902lowast(0) 119891 (0119882 (119911 (119905) 119911 (119905) 0) +2Re 119911 (119905) 119902 (0))
= 1198941205961205911119911 + 119902lowast(0) 1198910(119911 119911) ≜ 119894120596120591
1119911 + 119892 (119911 119911)
(64)
where
119892 (119911 119911) = 119902lowast(0) 1198910(119911 119911) = 119892
20
1199112
2
+ 11989211119911119911
+ 11989202
1199112
2
+ 11989221
1199112119911
2
+ sdot sdot sdot
(65)
By (62) we have 119909119905(120579) = (119909
1119905(120579) 1199092119905(120579) 1199093119905(120579))119879= 119882(119905 120579) +
119911119902(120579) + 119911 119902(120579) and then
1199091119905(0) = 119911 + 119911 +119882
(1)
20(0)
1199112
2
+119882(1)
11(0) 119911119911
+119882(1)
02(0)
1199112
2
+ 119900 (|(119911 119911)|3)
1199092119905(0) = 119911120572 + 119911 120572 +119882
(2)
20(0)
1199112
2
+119882(2)
11(0) 119911119911 +119882
(2)
02(0)
1199112
2
+ 119900 (|(119911 119911)|3)
1199093119905(0) = 119911120573 + 119911120573 +119882
(3)
20(0)
1199112
2
+119882(3)
11(0) 119911119911
+119882(3)
02(0)
1199112
2
+ 119900 (|(119911 119911)|3)
1199091119905(minus1) = 119911119890
minus1198941205961205911+ 1199111198901198941205961205911+119882(1)
20(minus1)
1199112
2
+119882(1)
11(minus1) 119911119911 +119882
(1)
02(minus1)
1199112
2
+ 119900 (|(119911 119911)|3)
1199092119905(minus1) = 119911120572119890
minus1198941205961205911+ 119911 120572119890
1198941205961205911+119882(2)
20(minus1)
1199112
2
+119882(2)
11(minus1) 119911119911 +119882
(2)
02(minus1)
1199112
2
+ 119900 (|(119911 119911)|3)
1199093119905(minus1) = 119911120573119890
minus1198941205961205911+ 119911120573119890
1205961205911+119882(3)
20(minus1)
1199112
2
+119882(3)
11(minus1) 119911119911 +119882
(3)
02(minus1)
1199112
2
+ 119900 (|(119911 119911)|3)
1199091119905(minus
120591lowast
2
12059110
) = 119911119890minus119894120596120591lowast
2+ 119911119890119894120596120591lowast
2+119882(1)
20(minus
120591lowast
2
1205911
)
1199112
2
+119882(1)
11(minus
120591lowast
2
1205911
)119911119911 +119882(1)
02(minus
120591lowast
2
1205911
)
1199112
2
+ 119900 (|(119911 119911)|3)
1199092119905(minus
120591lowast
2
1205911
) = 119911120572119890minus119894120596120591lowast
2+ 119911 120572119890
119894120596120591lowast
2+119882(2)
20(minus
120591lowast
2
1205911
)
1199112
2
+119882(2)
11(minus
120591lowast
2
1205911
)119911119911 +119882(2)
02(minus
120591lowast
2
1205911
)
1199112
2
+ 119900 (|(119911 119911)|3)
1199093119905(minus
120591lowast
2
1205911
) = 119911120573119890minus119894120596120591lowast
2+ 119911120573119890
120596120591lowast
2+119882(3)
20(minus
120591lowast
2
1205911
)
1199112
2
+119882(3)
11(minus
120591lowast
2
1205911
)119911119911 +119882(3)
02(minus
120591lowast
2
1205911
)
1199112
2
+ 119900 (|(119911 119911)|3)
(66)
Abstract and Applied Analysis 9
It follows together with (50) that
119892 (119911 119911)=119902lowast(0) 1198910(119911 119911) =119863120591
1(1 120572lowast 120573
lowast
) (119891(0)
1119891(0)
2119891(0)
3)
119879
= 1198631205911[minus (119886
11+ 1198973) 1206012
1(0) minus 119897
11206011(0) 1206012(0)
minus 11989721206012
1(0) 1206012(0) minus 119897
51206013
1(0) minus 119897
41206013
1(0) 1206012(0)
+ sdot sdot sdot ]
+ 120572lowast[11989761206012(0) 1206013(0) (1198977+ 11988622) 1206012
2(0)
+ 11989711206011(minus1) 120601
2(0)
+ 11989731206012
1(minus1) minus 119897
81206013
2(0) minus 119897
91206012
2(0) 1206013(0)
+ 11989721206012
1(minus1) 120601
2(0) + 119897
31206013
1(minus1) + sdot sdot sdot ]
+ 120573
lowast
[11989761206012(minus
120591lowast
2
1205911
)1206013(0) + 119897
71206012
2(minus
120591lowast
2
1205911
)
minus 119886331206012
3(0) + 119897
81206013
2(minus
120591lowast
2
1205911
) + sdot sdot sdot ]
(67)
Comparing the coefficients with (65) we have
11989220
= 1198631205911[minus2 (119886
11+ 1198973) minus 2120572119897
1]
+ 120572lowast[21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591
1
minus21198976120572120573 minus 2 (119897
7+ 11988622) 1205722]
+ 120573
lowast
[21198976120572120573119890minus119894120596120591lowast
2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591
lowast
2
minus2119886331205732]
11989211
= 1198631205911[minus2 (119886
11+ 1198973) minus 1198971(120572 + 120572)]
+ 120572lowast[1198971(1205721198901198941205961205911+ 120572119890minus1198941205961205911) minus 1198976(120572120573 + 120572120573)
minus2 (1198977+ 11988622) 120572120572 + 2119897
3]
+ 120573
lowast
[1198976(120573120572119890119894120596120591lowast
2+ 120572120573119890
minus119894120596120591lowast
2)
+ 21198977120572120572 minus 119886
33120573120573]
11989202
= 21198631205911[minus2 (119886
11+ 1198973) minus 21198971120572]
+ 120572lowast[minus21198976120572120573 minus 2 (119897
7+ 11988622) 1205722+ 211989711205721198901198941205961205911
+2119897311989021198941205961205911]
+ 120573
lowast
[21198976120572120573119890119894120596120591lowast
2+ 2119897712057221198902119894120596120591lowast
2minus 11988633120573
2
]
11989221
= 1198631205911[minus (119886
11+ 1198973) (2119882
(1)
20(0) + 4119882
(1)
11(0))
minus 1198971(2120572119882
(1)
11(0) + 120572119882
(1)
20(0) + 119882
(2)
20(0)
+ 2119882(2)
11(0))]
+ 120572lowast[minus1198976(2120573119882
(2)
11(0) + 120572119882
(3)
20(0) + 120573119882
(2)
20(0)
+ 2120572119882(3)
11(0)) minus (119897
7+ 11988622)
times (4120572119882(2)
11(0) + 2120572119882
(2)
20(0))
+ 1198971(2120572119882
(1)
11(minus1) + 120572119882
(1)
20(minus1)
+ 119882(2)
20(0) 1198901198941205961205911+ 2119882
(2)
11(0) 119890minus1198941205961205911)
+ 1198973(4119882(1)
11(minus1) 119890
minus1198941205961205911+2119882(1)
20(minus1) 119890
1198941205961205911)]
+ 120573
lowast
[1198976(2120573119882
(2)
11(minus
120591lowast
2
1205911
) + 120573119882(2)
20(minus
120591lowast
2
1205911
)
+ 120572119882(3)
20(0) 119890119894120596120591lowast
2+ 2120572119882
(3)
11(0) 119890minus119894120596120591lowast
2)
+ 1198977(4120572119882
(2)
11(minus
120591lowast
2
1205911
) 119890minus119894120596120591lowast
2
+2120572119882(2)
20(minus
120591lowast
2
1205911
) 119890119894120596120591lowast
2)
minus 11988633(4120573119882
(3)
11(0) + 2120573119882
(3)
20(0))]
(68)
where
11988220(120579) =
11989411989220
1205961205911
119902 (0) 1198901198941205961205911120579+
11989411989202
31205961205911
119902 (0) 119890minus1198941205961205911120579+ 119864111989021198941205961205911120579
11988211(120579) = minus
11989411989211
1205961205911
119902 (0) 1198901198941205961205911120579+
11989411989211
1205961205911
119902 (0) 119890minus1198941205961205911120579+ 1198642
1198641= 2
[[[[[
[
2119894120596 minus 11988711
minus11988712
0
minus11988821119890minus2119894120596120591
12119894120596 minus 119887
22minus11988723
0 minus11988932119890minus2119894120596120591
lowast
22119894120596 minus 119887
33
]]]]]
]
minus1
times
[[[[[
[
minus2 (11988611
+ 1198973) minus 2120572119897
1
21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591
1minus 21198976120572120573 minus 2 (119897
7+ 11988622) 1205722
21198976120572120573119890minus119894120596120591lowast
2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591
lowast
2minus 2119886331205732
]]]]]
]
10 Abstract and Applied Analysis
1198642= 2
[[[[
[
minus11988711
minus11988712
0
minus11988821
minus11988722
minus11988723
0 minus11988932
minus11988733
]]]]
]
minus1
times
[[[[
[
minus2 (11988611
+ 1198973) minus 1198971(120572 + 120572)
1198971(1205721198901198941205961205911+ 120572119890minus1198941205961205911) minus 1198976(120572120573 + 120572120573) minus 2 (119897
7+ 11988622) 120572120572 + 2119897
3
1198976(120573120572119890119894120596120591lowast
2+ 120572120573119890
minus119894120596120591lowast
2) + 21198977120572120572 minus 119886
33120573120573
]]]]
]
(69)
Thus we can determine11988220(120579) and119882
11(120579) Furthermore
we can determine each 119892119894119895by the parameters and delay in (3)
Thus we can compute the following values
1198881(0) =
119894
21205961205911
(1198922011989211
minus 2100381610038161003816100381611989211
1003816100381610038161003816
2
minus
1
3
100381610038161003816100381611989202
1003816100381610038161003816
2
) +
1
2
11989221
1205832= minus
Re 1198881(0)
Re 1205821015840 (1205911)
1198792= minus
Im 1198881(0) + 120583
2Im 120582
1015840(1205911)
1205961205911
1205732= 2Re 119888
1(0)
(70)
which determine the quantities of bifurcating periodic solu-tions in the center manifold at the critical value 120591
1 Suppose
Re1205821015840(1205911) gt 0 120583
2determines the directions of the Hopf
bifurcation if 1205832
gt 0(lt 0) then the Hopf bifurcationis supercritical (subcritical) and the bifurcation exists for120591 gt 120591
1(lt 1205911) 1205732determines the stability of the bifurcation
periodic solutions the bifurcating periodic solutions arestable (unstable) if 120573
2lt 0(gt 0) and119879
2determines the period
of the bifurcating periodic solutions the period increases(decreases) if 119879
2gt 0(lt 0)
4 Numerical Simulation
We consider system (3) by taking the following coefficients1198861
= 03 11988611
= 58889 11988612
= 1 1198981
= 1 1198862
=
01 11988621
= 27 11988622
= 12 11988623
= 12 1198982
= 1 1198863
=
02 11988632
= 25 11988633
= 12 We have the unique positiveequilibrium 119864
lowast= (00451 00357 00551)
By computation we get 11989810
= minus00104 12059611
= 0516411991111
= 02666 ℎ10158401(11991111) = 03402 120591
20
= 32348 FromTheorem 6 we know that when 120591
1= 0 the positive
equilibrium 119864lowastis locally asymptotically stable for 120591
2isin
[0 32348) When 1205912crosses 120591
20
the equilibrium 119864lowastloses its
stability and Hopf bifurcation occurs From the algorithm inSection 3 we have 120583
2= 56646 120573
2= minus31583 119879
2= 5445
whichmeans that the bifurcation is supercritical and periodicsolution is stable The trajectories and the phase graphs areshown in Figures 1 and 2
Regarding 1205911as a parameter and let 120591
2= 29 isin
[0 32348) we can observe that with 1205911increasing the pos-
itive equilibrium 119864lowastloses its stability and Hopf bifurcation
occurs (see Figures 3 and 4)
5 Global Continuationof Local Hopf Bifurcations
In this section we study the global continuation of peri-odic solutions bifurcating from the positive equilibrium(119864lowast 120591(119895)
1119896) (119896 = 1 2 6 119895 = 0 1 ) Throughout this
section we follow closely the notations in [24] and assumethat 120591
2= 120591lowast
2isin [0 120591
210
) regarding 1205911as a parameter For
simplification of notations setting 119911119905(119905) = (119909
1119905 1199092119905 1199093119905)119879 we
may rewrite system (3) as the following functional differentialequation
(119905) = 119865 (119911119905 1205911 119901) (71)
where 119911119905(120579) = (119909
1119905(120579) 1199092119905(120579) 1199093119905(120579))119879
= (1199091(119905 + 120579) 119909
2(119905 +
120579) 1199093(119905 + 120579))
119879 for 119905 ge 0 and 120579 isin [minus1205911 0] Since 119909
1(119905) 1199092(119905)
and 1199093(119905) denote the densities of the prey the predator and
the top predator respectively the positive solution of system(3) is of interest and its periodic solutions only arise in the firstquadrant Thus we consider system (3) only in the domain1198773
+= (1199091 1199092 1199093) isin 1198773 1199091gt 0 119909
2gt 0 119909
3gt 0 It is obvious
that (71) has a unique positive equilibrium 119864lowast(119909lowast
1 119909lowast
2 119909lowast
3) in
1198773
+under the assumption (119867
1)ndash(1198674) Following the work of
[24] we need to define
119883 = 119862([minus1205911 0] 119877
3
+)
Γ = 119862119897 (119911 1205911 119901) isin X times R times R+
119911 is a 119901-periodic solution of system (71)
N = (119911 1205911 119901) 119865 (119911 120591
1 119901) = 0
(72)
Let ℓ(119864lowast120591(119895)
11198962120587120596
(119895)
1119896)denote the connected component pass-
ing through (119864lowast 120591(119895)
1119896 2120587120596
(119895)
1119896) in Γ where 120591
(119895)
1119896is defined by
(43) We know that ℓ(119864lowast120591(119895)
11198962120587120596
(119895)
1119896)through (119864
lowast 120591(119895)
1119896 2120587120596
(119895)
1119896)
is nonemptyFor the benefit of readers we first state the global Hopf
bifurcation theory due to Wu [24] for functional differentialequations
Lemma 9 Assume that (119911lowast 120591 119901) is an isolated center satis-
fying the hypotheses (A1)ndash(A4) in [24] Denote by ℓ(119911lowast120591119901)
theconnected component of (119911
lowast 120591 119901) in Γ Then either
(i) ℓ(119911lowast120591119901)
is unbounded or(ii) ℓ(119911lowast120591119901)
is bounded ℓ(119911lowast120591119901)
cap Γ is finite and
sum
(119911120591119901)isinℓ(119911lowast120591119901)capN
120574119898(119911lowast 120591 119901) = 0 (73)
for all 119898 = 1 2 where 120574119898(119911lowast 120591 119901) is the 119898119905ℎ crossing
number of (119911lowast 120591 119901) if119898 isin 119869(119911
lowast 120591 119901) or it is zero if otherwise
Clearly if (ii) in Lemma 9 is not true then ℓ(119911lowast120591119901)
isunbounded Thus if the projections of ℓ
(119911lowast120591119901)
onto 119911-spaceand onto 119901-space are bounded then the projection of ℓ
(119911lowast120591119901)
onto 120591-space is unbounded Further if we can show that the
Abstract and Applied Analysis 11
0 200 400 60000451
00451
00452
t
x1(t)
t
0 200 400 60000356
00357
00358
x2(t)
t
0 200 400 6000055
00551
00552
x3(t)
00451
00451
00452
00356
00357
00357
003570055
0055
00551
00551
00552
x 1(t)
x2 (t)
x3(t)
Figure 1 The trajectories and the phase graph with 1205911= 0 120591
2= 29 lt 120591
20= 32348 119864
lowastis locally asymptotically stable
0 500 1000004
0045
005
0 500 1000002
004
006
0 500 10000
005
01
t
t
t
x1(t)
x2(t)
x3(t)
0044
0046
0048
002
003
004
005003
004
005
006
007
008
x 1(t)
x2 (t)
x3(t)
Figure 2 The trajectories and the phase graph with 1205911= 0 120591
2= 33 gt 120591
20= 32348 a periodic orbit bifurcate from 119864
lowast
12 Abstract and Applied Analysis
0 500 1000 1500
00451
00452
0 500 1000 15000035
0036
0037
0 500 1000 15000054
0055
0056
t
t
t
x1(t)
x2(t)
x3(t)
0045100451
0045200452
0035
00355
003600546
00548
0055
00552
00554
00556
00558
x1(t)
x2 (t)
x3(t)
Figure 3 The trajectories and the phase graph with 1205911= 09 120591
2= 29 119864
lowastis locally asymptotically stable
0 500 1000 15000044
0045
0046
0 500 1000 1500002
003
004
0 500 1000 1500004
006
008
t
t
t
x1(t)
x2(t)
x3(t)
00445
0045
00455
0032
0034
0036
00380052
0054
0056
0058
006
0062
0064
x 1(t)
x2 (t)
x3(t)
Figure 4 The trajectories and the phase graph with 1205911= 12 120591
2= 29 a periodic orbit bifurcate from 119864
lowast
Abstract and Applied Analysis 13
projection of ℓ(119911lowast120591119901)
onto 120591-space is away from zero thenthe projection of ℓ
(119911lowast120591119901)
onto 120591-space must include interval[120591 infin) Following this ideal we can prove our results on theglobal continuation of local Hopf bifurcation
Lemma 10 If the conditions (1198671)ndash(1198674) hold then all nontriv-
ial periodic solutions of system (71) with initial conditions
1199091(120579) = 120593 (120579) ge 0 119909
2(120579) = 120595 (120579) ge 0
1199093(120579) = 120601 (120579) ge 0 120579 isin [minus120591
1 0)
120593 (0) gt 0 120595 (0) gt 0 120601 (0) gt 0
(74)
are uniformly bounded
Proof Suppose that (1199091(119905) 1199092(119905) 1199093(119905)) are nonconstant peri-
odic solutions of system (3) and define
1199091(1205851) = min 119909
1(119905) 119909
1(1205781) = max 119909
1(119905)
1199092(1205852) = min 119909
2(119905) 119909
2(1205782) = max 119909
2(119905)
1199093(1205853) = min 119909
3(119905) 119909
3(1205783) = max 119909
3(119905)
(75)
It follows from system (3) that
1199091(119905) = 119909
1(0) expint
119905
0
[1198861minus 119886111199091(119904) minus
119886121199092(119904)
1198981+ 1199091(119904)
] 119889119904
1199092(119905) = 119909
2(0) expint
119905
0
[minus1198862+
119886211199091(119904 minus 1205911)
1198981+ 1199091(119904 minus 1205911)
minus 119886221199092(119904) minus
119886231199093(119904)
1198982+ 1199092(119904)
] 119889119904
1199093(119905) = 119909
3(0) expint
119905
0
[minus1198863+
119886321199092(119904 minus 120591
lowast
2)
1198982+ 1199092(119904 minus 120591
lowast
2)
minus 119886331199093(119904)] 119889119904
(76)
which implies that the solutions of system (3) cannot crossthe 119909119894-axis (119894 = 1 2 3) Thus the nonconstant periodic orbits
must be located in the interior of first quadrant It followsfrom initial data of system (3) that 119909
1(119905) gt 0 119909
2(119905) gt
0 1199093(119905) gt 0 for 119905 ge 0
From the first equation of system (3) we can get
0 = 1198861minus 119886111199091(1205781) minus
119886121199092(1205781)
1198981+ 1199091(1205781)
le 1198861minus 119886111199091(1205781) (77)
thus we have
1199091(1205781) le
1198861
11988611
(78)
From the second equation of (3) we obtain
0 = minus 1198862+
119886211199091(1205782minus 1205911)
1198981+ 1199091(1205782minus 1205911)
minus 119886221199092(1205782)
minus
119886231199093(1205782)
1198982+ 1199092(1205782)
le minus1198862+
11988621(119886111988611)
1198981+ (119886111988611)
minus 119886221199092(1205782)
(79)
therefore one can get
1199092(1205782) le
minus1198862(119886111198981+ 1198861) + 119886111988621
11988622(119886111198981+ 1198861)
≜ 1198721 (80)
Applying the third equation of system (3) we know
0 = minus1198863+
119886321199092(1205783minus 120591lowast
2)
1198982+ 1199092(1205783minus 120591lowast
2)
minus 119886331199093(1205783)
le minus1198863+
119886321198721
1198982+1198721
minus 119886331199093(1205783)
(81)
It follows that
1199093(1205783) le
minus1198863(1198982+1198721) + 119886321198721
11988633(1198982+1198721)
≜ 1198722 (82)
This shows that the nontrivial periodic solution of system (3)is uniformly bounded and the proof is complete
Lemma 11 If the conditions (1198671)ndash(1198674) and
(H7) 11988622
minus (119886211198982) gt 0 [(119886
11minus 11988611198981)11988622
minus
(11988612119886211198982
1)]11988633
minus (11988611
minus (11988611198981))[(119886211198982)11988633
+
(11988623119886321198982
2)] gt 0
hold then system (3) has no nontrivial 1205911-periodic solution
Proof Suppose for a contradiction that system (3) has non-trivial periodic solution with period 120591
1 Then the following
system (83) of ordinary differential equations has nontrivialperiodic solution
1198891199091
119889119905
= 1199091(119905) [1198861minus 119886111199091(119905) minus
119886121199092(119905)
1198981+ 1199091(119905)
]
1198891199092
119889119905
= 1199092(119905) [minus119886
2+
119886211199091(119905)
1198981+ 1199091(119905)
minus 119886221199092(119905) minus
119886231199093(119905)
1198982+ 1199092(119905)
]
1198891199093
119889119905
= 1199093(119905) [minus119886
3+
119886321199092(119905 minus 120591
lowast
2)
1198982+ 1199092(119905 minus 120591
lowast
2)
minus 119886331199093(119905)]
(83)
which has the same equilibria to system (3) that is
1198641= (0 0 0) 119864
2= (
1198861
11988611
0 0)
1198643= (1199091 1199092 0) 119864
lowast= (119909lowast
1 119909lowast
2 119909lowast
3)
(84)
Note that 119909119894-axis (119894 = 1 2 3) are the invariable manifold of
system (83) and the orbits of system (83) do not intersect each
14 Abstract and Applied Analysis
other Thus there are no solutions crossing the coordinateaxes On the other hand note the fact that if system (83) hasa periodic solution then there must be the equilibrium in itsinterior and that 119864
1 1198642 1198643are located on the coordinate
axis Thus we conclude that the periodic orbit of system (83)must lie in the first quadrant If (119867
7) holds it is well known
that the positive equilibrium 119864lowastis globally asymptotically
stable in the first quadrant (see [1])Thus there is no periodicorbit in the first quadrant too The above discussion meansthat (83) does not have any nontrivial periodic solution It isa contradiction Therefore the lemma is confirmed
Theorem 12 Suppose the conditions of Theorem 8 and (1198677)
hold let120596119896and 120591(119895)1119896
be defined in Section 2 then when 1205911gt 120591(119895)
1119896
system (3) has at least 119895 minus 1 periodic solutions
Proof It is sufficient to prove that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is [120591
1 +infin) for each 119895 ge 1 where
1205911le 120591(119895)
1119896
In following we prove that the hypotheses (A1)ndash(A4) in[24] hold
(1) From system (3) we know easily that the followingconditions hold
(A1) 119865 isin 1198622(1198773
+times119877+times119877+) where 119865 = 119865|
1198773
+times119877+times119877+
rarr
1198773
+
(A3) 119865(120601 1205911 119901) is differential with respect to 120601
(2) It follows from system (3) that
119863119911119865 (119911 120591
1 119901) =
[[[[[[[[[[[
[
1198861minus 2119886111199091minus
1198861211989811199092
(1198981+ 1199091)2
minus
119886121199091
1198981+ 1199091
0
1198862111989811199092
(1198981+ 1199091)2
minus1198862+
119886211199091
1198981+ 1199091
minus 2119886221199092minus
1198862311989821199093
(1198982+ 1199092)2
minus
119886231199092
1198982+ 1199092
0
1198863211989821199093
(1198982+ 1199092)2
minus1198863+
119886321199092
1198982+ 1199092
minus 2119886331199093
]]]]]]]]]]]
]
(85)
Then under the assumption (1198671)ndash(1198674) we have
det119863119911119865 (119911lowast 1205911 119901)
= det
[[[[[[[[
[
minus11988611119909lowast
1+
11988612119909lowast
1119909lowast
2
(1198981+ 119909lowast
1)2
11988612119909lowast
1
1198981+ 119909lowast
1
0
119886211198981119909lowast
2
(1198981+ 119909lowast
1)2
minus11988622119909lowast
2+
11988623119909lowast
2119909lowast
3
(1198982+ 119909lowast
2)2
minus
11988623119909lowast
2
1198982+ 119909lowast
2
0
119886321198982119909lowast
3
(1198982+ 119909lowast
2)2
minus11988633119909lowast
3
]]]]]]]]
]
= minus
1198862
21199101lowast1199102lowast
(1 + 11988721199101lowast)3[minus119909lowast+
11988611198871119909lowast1199101lowast
(1 + 1198871119909lowast)2] = 0
(86)
From (86) we know that the hypothesis (A2) in [24]is satisfied
(3) The characteristic matrix of (71) at a stationarysolution (119911 120591
0 1199010) where 119911 = (119911
(1) 119911(2) 119911(3)) isin 119877
3
takes the following form
Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863
120601119865 (119911 120591
1 119901) (119890
120582119868) (87)
that is
Δ (119911 1205911 119901) (120582)
=
[[[[[[[[[[[[[[[[[
[
120582 minus 1198861+ 211988611119911(1)
+
119886121198981119911(2)
(1198981+ 119911(1))
2
11988612119911(1)
1198981+ 119911(1)
0
minus
119886211198981119911(2)
(1198981+ 119911(1))
2119890minus1205821205911
120582 + 1198862minus
11988621119911(1)
1198981+ 119911(1)
+ 211988622119911(2)
+
119886231198982119911(3)
(1198982+ 119911(2))
2
11988623119911(2)
1198982+ 119911(2)
0 minus
119886321198982119911(3)
(1198982+ 119911(2))
2119890minus120582120591lowast
2120582 + 1198863minus
11988632119911(2)
1198982+ 119911(2)
+ 211988633119911(3)
]]]]]]]]]]]]]]]]]
]
(88)
Abstract and Applied Analysis 15
From (88) we have
det (Δ (119864lowast 1205911 119901) (120582))
= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902
0+ 1199030)] 119890minus1205821205911
(89)
Note that (89) is the same as (20) from the discussion inSection 2 about the local Hopf bifurcation it is easy to verifythat (119864
lowast 120591(119895)
1119896 2120587120596
119896) is an isolated center and there exist 120598 gt
0 120575 gt 0 and a smooth curve 120582 (120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575) rarr C
such that det(Δ(120582(1205911))) = 0 |120582(120591
1) minus 120596119896| lt 120598 for all 120591
1isin
[120591(119895)
1119896minus 120575 120591(119895)
1119896+ 120575] and
120582 (120591(119895)
1119896) = 120596119896119894
119889Re 120582 (1205911)
1198891205911
1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)
1119896
gt 0 (90)
Let
Ω1205982120587120596
119896
= (120578 119901) 0 lt 120578 lt 120598
10038161003816100381610038161003816100381610038161003816
119901 minus
2120587
120596119896
10038161003816100381610038161003816100381610038161003816
lt 120598 (91)
It is easy to see that on [120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575] times 120597Ω
1205982120587120596119896
det(Δ(119864
lowast 1205911 119901)(120578 + (2120587119901)119894)) = 0 if and only if 120578 = 0
1205911= 120591(119895)
1119896 119901 = 2120587120596
119896 119896 = 1 2 3 119895 = 0 1 2
Therefore the hypothesis (A4) in [24] is satisfiedIf we define
119867plusmn(119864lowast 120591(119895)
1119896
2120587
120596119896
) (120578 119901)
= det(Δ (119864lowast 120591(119895)
1119896plusmn 120575 119901) (120578 +
2120587
119901
119894))
(92)
then we have the crossing number of isolated center(119864lowast 120591(119895)
1119896 2120587120596
119896) as follows
120574(119864lowast 120591(119895)
1119896
2120587
120596119896
) = deg119861(119867minus(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
minus deg119861(119867+(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
= minus1
(93)
Thus we have
sum
(1199111205911119901)isinC
(119864lowast120591(119895)
11198962120587120596119896)
120574 (119911 1205911 119901) lt 0
(94)
where (119911 1205911 119901) has all or parts of the form
(119864lowast 120591(119896)
1119895 2120587120596
119896) (119895 = 0 1 ) It follows from
Lemma 9 that the connected component ℓ(119864lowast120591(119895)
11198962120587120596
119896)
through (119864lowast 120591(119895)
1119896 2120587120596
119896) is unbounded for each center
(119911lowast 1205911 119901) (119895 = 0 1 ) From the discussion in Section 2
we have
120591(119895)
1119896=
1
120596119896
times arccos(((11990121205962
119896minus1199010) (1199030+1199020)
+ 1205962(1205962minus1199011) (1199031+1199021))
times ((1199030+1199020)2
+ (1199031+1199021)2
1205962
119896)
minus1
) +2119895120587
(95)
where 119896 = 1 2 3 119895 = 0 1 Thus one can get 2120587120596119896le 120591(119895)
1119896
for 119895 ge 1Now we prove that the projection of ℓ
(119864lowast120591(119895)
11198962120587120596
119896)onto
1205911-space is [120591
1 +infin) where 120591
1le 120591(119895)
1119896 Clearly it follows
from the proof of Lemma 11 that system (3) with 1205911
= 0
has no nontrivial periodic solution Hence the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is away from zero
For a contradiction we suppose that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is bounded this means that the
projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is included in a
interval (0 120591lowast) Noticing 2120587120596119896lt 120591119895
1119896and applying Lemma 11
we have 119901 lt 120591lowast for (119911(119905) 120591
1 119901) belonging to ℓ
(119864lowast120591(119895)
11198962120587120596
119896)
This implies that the projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 119901-
space is bounded Then applying Lemma 10 we get thatthe connected component ℓ
(119864lowast120591(119895)
11198962120587120596
119896)is bounded This
contradiction completes the proof
6 Conclusion
In this paper we take our attention to the stability and Hopfbifurcation analysis of a predator-prey systemwithMichaelis-Menten type functional response and two unequal delaysWe obtained some conditions for local stability and Hopfbifurcation occurring When 120591
1= 1205912 we derived the explicit
formulas to determine the properties of periodic solutionsby the normal form method and center manifold theoremSpecially the global existence results of periodic solutionsbifurcating from Hopf bifurcations are also established byusing a global Hopf bifurcation result due to Wu [24]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to thank the anonymous referees fortheir careful reading and constructive suggestions which leadto truly significant improvement of the paper This researchis supported by the National Natural Science Foundation ofChina (no 11061016)
16 Abstract and Applied Analysis
References
[1] R Xu and M A J Chaplain ldquoPersistence and global stabilityin a delayed predator-prey system with Michaelis-Menten typefunctional responserdquo Applied Mathematics and Computationvol 130 no 2-3 pp 441ndash455 2002
[2] Y L Song Y H Peng and J JWei ldquoBifurcations for a predator-prey system with two delaysrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 466ndash479 2008
[3] G-P Hu W-T Li and X-P Yan ldquoHopf bifurcations in apredator-prey system with multiple delaysrdquo Chaos Solitons ampFractals vol 42 no 2 pp 1273ndash1285 2009
[4] M Liao X Tang and C Xu ldquoBifurcation analysis for a three-species predator-prey systemwith two delaysrdquoCommunicationsin Nonlinear Science and Numerical Simulation vol 17 no 1 pp183ndash194 2012
[5] C Xu and P Li ldquoDynamical analysis in a delayed predator-prey model with two delaysrdquo Discrete Dynamics in Nature andSociety vol 2012 Article ID 652947 22 pages 2012
[6] J-F Zhang ldquoStability and bifurcation periodic solutions ina Lotka-Volterra competition system with multiple delaysrdquoNonlinear Dynamics vol 70 no 1 pp 849ndash860 2012
[7] J Xia Z Yu and R Yuan ldquoStability and Hopf bifurcation ina symmetric Lotka-Volterra predator-prey system with delaysrdquoElectronic Journal of Differential Equations vol 2013 no 9 pp1ndash16 2013
[8] H Wan and J Cui ldquoA malaria model with two delaysrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 601265 8pages 2013
[9] M Zhao ldquoHopf bifurcation analysis for a semiratio-dependentpredator-prey system with two delaysrdquo Abstract and AppliedAnalysis vol 2013 Article ID 495072 13 pages 2013
[10] S Feyissa and S Banerjee ldquoDelay-induced oscillatory dynamicsin humoral mediated immune response with two time delaysrdquoNonlinear Analysis Real World Applications vol 14 no 1 pp35ndash52 2013
[11] G Zhang Y Shen andB Chen ldquoHopf bifurcation of a predator-prey systemwith predator harvesting and twodelaysrdquoNonlinearDynamics vol 73 no 4 pp 2119ndash2131 2013
[12] X-Y Meng H-F Huo and H Xiang ldquoHopf bifurcation in athree-species systemwith delaysrdquo Journal of AppliedMathemat-ics and Computing vol 35 no 1-2 pp 635ndash661 2011
[13] J-F Zhang ldquoGlobal existence of bifurcated periodic solutions ina commensalism model with delaysrdquo Applied Mathematics andComputation vol 218 no 23 pp 11688ndash11699 2012
[14] Y Wang W Jiang and H Wang ldquoStability and global Hopfbifurcation in toxic phytoplankton-zooplankton model withdelay and selective harvestingrdquoNonlinear Dynamics vol 73 no1-2 pp 881ndash896 2013
[15] Y Ma ldquoGlobal Hopf bifurcation in the Leslie-Gower predator-prey model with two delaysrdquo Nonlinear Analysis Real WorldApplications vol 13 no 1 pp 370ndash375 2012
[16] X-Y Meng H-F Huo and X-B Zhang ldquoStability and globalHopf bifurcation in a delayed food web consisting of a preyand two predatorsrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4335ndash4348 2011
[17] Y Song J Wei andM Han ldquoLocal and global Hopf bifurcationin a delayed hematopoiesis modelrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol14 no 11 pp 3909ndash3919 2004
[18] Y Song and J Wei ldquoLocal Hopf bifurcation and global periodicsolutions in a delayed predator-prey systemrdquo Journal of Mathe-matical Analysis and Applications vol 301 no 1 pp 1ndash21 2005
[19] J Wei and M Y Li ldquoGlobal existence of periodic solutions in atri-neuron network model with delaysrdquo Physica D vol 198 no1-2 pp 106ndash119 2004
[20] T Zhao Y Kuang and H L Smith ldquoGlobal existence of peri-odic solutions in a class of delayed Gause-type predator-preysystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 28 no 8 pp 1373ndash1394 1997
[21] S J Gao L S Chen and Z D Teng ldquoHopf bifurcation andglobal stability for a delayed predator-prey system with stagestructure for predatorrdquo Applied Mathematics and Computationvol 202 no 2 pp 721ndash729 2008
[22] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977
[23] B Hassard D Kazarinoff and Y WanTheory and Applicationsof Hopf Bifurcation Cambridge University Press CambridgeUK 1981
[24] J H Wu ldquoSymmetric functional-differential equations andneural networks with memoryrdquo Transactions of the AmericanMathematical Society vol 350 no 12 pp 4799ndash4838 1998
[25] V Hutson ldquoThe existence of an equilibrium for permanentsystemsrdquo The Rocky Mountain Journal of Mathematics vol 20no 4 pp 1033ndash1040 1990
[26] S G Ruan and J J Wei ldquoOn the zeros of transcendentalfunctions with applications to stability of delay differentialequations with two delaysrdquo Dynamics of Continuous Discreteamp Impulsive Systems A vol 10 no 6 pp 863ndash874 2003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Abstract and Applied Analysis
Case d Consider1205911gt 0 1205912gt 0 1205911
= 1205912
The associated characteristic equation of system (3) is
1205823+11990121205822+1199011120582+1199010+ (1199021120582 + 1199020) 119890minus1205821205911+ (1199031120582+1199030) 119890minus1205821205912=0
(41)
We consider (41) with 1205912= 120591lowast
2in its stable interval [0 120591
210
)Regard 120591
1as a parameter
Let 120582 = 119894120596 (120596 gt 0) be a root of (41) then we have
minus 1198941205963minus 11990121205962+ 1198941199011120596 + 1199010+ (1198941199021120596 + 1199020) (cos120596120591
1minus 119894 sin120596120591
1)
+ (1199030+ 1198941199031120596) (cos120596120591lowast
2minus 119894 sin120596120591
lowast
2) = 0
(42)
Separating the real and imaginary parts we have
1205963minus 1199011120596 minus 1199031120596 cos120596120591lowast
2+ 1199030sin120596120591
lowast
2
= 1199021120596 cos120596120591
1minus 1199020sin120596120591
1
11990121205962minus 1199010minus 1199030cos120596120591lowast
2minus 1199031120596 sin120596120591
lowast
2
= 1199020cos120596120591
1+ 1199021120596 sin120596120591
1
(43)
It follows that
1205966+ 119898331205964+ 119898321205963+ 119898311205962+ 11989830
= 0 (44)
where
11989833
= 1199012
2minus 21199011minus 21199031cos120596120591lowast
2
11989832
= 2 (1199030minus 11990121199031) sin120596120591
lowast
2
11989831
= 1199012
1minus 211990101199012minus 2 (119901
21199030minus 11990111199031) cos120596120591lowast
2+ 1199032
1minus 1199022
1
11989830
= 1199012
0+ 211990101199030cos120596120591lowast
2+ 1199032
0minus 1199022
0
(45)
Denote 119865(120596) = 1205966+ 119898331205964+ 119898321205963+ 119898311205962+ 11989830 If
11989830
lt 0 then
119865 (0) lt 0 lim120596rarr+infin
119865 (120596) = +infin (46)
We can obtain that (44) has at most six positive roots1205961 1205962 120596
6 For every fixed 120596
119896 119896 = 1 2 6 there exists
a sequence 120591(119895)1119896
| 119895 = 0 1 2 3 such that (43) holdsLet
12059110
= min 120591(119895)
1119896| 119896 = 1 2 6 119895 = 0 1 2 3 (47)
When 1205911
= 120591(119895)
1119896 (41) has a pair of purely imaginary roots
plusmn119894120596(119895)
1119896for 120591lowast2isin [0 120591
210
)In the following we assume that
(H6) ((119889Re(120582))119889120591
1)|120582=plusmn119894120596
(119895)
1119896
= 0
Thus by the general Hopf bifurcation theorem for FDEsin Hale [22] we have the following result on the stability andHopf bifurcation in system (3)
Theorem 8 For 1205911
gt 0 1205912
gt 0 1205911
= 1205912 suppose that
(1198671)ndash(1198676) is satisfied If 119898
30lt 0 and 120591
lowast
2isin [0 120591
210
] thenthe positive equilibrium 119864
lowastis locally asymptotically stable for
1205911
isin [0 12059110) System (3) undergoes Hopf bifurcations at the
positive equilibrium 119864lowastfor 1205911= 120591(119895)
1119896
3 Direction and Stability ofthe Hopf Bifurcation
In Section 2 we obtain the conditions underwhich system (3)undergoes the Hopf bifurcation at the positive equilibrium119864lowast In this section we consider with 120591
2= 120591lowast
2isin [0 120591
210
)
and regard 1205911as a parameter We will derive the explicit
formulas determining the direction stability and period ofthese periodic solutions bifurcating from equilibrium 119864
lowastat
the critical values 1205911by using the normal form and the center
manifold theory developed by Hassard et al [23] Withoutloss of generality denote any one of these critical values 120591
1=
120591(119895)
1119896(119896 = 1 2 6 119895 = 0 1 2 ) by 120591
1 at which (43) has a
pair of purely imaginary roots plusmn119894120596 and system (3) undergoesHopf bifurcation from 119864
lowast
Throughout this section we always assume that 120591lowast2lt 12059110
Let 1199061= 1199091minus 119909lowast
1 1199062= 1199091minus 119909lowast
2 1199063= 1199092minus 119909lowast
3 119905 = 120591
1119905
and 120583 = 1205911minus 1205911 120583 isin R Then 120583 = 0 is the Hopf bifurcation
value of system (3) System (3) may be written as a functionaldifferential equation inC([minus1 0]R3)
(119905) = 119871120583(119906119905) + 119891 (120583 119906
119905) (48)
where 119906 = (1199061 1199062 1199063)119879isin R3 and
119871120583(120601) = (120591
1+ 120583) 119861[
[
1206011(0)
1206012(0)
1206013(0)
]
]
+ (1205911+ 120583)119862[
[
1206011(minus1)
1206012(minus1)
1206013(minus1)
]
]
+ (1205911+ 120583)119863
[[[[[[[[[[
[
1206011(minus
120591lowast
2
1205911
)
1206012(minus
120591lowast
2
1205911
)
1206013(minus
120591lowast
2
1205911
)
]]]]]]]]]]
]
(49)
119891 (120583 120601) = (1205911+ 120583)[
[
1198911
1198912
1198913
]
]
(50)
Abstract and Applied Analysis 7
where 120601 = (1206011 1206012 1206013)119879isin C([minus1 0]R3) and
119861 = [
[
11988711
11988712
0
0 11988722
11988723
0 0 11988733
]
]
119862 = [
[
0 0 0
11988821
0 0
0 0 0
]
]
119863 = [
[
0 0 0
0 0 0
0 11988932
0
]
]
1198911= minus (119886
11+ 1198973) 1206012
1(0) minus 119897
11206011(0) 1206012(0) minus 119897
21206012
1(0) 1206012(0)
minus 11989751206013
1(0) minus 119897
41206013
1(0) 1206012(0) + sdot sdot sdot
1198912= minus11989761206012(0) 1206013(0) minus (119897
7+ 11988622) 1206012
2(0) + 119897
11206011(minus1) 120601
2(0)
+ 11989731206012
1(minus1) minus 119897
81206013
2(0) minus 119897
91206012
2(0) 1206013(0)
+ 11989721206012
1(minus1) 120601
2(0) + 119897
31206013
1(minus1) + 119897
41206013
1(minus1) 120601
2(0)
minus 119897101206013
2(0) 1206013(0) + sdot sdot sdot
1198913= 11989761206012(minus
120591lowast
2
1205911
)1206013(0) + 119897
71206012
2(minus
120591lowast
2
1205911
) minus 119886331206012
3(0)
+ 11989791206012
2(minus
120591lowast
2
1205911
)1206013(0) + 119897
81206013
2(minus
120591lowast
2
1205911
)
+ 119897101206013
2(minus
120591lowast
2
1205911
)1206013(0) + sdot sdot sdot
1199011(119909) =
1198861119909
1 + 1198871119909
1199012(119909) =
1198862119909
1 + 1198872119909
1198971= 1199011015840
1(119909lowast)
1198972=
1
2
11990110158401015840
1(119909lowast) 1198973=
1
2
11990110158401015840
1(119909lowast) 1199101lowast
1198974=
1
3
119901101584010158401015840
1(119909lowast) 1198975=
1
3
119901101584010158401015840
1(119909lowast) 1199101lowast
1198976= 1199011015840
2(1199101lowast) 1198977=
1
2
11990110158401015840
2(1199101lowast) 1199102lowast
1198978=
1
3
119901101584010158401015840
2(1199101lowast) 1199102lowast 1198979=
1
2
11990110158401015840
2(1199101lowast)
11989710
=
1
3
119901101584010158401015840
2(1199101lowast)
(51)
Obviously 119871120583(120601) is a continuous linear function mapping
C([minus1 0]R3) intoR3 By the Riesz representation theoremthere exists a 3 times 3 matrix function 120578(120579 120583) (minus1 ⩽ 120579 ⩽ 0)whose elements are of bounded variation such that
119871120583120601 = int
0
minus1
119889120578 (120579 120583) 120601 (120579) for 120601 isin C ([minus1 0] R3) (52)
In fact we can choose
119889120578 (120579 120583) = (1205911+ 120583) [119861120575 (120579) + 119862120575 (120579 + 1) + 119863120575(120579 +
120591lowast
2
1205911
)]
(53)
where 120575 is Dirac-delta function For 120601 isin C([minus1 0]R3)define
119860 (120583) 120601 =
119889120601 (120579)
119889120579
120579 isin [minus1 0)
int
0
minus1
119889120578 (119904 120583) 120601 (119904) 120579 = 0
119877 (120583) 120601 =
0 120579 isin [minus1 0)
119891 (120583 120601) 120579 = 0
(54)
Then when 120579 = 0 the system is equivalent to
119905= 119860 (120583) 119909
119905+ 119877 (120583) 119909
119905 (55)
where 119909119905(120579) = 119909(119905+120579) 120579 isin [minus1 0] For120595 isin C1([0 1] (R3)
lowast)
define
119860lowast120595 (119904) =
minus
119889120595 (119904)
119889119904
119904 isin (0 1]
int
0
minus1
119889120578119879(119905 0) 120595 (minus119905) 119904 = 0
(56)
and a bilinear inner product
⟨120595 (119904) 120601 (120579)⟩=120595 (0) 120601 (0)minusint
0
minus1
int
120579
120585=0
120595 (120585minus120579) 119889120578 (120579) 120601 (120585) 119889120585
(57)
where 120578(120579) = 120578(120579 0) Let 119860 = 119860(0) then 119860 and 119860lowast
are adjoint operators By the discussion in Section 2 weknow that plusmn119894120596120591
1are eigenvalues of 119860 Thus they are also
eigenvalues of 119860lowast We first need to compute the eigenvectorof 119860 and 119860
lowast corresponding to 1198941205961205911and minus119894120596120591
1 respectively
Suppose that 119902(120579) = (1 120572 120573)1198791198901198941205791205961205911 is the eigenvector of 119860
corresponding to 1198941205961205911 Then 119860119902(120579) = 119894120596120591
1119902(120579) From the
definition of 119860119871120583(120601) and 120578(120579 120583) we can easily obtain 119902(120579) =
(1 120572 120573)1198791198901198941205791205961205911 where
120572 =
119894120596 minus 11988711
11988712
120573 =
11988932(119894120596 minus 119887
11)
11988712(119894120596 minus 119887
33) 119890119894120596120591lowast
2
(58)
and 119902(0) = (1 120572 120573)119879 Similarly let 119902lowast(119904) = 119863(1 120572
lowast 120573lowast)1198901198941199041205961205911
be the eigenvector of 119860lowast corresponding to minus119894120596120591
1 By the
definition of 119860lowast we can compute
120572lowast=
minus119894120596 minus 11988711
119888211198901198941205961205911
120573lowast=
11988723(minus119894120596 minus 119887
11)
11988821(119894120596 minus 119887
33) 1198901198941205961205911
(59)
From (57) we have⟨119902lowast(119904) 119902 (120579)⟩
= 119863(1 120572lowast 120573
lowast
) (1 120572 120573)119879
minus int
0
minus1
int
120579
120585=0
119863(1 120572lowast 120573
lowast
) 119890minus1198941205961205911(120585minus120579)
119889120578 (120579)
times (1 120572 120573)119879
1198901198941205961205911120585119889120585
= 1198631 + 120572120572lowast+ 120573120573
lowast
+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573
lowast
120591lowast
2119890minus119894120596120591lowast
2
(60)
8 Abstract and Applied Analysis
Thus we can choose
119863 = 1 + 120572120572lowast+ 120573120573
lowast
+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573
lowast
120591lowast
2119890minus119894120596120591lowast
2
minus1
(61)
such that ⟨119902lowast(119904) 119902(120579)⟩ = 1 ⟨119902lowast(119904) 119902(120579)⟩ = 0
In the remainder of this section we follow the ideas inHassard et al [23] and use the same notations as there tocompute the coordinates describing the center manifold 119862
0
at 120583 = 0 Let 119909119905be the solution of (48) when 120583 = 0 Define
119911 (119905) = ⟨119902lowast 119909119905⟩ 119882 (119905 120579) = 119909
119905(120579) minus 2Re 119911 (119905) 119902 (120579)
(62)
On the center manifold 1198620 we have
119882(119905 120579) = 119882 (119911 (119905) 119911 (119905) 120579)
= 11988220(120579)
1199112
2
+11988211(120579) 119911119911 +119882
02(120579)
1199112
2
+11988230(120579)
1199113
6
+ sdot sdot sdot
(63)
where 119911 and 119911 are local coordinates for center manifold 1198620in
the direction of 119902 and 119902 Note that 119882 is real if 119909119905is real We
consider only real solutions For the solution 119909119905isin 1198620of (48)
since 120583 = 0 we have
= 1198941205961205911119911 + ⟨119902
lowast(120579) 119891 (0119882 (119911 (119905) 119911 (119905) 120579)
+ 2Re 119911 (119905) 119902 (120579))⟩
= 1198941205961205911119911+119902lowast(0) 119891 (0119882 (119911 (119905) 119911 (119905) 0) +2Re 119911 (119905) 119902 (0))
= 1198941205961205911119911 + 119902lowast(0) 1198910(119911 119911) ≜ 119894120596120591
1119911 + 119892 (119911 119911)
(64)
where
119892 (119911 119911) = 119902lowast(0) 1198910(119911 119911) = 119892
20
1199112
2
+ 11989211119911119911
+ 11989202
1199112
2
+ 11989221
1199112119911
2
+ sdot sdot sdot
(65)
By (62) we have 119909119905(120579) = (119909
1119905(120579) 1199092119905(120579) 1199093119905(120579))119879= 119882(119905 120579) +
119911119902(120579) + 119911 119902(120579) and then
1199091119905(0) = 119911 + 119911 +119882
(1)
20(0)
1199112
2
+119882(1)
11(0) 119911119911
+119882(1)
02(0)
1199112
2
+ 119900 (|(119911 119911)|3)
1199092119905(0) = 119911120572 + 119911 120572 +119882
(2)
20(0)
1199112
2
+119882(2)
11(0) 119911119911 +119882
(2)
02(0)
1199112
2
+ 119900 (|(119911 119911)|3)
1199093119905(0) = 119911120573 + 119911120573 +119882
(3)
20(0)
1199112
2
+119882(3)
11(0) 119911119911
+119882(3)
02(0)
1199112
2
+ 119900 (|(119911 119911)|3)
1199091119905(minus1) = 119911119890
minus1198941205961205911+ 1199111198901198941205961205911+119882(1)
20(minus1)
1199112
2
+119882(1)
11(minus1) 119911119911 +119882
(1)
02(minus1)
1199112
2
+ 119900 (|(119911 119911)|3)
1199092119905(minus1) = 119911120572119890
minus1198941205961205911+ 119911 120572119890
1198941205961205911+119882(2)
20(minus1)
1199112
2
+119882(2)
11(minus1) 119911119911 +119882
(2)
02(minus1)
1199112
2
+ 119900 (|(119911 119911)|3)
1199093119905(minus1) = 119911120573119890
minus1198941205961205911+ 119911120573119890
1205961205911+119882(3)
20(minus1)
1199112
2
+119882(3)
11(minus1) 119911119911 +119882
(3)
02(minus1)
1199112
2
+ 119900 (|(119911 119911)|3)
1199091119905(minus
120591lowast
2
12059110
) = 119911119890minus119894120596120591lowast
2+ 119911119890119894120596120591lowast
2+119882(1)
20(minus
120591lowast
2
1205911
)
1199112
2
+119882(1)
11(minus
120591lowast
2
1205911
)119911119911 +119882(1)
02(minus
120591lowast
2
1205911
)
1199112
2
+ 119900 (|(119911 119911)|3)
1199092119905(minus
120591lowast
2
1205911
) = 119911120572119890minus119894120596120591lowast
2+ 119911 120572119890
119894120596120591lowast
2+119882(2)
20(minus
120591lowast
2
1205911
)
1199112
2
+119882(2)
11(minus
120591lowast
2
1205911
)119911119911 +119882(2)
02(minus
120591lowast
2
1205911
)
1199112
2
+ 119900 (|(119911 119911)|3)
1199093119905(minus
120591lowast
2
1205911
) = 119911120573119890minus119894120596120591lowast
2+ 119911120573119890
120596120591lowast
2+119882(3)
20(minus
120591lowast
2
1205911
)
1199112
2
+119882(3)
11(minus
120591lowast
2
1205911
)119911119911 +119882(3)
02(minus
120591lowast
2
1205911
)
1199112
2
+ 119900 (|(119911 119911)|3)
(66)
Abstract and Applied Analysis 9
It follows together with (50) that
119892 (119911 119911)=119902lowast(0) 1198910(119911 119911) =119863120591
1(1 120572lowast 120573
lowast
) (119891(0)
1119891(0)
2119891(0)
3)
119879
= 1198631205911[minus (119886
11+ 1198973) 1206012
1(0) minus 119897
11206011(0) 1206012(0)
minus 11989721206012
1(0) 1206012(0) minus 119897
51206013
1(0) minus 119897
41206013
1(0) 1206012(0)
+ sdot sdot sdot ]
+ 120572lowast[11989761206012(0) 1206013(0) (1198977+ 11988622) 1206012
2(0)
+ 11989711206011(minus1) 120601
2(0)
+ 11989731206012
1(minus1) minus 119897
81206013
2(0) minus 119897
91206012
2(0) 1206013(0)
+ 11989721206012
1(minus1) 120601
2(0) + 119897
31206013
1(minus1) + sdot sdot sdot ]
+ 120573
lowast
[11989761206012(minus
120591lowast
2
1205911
)1206013(0) + 119897
71206012
2(minus
120591lowast
2
1205911
)
minus 119886331206012
3(0) + 119897
81206013
2(minus
120591lowast
2
1205911
) + sdot sdot sdot ]
(67)
Comparing the coefficients with (65) we have
11989220
= 1198631205911[minus2 (119886
11+ 1198973) minus 2120572119897
1]
+ 120572lowast[21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591
1
minus21198976120572120573 minus 2 (119897
7+ 11988622) 1205722]
+ 120573
lowast
[21198976120572120573119890minus119894120596120591lowast
2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591
lowast
2
minus2119886331205732]
11989211
= 1198631205911[minus2 (119886
11+ 1198973) minus 1198971(120572 + 120572)]
+ 120572lowast[1198971(1205721198901198941205961205911+ 120572119890minus1198941205961205911) minus 1198976(120572120573 + 120572120573)
minus2 (1198977+ 11988622) 120572120572 + 2119897
3]
+ 120573
lowast
[1198976(120573120572119890119894120596120591lowast
2+ 120572120573119890
minus119894120596120591lowast
2)
+ 21198977120572120572 minus 119886
33120573120573]
11989202
= 21198631205911[minus2 (119886
11+ 1198973) minus 21198971120572]
+ 120572lowast[minus21198976120572120573 minus 2 (119897
7+ 11988622) 1205722+ 211989711205721198901198941205961205911
+2119897311989021198941205961205911]
+ 120573
lowast
[21198976120572120573119890119894120596120591lowast
2+ 2119897712057221198902119894120596120591lowast
2minus 11988633120573
2
]
11989221
= 1198631205911[minus (119886
11+ 1198973) (2119882
(1)
20(0) + 4119882
(1)
11(0))
minus 1198971(2120572119882
(1)
11(0) + 120572119882
(1)
20(0) + 119882
(2)
20(0)
+ 2119882(2)
11(0))]
+ 120572lowast[minus1198976(2120573119882
(2)
11(0) + 120572119882
(3)
20(0) + 120573119882
(2)
20(0)
+ 2120572119882(3)
11(0)) minus (119897
7+ 11988622)
times (4120572119882(2)
11(0) + 2120572119882
(2)
20(0))
+ 1198971(2120572119882
(1)
11(minus1) + 120572119882
(1)
20(minus1)
+ 119882(2)
20(0) 1198901198941205961205911+ 2119882
(2)
11(0) 119890minus1198941205961205911)
+ 1198973(4119882(1)
11(minus1) 119890
minus1198941205961205911+2119882(1)
20(minus1) 119890
1198941205961205911)]
+ 120573
lowast
[1198976(2120573119882
(2)
11(minus
120591lowast
2
1205911
) + 120573119882(2)
20(minus
120591lowast
2
1205911
)
+ 120572119882(3)
20(0) 119890119894120596120591lowast
2+ 2120572119882
(3)
11(0) 119890minus119894120596120591lowast
2)
+ 1198977(4120572119882
(2)
11(minus
120591lowast
2
1205911
) 119890minus119894120596120591lowast
2
+2120572119882(2)
20(minus
120591lowast
2
1205911
) 119890119894120596120591lowast
2)
minus 11988633(4120573119882
(3)
11(0) + 2120573119882
(3)
20(0))]
(68)
where
11988220(120579) =
11989411989220
1205961205911
119902 (0) 1198901198941205961205911120579+
11989411989202
31205961205911
119902 (0) 119890minus1198941205961205911120579+ 119864111989021198941205961205911120579
11988211(120579) = minus
11989411989211
1205961205911
119902 (0) 1198901198941205961205911120579+
11989411989211
1205961205911
119902 (0) 119890minus1198941205961205911120579+ 1198642
1198641= 2
[[[[[
[
2119894120596 minus 11988711
minus11988712
0
minus11988821119890minus2119894120596120591
12119894120596 minus 119887
22minus11988723
0 minus11988932119890minus2119894120596120591
lowast
22119894120596 minus 119887
33
]]]]]
]
minus1
times
[[[[[
[
minus2 (11988611
+ 1198973) minus 2120572119897
1
21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591
1minus 21198976120572120573 minus 2 (119897
7+ 11988622) 1205722
21198976120572120573119890minus119894120596120591lowast
2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591
lowast
2minus 2119886331205732
]]]]]
]
10 Abstract and Applied Analysis
1198642= 2
[[[[
[
minus11988711
minus11988712
0
minus11988821
minus11988722
minus11988723
0 minus11988932
minus11988733
]]]]
]
minus1
times
[[[[
[
minus2 (11988611
+ 1198973) minus 1198971(120572 + 120572)
1198971(1205721198901198941205961205911+ 120572119890minus1198941205961205911) minus 1198976(120572120573 + 120572120573) minus 2 (119897
7+ 11988622) 120572120572 + 2119897
3
1198976(120573120572119890119894120596120591lowast
2+ 120572120573119890
minus119894120596120591lowast
2) + 21198977120572120572 minus 119886
33120573120573
]]]]
]
(69)
Thus we can determine11988220(120579) and119882
11(120579) Furthermore
we can determine each 119892119894119895by the parameters and delay in (3)
Thus we can compute the following values
1198881(0) =
119894
21205961205911
(1198922011989211
minus 2100381610038161003816100381611989211
1003816100381610038161003816
2
minus
1
3
100381610038161003816100381611989202
1003816100381610038161003816
2
) +
1
2
11989221
1205832= minus
Re 1198881(0)
Re 1205821015840 (1205911)
1198792= minus
Im 1198881(0) + 120583
2Im 120582
1015840(1205911)
1205961205911
1205732= 2Re 119888
1(0)
(70)
which determine the quantities of bifurcating periodic solu-tions in the center manifold at the critical value 120591
1 Suppose
Re1205821015840(1205911) gt 0 120583
2determines the directions of the Hopf
bifurcation if 1205832
gt 0(lt 0) then the Hopf bifurcationis supercritical (subcritical) and the bifurcation exists for120591 gt 120591
1(lt 1205911) 1205732determines the stability of the bifurcation
periodic solutions the bifurcating periodic solutions arestable (unstable) if 120573
2lt 0(gt 0) and119879
2determines the period
of the bifurcating periodic solutions the period increases(decreases) if 119879
2gt 0(lt 0)
4 Numerical Simulation
We consider system (3) by taking the following coefficients1198861
= 03 11988611
= 58889 11988612
= 1 1198981
= 1 1198862
=
01 11988621
= 27 11988622
= 12 11988623
= 12 1198982
= 1 1198863
=
02 11988632
= 25 11988633
= 12 We have the unique positiveequilibrium 119864
lowast= (00451 00357 00551)
By computation we get 11989810
= minus00104 12059611
= 0516411991111
= 02666 ℎ10158401(11991111) = 03402 120591
20
= 32348 FromTheorem 6 we know that when 120591
1= 0 the positive
equilibrium 119864lowastis locally asymptotically stable for 120591
2isin
[0 32348) When 1205912crosses 120591
20
the equilibrium 119864lowastloses its
stability and Hopf bifurcation occurs From the algorithm inSection 3 we have 120583
2= 56646 120573
2= minus31583 119879
2= 5445
whichmeans that the bifurcation is supercritical and periodicsolution is stable The trajectories and the phase graphs areshown in Figures 1 and 2
Regarding 1205911as a parameter and let 120591
2= 29 isin
[0 32348) we can observe that with 1205911increasing the pos-
itive equilibrium 119864lowastloses its stability and Hopf bifurcation
occurs (see Figures 3 and 4)
5 Global Continuationof Local Hopf Bifurcations
In this section we study the global continuation of peri-odic solutions bifurcating from the positive equilibrium(119864lowast 120591(119895)
1119896) (119896 = 1 2 6 119895 = 0 1 ) Throughout this
section we follow closely the notations in [24] and assumethat 120591
2= 120591lowast
2isin [0 120591
210
) regarding 1205911as a parameter For
simplification of notations setting 119911119905(119905) = (119909
1119905 1199092119905 1199093119905)119879 we
may rewrite system (3) as the following functional differentialequation
(119905) = 119865 (119911119905 1205911 119901) (71)
where 119911119905(120579) = (119909
1119905(120579) 1199092119905(120579) 1199093119905(120579))119879
= (1199091(119905 + 120579) 119909
2(119905 +
120579) 1199093(119905 + 120579))
119879 for 119905 ge 0 and 120579 isin [minus1205911 0] Since 119909
1(119905) 1199092(119905)
and 1199093(119905) denote the densities of the prey the predator and
the top predator respectively the positive solution of system(3) is of interest and its periodic solutions only arise in the firstquadrant Thus we consider system (3) only in the domain1198773
+= (1199091 1199092 1199093) isin 1198773 1199091gt 0 119909
2gt 0 119909
3gt 0 It is obvious
that (71) has a unique positive equilibrium 119864lowast(119909lowast
1 119909lowast
2 119909lowast
3) in
1198773
+under the assumption (119867
1)ndash(1198674) Following the work of
[24] we need to define
119883 = 119862([minus1205911 0] 119877
3
+)
Γ = 119862119897 (119911 1205911 119901) isin X times R times R+
119911 is a 119901-periodic solution of system (71)
N = (119911 1205911 119901) 119865 (119911 120591
1 119901) = 0
(72)
Let ℓ(119864lowast120591(119895)
11198962120587120596
(119895)
1119896)denote the connected component pass-
ing through (119864lowast 120591(119895)
1119896 2120587120596
(119895)
1119896) in Γ where 120591
(119895)
1119896is defined by
(43) We know that ℓ(119864lowast120591(119895)
11198962120587120596
(119895)
1119896)through (119864
lowast 120591(119895)
1119896 2120587120596
(119895)
1119896)
is nonemptyFor the benefit of readers we first state the global Hopf
bifurcation theory due to Wu [24] for functional differentialequations
Lemma 9 Assume that (119911lowast 120591 119901) is an isolated center satis-
fying the hypotheses (A1)ndash(A4) in [24] Denote by ℓ(119911lowast120591119901)
theconnected component of (119911
lowast 120591 119901) in Γ Then either
(i) ℓ(119911lowast120591119901)
is unbounded or(ii) ℓ(119911lowast120591119901)
is bounded ℓ(119911lowast120591119901)
cap Γ is finite and
sum
(119911120591119901)isinℓ(119911lowast120591119901)capN
120574119898(119911lowast 120591 119901) = 0 (73)
for all 119898 = 1 2 where 120574119898(119911lowast 120591 119901) is the 119898119905ℎ crossing
number of (119911lowast 120591 119901) if119898 isin 119869(119911
lowast 120591 119901) or it is zero if otherwise
Clearly if (ii) in Lemma 9 is not true then ℓ(119911lowast120591119901)
isunbounded Thus if the projections of ℓ
(119911lowast120591119901)
onto 119911-spaceand onto 119901-space are bounded then the projection of ℓ
(119911lowast120591119901)
onto 120591-space is unbounded Further if we can show that the
Abstract and Applied Analysis 11
0 200 400 60000451
00451
00452
t
x1(t)
t
0 200 400 60000356
00357
00358
x2(t)
t
0 200 400 6000055
00551
00552
x3(t)
00451
00451
00452
00356
00357
00357
003570055
0055
00551
00551
00552
x 1(t)
x2 (t)
x3(t)
Figure 1 The trajectories and the phase graph with 1205911= 0 120591
2= 29 lt 120591
20= 32348 119864
lowastis locally asymptotically stable
0 500 1000004
0045
005
0 500 1000002
004
006
0 500 10000
005
01
t
t
t
x1(t)
x2(t)
x3(t)
0044
0046
0048
002
003
004
005003
004
005
006
007
008
x 1(t)
x2 (t)
x3(t)
Figure 2 The trajectories and the phase graph with 1205911= 0 120591
2= 33 gt 120591
20= 32348 a periodic orbit bifurcate from 119864
lowast
12 Abstract and Applied Analysis
0 500 1000 1500
00451
00452
0 500 1000 15000035
0036
0037
0 500 1000 15000054
0055
0056
t
t
t
x1(t)
x2(t)
x3(t)
0045100451
0045200452
0035
00355
003600546
00548
0055
00552
00554
00556
00558
x1(t)
x2 (t)
x3(t)
Figure 3 The trajectories and the phase graph with 1205911= 09 120591
2= 29 119864
lowastis locally asymptotically stable
0 500 1000 15000044
0045
0046
0 500 1000 1500002
003
004
0 500 1000 1500004
006
008
t
t
t
x1(t)
x2(t)
x3(t)
00445
0045
00455
0032
0034
0036
00380052
0054
0056
0058
006
0062
0064
x 1(t)
x2 (t)
x3(t)
Figure 4 The trajectories and the phase graph with 1205911= 12 120591
2= 29 a periodic orbit bifurcate from 119864
lowast
Abstract and Applied Analysis 13
projection of ℓ(119911lowast120591119901)
onto 120591-space is away from zero thenthe projection of ℓ
(119911lowast120591119901)
onto 120591-space must include interval[120591 infin) Following this ideal we can prove our results on theglobal continuation of local Hopf bifurcation
Lemma 10 If the conditions (1198671)ndash(1198674) hold then all nontriv-
ial periodic solutions of system (71) with initial conditions
1199091(120579) = 120593 (120579) ge 0 119909
2(120579) = 120595 (120579) ge 0
1199093(120579) = 120601 (120579) ge 0 120579 isin [minus120591
1 0)
120593 (0) gt 0 120595 (0) gt 0 120601 (0) gt 0
(74)
are uniformly bounded
Proof Suppose that (1199091(119905) 1199092(119905) 1199093(119905)) are nonconstant peri-
odic solutions of system (3) and define
1199091(1205851) = min 119909
1(119905) 119909
1(1205781) = max 119909
1(119905)
1199092(1205852) = min 119909
2(119905) 119909
2(1205782) = max 119909
2(119905)
1199093(1205853) = min 119909
3(119905) 119909
3(1205783) = max 119909
3(119905)
(75)
It follows from system (3) that
1199091(119905) = 119909
1(0) expint
119905
0
[1198861minus 119886111199091(119904) minus
119886121199092(119904)
1198981+ 1199091(119904)
] 119889119904
1199092(119905) = 119909
2(0) expint
119905
0
[minus1198862+
119886211199091(119904 minus 1205911)
1198981+ 1199091(119904 minus 1205911)
minus 119886221199092(119904) minus
119886231199093(119904)
1198982+ 1199092(119904)
] 119889119904
1199093(119905) = 119909
3(0) expint
119905
0
[minus1198863+
119886321199092(119904 minus 120591
lowast
2)
1198982+ 1199092(119904 minus 120591
lowast
2)
minus 119886331199093(119904)] 119889119904
(76)
which implies that the solutions of system (3) cannot crossthe 119909119894-axis (119894 = 1 2 3) Thus the nonconstant periodic orbits
must be located in the interior of first quadrant It followsfrom initial data of system (3) that 119909
1(119905) gt 0 119909
2(119905) gt
0 1199093(119905) gt 0 for 119905 ge 0
From the first equation of system (3) we can get
0 = 1198861minus 119886111199091(1205781) minus
119886121199092(1205781)
1198981+ 1199091(1205781)
le 1198861minus 119886111199091(1205781) (77)
thus we have
1199091(1205781) le
1198861
11988611
(78)
From the second equation of (3) we obtain
0 = minus 1198862+
119886211199091(1205782minus 1205911)
1198981+ 1199091(1205782minus 1205911)
minus 119886221199092(1205782)
minus
119886231199093(1205782)
1198982+ 1199092(1205782)
le minus1198862+
11988621(119886111988611)
1198981+ (119886111988611)
minus 119886221199092(1205782)
(79)
therefore one can get
1199092(1205782) le
minus1198862(119886111198981+ 1198861) + 119886111988621
11988622(119886111198981+ 1198861)
≜ 1198721 (80)
Applying the third equation of system (3) we know
0 = minus1198863+
119886321199092(1205783minus 120591lowast
2)
1198982+ 1199092(1205783minus 120591lowast
2)
minus 119886331199093(1205783)
le minus1198863+
119886321198721
1198982+1198721
minus 119886331199093(1205783)
(81)
It follows that
1199093(1205783) le
minus1198863(1198982+1198721) + 119886321198721
11988633(1198982+1198721)
≜ 1198722 (82)
This shows that the nontrivial periodic solution of system (3)is uniformly bounded and the proof is complete
Lemma 11 If the conditions (1198671)ndash(1198674) and
(H7) 11988622
minus (119886211198982) gt 0 [(119886
11minus 11988611198981)11988622
minus
(11988612119886211198982
1)]11988633
minus (11988611
minus (11988611198981))[(119886211198982)11988633
+
(11988623119886321198982
2)] gt 0
hold then system (3) has no nontrivial 1205911-periodic solution
Proof Suppose for a contradiction that system (3) has non-trivial periodic solution with period 120591
1 Then the following
system (83) of ordinary differential equations has nontrivialperiodic solution
1198891199091
119889119905
= 1199091(119905) [1198861minus 119886111199091(119905) minus
119886121199092(119905)
1198981+ 1199091(119905)
]
1198891199092
119889119905
= 1199092(119905) [minus119886
2+
119886211199091(119905)
1198981+ 1199091(119905)
minus 119886221199092(119905) minus
119886231199093(119905)
1198982+ 1199092(119905)
]
1198891199093
119889119905
= 1199093(119905) [minus119886
3+
119886321199092(119905 minus 120591
lowast
2)
1198982+ 1199092(119905 minus 120591
lowast
2)
minus 119886331199093(119905)]
(83)
which has the same equilibria to system (3) that is
1198641= (0 0 0) 119864
2= (
1198861
11988611
0 0)
1198643= (1199091 1199092 0) 119864
lowast= (119909lowast
1 119909lowast
2 119909lowast
3)
(84)
Note that 119909119894-axis (119894 = 1 2 3) are the invariable manifold of
system (83) and the orbits of system (83) do not intersect each
14 Abstract and Applied Analysis
other Thus there are no solutions crossing the coordinateaxes On the other hand note the fact that if system (83) hasa periodic solution then there must be the equilibrium in itsinterior and that 119864
1 1198642 1198643are located on the coordinate
axis Thus we conclude that the periodic orbit of system (83)must lie in the first quadrant If (119867
7) holds it is well known
that the positive equilibrium 119864lowastis globally asymptotically
stable in the first quadrant (see [1])Thus there is no periodicorbit in the first quadrant too The above discussion meansthat (83) does not have any nontrivial periodic solution It isa contradiction Therefore the lemma is confirmed
Theorem 12 Suppose the conditions of Theorem 8 and (1198677)
hold let120596119896and 120591(119895)1119896
be defined in Section 2 then when 1205911gt 120591(119895)
1119896
system (3) has at least 119895 minus 1 periodic solutions
Proof It is sufficient to prove that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is [120591
1 +infin) for each 119895 ge 1 where
1205911le 120591(119895)
1119896
In following we prove that the hypotheses (A1)ndash(A4) in[24] hold
(1) From system (3) we know easily that the followingconditions hold
(A1) 119865 isin 1198622(1198773
+times119877+times119877+) where 119865 = 119865|
1198773
+times119877+times119877+
rarr
1198773
+
(A3) 119865(120601 1205911 119901) is differential with respect to 120601
(2) It follows from system (3) that
119863119911119865 (119911 120591
1 119901) =
[[[[[[[[[[[
[
1198861minus 2119886111199091minus
1198861211989811199092
(1198981+ 1199091)2
minus
119886121199091
1198981+ 1199091
0
1198862111989811199092
(1198981+ 1199091)2
minus1198862+
119886211199091
1198981+ 1199091
minus 2119886221199092minus
1198862311989821199093
(1198982+ 1199092)2
minus
119886231199092
1198982+ 1199092
0
1198863211989821199093
(1198982+ 1199092)2
minus1198863+
119886321199092
1198982+ 1199092
minus 2119886331199093
]]]]]]]]]]]
]
(85)
Then under the assumption (1198671)ndash(1198674) we have
det119863119911119865 (119911lowast 1205911 119901)
= det
[[[[[[[[
[
minus11988611119909lowast
1+
11988612119909lowast
1119909lowast
2
(1198981+ 119909lowast
1)2
11988612119909lowast
1
1198981+ 119909lowast
1
0
119886211198981119909lowast
2
(1198981+ 119909lowast
1)2
minus11988622119909lowast
2+
11988623119909lowast
2119909lowast
3
(1198982+ 119909lowast
2)2
minus
11988623119909lowast
2
1198982+ 119909lowast
2
0
119886321198982119909lowast
3
(1198982+ 119909lowast
2)2
minus11988633119909lowast
3
]]]]]]]]
]
= minus
1198862
21199101lowast1199102lowast
(1 + 11988721199101lowast)3[minus119909lowast+
11988611198871119909lowast1199101lowast
(1 + 1198871119909lowast)2] = 0
(86)
From (86) we know that the hypothesis (A2) in [24]is satisfied
(3) The characteristic matrix of (71) at a stationarysolution (119911 120591
0 1199010) where 119911 = (119911
(1) 119911(2) 119911(3)) isin 119877
3
takes the following form
Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863
120601119865 (119911 120591
1 119901) (119890
120582119868) (87)
that is
Δ (119911 1205911 119901) (120582)
=
[[[[[[[[[[[[[[[[[
[
120582 minus 1198861+ 211988611119911(1)
+
119886121198981119911(2)
(1198981+ 119911(1))
2
11988612119911(1)
1198981+ 119911(1)
0
minus
119886211198981119911(2)
(1198981+ 119911(1))
2119890minus1205821205911
120582 + 1198862minus
11988621119911(1)
1198981+ 119911(1)
+ 211988622119911(2)
+
119886231198982119911(3)
(1198982+ 119911(2))
2
11988623119911(2)
1198982+ 119911(2)
0 minus
119886321198982119911(3)
(1198982+ 119911(2))
2119890minus120582120591lowast
2120582 + 1198863minus
11988632119911(2)
1198982+ 119911(2)
+ 211988633119911(3)
]]]]]]]]]]]]]]]]]
]
(88)
Abstract and Applied Analysis 15
From (88) we have
det (Δ (119864lowast 1205911 119901) (120582))
= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902
0+ 1199030)] 119890minus1205821205911
(89)
Note that (89) is the same as (20) from the discussion inSection 2 about the local Hopf bifurcation it is easy to verifythat (119864
lowast 120591(119895)
1119896 2120587120596
119896) is an isolated center and there exist 120598 gt
0 120575 gt 0 and a smooth curve 120582 (120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575) rarr C
such that det(Δ(120582(1205911))) = 0 |120582(120591
1) minus 120596119896| lt 120598 for all 120591
1isin
[120591(119895)
1119896minus 120575 120591(119895)
1119896+ 120575] and
120582 (120591(119895)
1119896) = 120596119896119894
119889Re 120582 (1205911)
1198891205911
1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)
1119896
gt 0 (90)
Let
Ω1205982120587120596
119896
= (120578 119901) 0 lt 120578 lt 120598
10038161003816100381610038161003816100381610038161003816
119901 minus
2120587
120596119896
10038161003816100381610038161003816100381610038161003816
lt 120598 (91)
It is easy to see that on [120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575] times 120597Ω
1205982120587120596119896
det(Δ(119864
lowast 1205911 119901)(120578 + (2120587119901)119894)) = 0 if and only if 120578 = 0
1205911= 120591(119895)
1119896 119901 = 2120587120596
119896 119896 = 1 2 3 119895 = 0 1 2
Therefore the hypothesis (A4) in [24] is satisfiedIf we define
119867plusmn(119864lowast 120591(119895)
1119896
2120587
120596119896
) (120578 119901)
= det(Δ (119864lowast 120591(119895)
1119896plusmn 120575 119901) (120578 +
2120587
119901
119894))
(92)
then we have the crossing number of isolated center(119864lowast 120591(119895)
1119896 2120587120596
119896) as follows
120574(119864lowast 120591(119895)
1119896
2120587
120596119896
) = deg119861(119867minus(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
minus deg119861(119867+(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
= minus1
(93)
Thus we have
sum
(1199111205911119901)isinC
(119864lowast120591(119895)
11198962120587120596119896)
120574 (119911 1205911 119901) lt 0
(94)
where (119911 1205911 119901) has all or parts of the form
(119864lowast 120591(119896)
1119895 2120587120596
119896) (119895 = 0 1 ) It follows from
Lemma 9 that the connected component ℓ(119864lowast120591(119895)
11198962120587120596
119896)
through (119864lowast 120591(119895)
1119896 2120587120596
119896) is unbounded for each center
(119911lowast 1205911 119901) (119895 = 0 1 ) From the discussion in Section 2
we have
120591(119895)
1119896=
1
120596119896
times arccos(((11990121205962
119896minus1199010) (1199030+1199020)
+ 1205962(1205962minus1199011) (1199031+1199021))
times ((1199030+1199020)2
+ (1199031+1199021)2
1205962
119896)
minus1
) +2119895120587
(95)
where 119896 = 1 2 3 119895 = 0 1 Thus one can get 2120587120596119896le 120591(119895)
1119896
for 119895 ge 1Now we prove that the projection of ℓ
(119864lowast120591(119895)
11198962120587120596
119896)onto
1205911-space is [120591
1 +infin) where 120591
1le 120591(119895)
1119896 Clearly it follows
from the proof of Lemma 11 that system (3) with 1205911
= 0
has no nontrivial periodic solution Hence the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is away from zero
For a contradiction we suppose that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is bounded this means that the
projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is included in a
interval (0 120591lowast) Noticing 2120587120596119896lt 120591119895
1119896and applying Lemma 11
we have 119901 lt 120591lowast for (119911(119905) 120591
1 119901) belonging to ℓ
(119864lowast120591(119895)
11198962120587120596
119896)
This implies that the projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 119901-
space is bounded Then applying Lemma 10 we get thatthe connected component ℓ
(119864lowast120591(119895)
11198962120587120596
119896)is bounded This
contradiction completes the proof
6 Conclusion
In this paper we take our attention to the stability and Hopfbifurcation analysis of a predator-prey systemwithMichaelis-Menten type functional response and two unequal delaysWe obtained some conditions for local stability and Hopfbifurcation occurring When 120591
1= 1205912 we derived the explicit
formulas to determine the properties of periodic solutionsby the normal form method and center manifold theoremSpecially the global existence results of periodic solutionsbifurcating from Hopf bifurcations are also established byusing a global Hopf bifurcation result due to Wu [24]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to thank the anonymous referees fortheir careful reading and constructive suggestions which leadto truly significant improvement of the paper This researchis supported by the National Natural Science Foundation ofChina (no 11061016)
16 Abstract and Applied Analysis
References
[1] R Xu and M A J Chaplain ldquoPersistence and global stabilityin a delayed predator-prey system with Michaelis-Menten typefunctional responserdquo Applied Mathematics and Computationvol 130 no 2-3 pp 441ndash455 2002
[2] Y L Song Y H Peng and J JWei ldquoBifurcations for a predator-prey system with two delaysrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 466ndash479 2008
[3] G-P Hu W-T Li and X-P Yan ldquoHopf bifurcations in apredator-prey system with multiple delaysrdquo Chaos Solitons ampFractals vol 42 no 2 pp 1273ndash1285 2009
[4] M Liao X Tang and C Xu ldquoBifurcation analysis for a three-species predator-prey systemwith two delaysrdquoCommunicationsin Nonlinear Science and Numerical Simulation vol 17 no 1 pp183ndash194 2012
[5] C Xu and P Li ldquoDynamical analysis in a delayed predator-prey model with two delaysrdquo Discrete Dynamics in Nature andSociety vol 2012 Article ID 652947 22 pages 2012
[6] J-F Zhang ldquoStability and bifurcation periodic solutions ina Lotka-Volterra competition system with multiple delaysrdquoNonlinear Dynamics vol 70 no 1 pp 849ndash860 2012
[7] J Xia Z Yu and R Yuan ldquoStability and Hopf bifurcation ina symmetric Lotka-Volterra predator-prey system with delaysrdquoElectronic Journal of Differential Equations vol 2013 no 9 pp1ndash16 2013
[8] H Wan and J Cui ldquoA malaria model with two delaysrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 601265 8pages 2013
[9] M Zhao ldquoHopf bifurcation analysis for a semiratio-dependentpredator-prey system with two delaysrdquo Abstract and AppliedAnalysis vol 2013 Article ID 495072 13 pages 2013
[10] S Feyissa and S Banerjee ldquoDelay-induced oscillatory dynamicsin humoral mediated immune response with two time delaysrdquoNonlinear Analysis Real World Applications vol 14 no 1 pp35ndash52 2013
[11] G Zhang Y Shen andB Chen ldquoHopf bifurcation of a predator-prey systemwith predator harvesting and twodelaysrdquoNonlinearDynamics vol 73 no 4 pp 2119ndash2131 2013
[12] X-Y Meng H-F Huo and H Xiang ldquoHopf bifurcation in athree-species systemwith delaysrdquo Journal of AppliedMathemat-ics and Computing vol 35 no 1-2 pp 635ndash661 2011
[13] J-F Zhang ldquoGlobal existence of bifurcated periodic solutions ina commensalism model with delaysrdquo Applied Mathematics andComputation vol 218 no 23 pp 11688ndash11699 2012
[14] Y Wang W Jiang and H Wang ldquoStability and global Hopfbifurcation in toxic phytoplankton-zooplankton model withdelay and selective harvestingrdquoNonlinear Dynamics vol 73 no1-2 pp 881ndash896 2013
[15] Y Ma ldquoGlobal Hopf bifurcation in the Leslie-Gower predator-prey model with two delaysrdquo Nonlinear Analysis Real WorldApplications vol 13 no 1 pp 370ndash375 2012
[16] X-Y Meng H-F Huo and X-B Zhang ldquoStability and globalHopf bifurcation in a delayed food web consisting of a preyand two predatorsrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4335ndash4348 2011
[17] Y Song J Wei andM Han ldquoLocal and global Hopf bifurcationin a delayed hematopoiesis modelrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol14 no 11 pp 3909ndash3919 2004
[18] Y Song and J Wei ldquoLocal Hopf bifurcation and global periodicsolutions in a delayed predator-prey systemrdquo Journal of Mathe-matical Analysis and Applications vol 301 no 1 pp 1ndash21 2005
[19] J Wei and M Y Li ldquoGlobal existence of periodic solutions in atri-neuron network model with delaysrdquo Physica D vol 198 no1-2 pp 106ndash119 2004
[20] T Zhao Y Kuang and H L Smith ldquoGlobal existence of peri-odic solutions in a class of delayed Gause-type predator-preysystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 28 no 8 pp 1373ndash1394 1997
[21] S J Gao L S Chen and Z D Teng ldquoHopf bifurcation andglobal stability for a delayed predator-prey system with stagestructure for predatorrdquo Applied Mathematics and Computationvol 202 no 2 pp 721ndash729 2008
[22] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977
[23] B Hassard D Kazarinoff and Y WanTheory and Applicationsof Hopf Bifurcation Cambridge University Press CambridgeUK 1981
[24] J H Wu ldquoSymmetric functional-differential equations andneural networks with memoryrdquo Transactions of the AmericanMathematical Society vol 350 no 12 pp 4799ndash4838 1998
[25] V Hutson ldquoThe existence of an equilibrium for permanentsystemsrdquo The Rocky Mountain Journal of Mathematics vol 20no 4 pp 1033ndash1040 1990
[26] S G Ruan and J J Wei ldquoOn the zeros of transcendentalfunctions with applications to stability of delay differentialequations with two delaysrdquo Dynamics of Continuous Discreteamp Impulsive Systems A vol 10 no 6 pp 863ndash874 2003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 7
where 120601 = (1206011 1206012 1206013)119879isin C([minus1 0]R3) and
119861 = [
[
11988711
11988712
0
0 11988722
11988723
0 0 11988733
]
]
119862 = [
[
0 0 0
11988821
0 0
0 0 0
]
]
119863 = [
[
0 0 0
0 0 0
0 11988932
0
]
]
1198911= minus (119886
11+ 1198973) 1206012
1(0) minus 119897
11206011(0) 1206012(0) minus 119897
21206012
1(0) 1206012(0)
minus 11989751206013
1(0) minus 119897
41206013
1(0) 1206012(0) + sdot sdot sdot
1198912= minus11989761206012(0) 1206013(0) minus (119897
7+ 11988622) 1206012
2(0) + 119897
11206011(minus1) 120601
2(0)
+ 11989731206012
1(minus1) minus 119897
81206013
2(0) minus 119897
91206012
2(0) 1206013(0)
+ 11989721206012
1(minus1) 120601
2(0) + 119897
31206013
1(minus1) + 119897
41206013
1(minus1) 120601
2(0)
minus 119897101206013
2(0) 1206013(0) + sdot sdot sdot
1198913= 11989761206012(minus
120591lowast
2
1205911
)1206013(0) + 119897
71206012
2(minus
120591lowast
2
1205911
) minus 119886331206012
3(0)
+ 11989791206012
2(minus
120591lowast
2
1205911
)1206013(0) + 119897
81206013
2(minus
120591lowast
2
1205911
)
+ 119897101206013
2(minus
120591lowast
2
1205911
)1206013(0) + sdot sdot sdot
1199011(119909) =
1198861119909
1 + 1198871119909
1199012(119909) =
1198862119909
1 + 1198872119909
1198971= 1199011015840
1(119909lowast)
1198972=
1
2
11990110158401015840
1(119909lowast) 1198973=
1
2
11990110158401015840
1(119909lowast) 1199101lowast
1198974=
1
3
119901101584010158401015840
1(119909lowast) 1198975=
1
3
119901101584010158401015840
1(119909lowast) 1199101lowast
1198976= 1199011015840
2(1199101lowast) 1198977=
1
2
11990110158401015840
2(1199101lowast) 1199102lowast
1198978=
1
3
119901101584010158401015840
2(1199101lowast) 1199102lowast 1198979=
1
2
11990110158401015840
2(1199101lowast)
11989710
=
1
3
119901101584010158401015840
2(1199101lowast)
(51)
Obviously 119871120583(120601) is a continuous linear function mapping
C([minus1 0]R3) intoR3 By the Riesz representation theoremthere exists a 3 times 3 matrix function 120578(120579 120583) (minus1 ⩽ 120579 ⩽ 0)whose elements are of bounded variation such that
119871120583120601 = int
0
minus1
119889120578 (120579 120583) 120601 (120579) for 120601 isin C ([minus1 0] R3) (52)
In fact we can choose
119889120578 (120579 120583) = (1205911+ 120583) [119861120575 (120579) + 119862120575 (120579 + 1) + 119863120575(120579 +
120591lowast
2
1205911
)]
(53)
where 120575 is Dirac-delta function For 120601 isin C([minus1 0]R3)define
119860 (120583) 120601 =
119889120601 (120579)
119889120579
120579 isin [minus1 0)
int
0
minus1
119889120578 (119904 120583) 120601 (119904) 120579 = 0
119877 (120583) 120601 =
0 120579 isin [minus1 0)
119891 (120583 120601) 120579 = 0
(54)
Then when 120579 = 0 the system is equivalent to
119905= 119860 (120583) 119909
119905+ 119877 (120583) 119909
119905 (55)
where 119909119905(120579) = 119909(119905+120579) 120579 isin [minus1 0] For120595 isin C1([0 1] (R3)
lowast)
define
119860lowast120595 (119904) =
minus
119889120595 (119904)
119889119904
119904 isin (0 1]
int
0
minus1
119889120578119879(119905 0) 120595 (minus119905) 119904 = 0
(56)
and a bilinear inner product
⟨120595 (119904) 120601 (120579)⟩=120595 (0) 120601 (0)minusint
0
minus1
int
120579
120585=0
120595 (120585minus120579) 119889120578 (120579) 120601 (120585) 119889120585
(57)
where 120578(120579) = 120578(120579 0) Let 119860 = 119860(0) then 119860 and 119860lowast
are adjoint operators By the discussion in Section 2 weknow that plusmn119894120596120591
1are eigenvalues of 119860 Thus they are also
eigenvalues of 119860lowast We first need to compute the eigenvectorof 119860 and 119860
lowast corresponding to 1198941205961205911and minus119894120596120591
1 respectively
Suppose that 119902(120579) = (1 120572 120573)1198791198901198941205791205961205911 is the eigenvector of 119860
corresponding to 1198941205961205911 Then 119860119902(120579) = 119894120596120591
1119902(120579) From the
definition of 119860119871120583(120601) and 120578(120579 120583) we can easily obtain 119902(120579) =
(1 120572 120573)1198791198901198941205791205961205911 where
120572 =
119894120596 minus 11988711
11988712
120573 =
11988932(119894120596 minus 119887
11)
11988712(119894120596 minus 119887
33) 119890119894120596120591lowast
2
(58)
and 119902(0) = (1 120572 120573)119879 Similarly let 119902lowast(119904) = 119863(1 120572
lowast 120573lowast)1198901198941199041205961205911
be the eigenvector of 119860lowast corresponding to minus119894120596120591
1 By the
definition of 119860lowast we can compute
120572lowast=
minus119894120596 minus 11988711
119888211198901198941205961205911
120573lowast=
11988723(minus119894120596 minus 119887
11)
11988821(119894120596 minus 119887
33) 1198901198941205961205911
(59)
From (57) we have⟨119902lowast(119904) 119902 (120579)⟩
= 119863(1 120572lowast 120573
lowast
) (1 120572 120573)119879
minus int
0
minus1
int
120579
120585=0
119863(1 120572lowast 120573
lowast
) 119890minus1198941205961205911(120585minus120579)
119889120578 (120579)
times (1 120572 120573)119879
1198901198941205961205911120585119889120585
= 1198631 + 120572120572lowast+ 120573120573
lowast
+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573
lowast
120591lowast
2119890minus119894120596120591lowast
2
(60)
8 Abstract and Applied Analysis
Thus we can choose
119863 = 1 + 120572120572lowast+ 120573120573
lowast
+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573
lowast
120591lowast
2119890minus119894120596120591lowast
2
minus1
(61)
such that ⟨119902lowast(119904) 119902(120579)⟩ = 1 ⟨119902lowast(119904) 119902(120579)⟩ = 0
In the remainder of this section we follow the ideas inHassard et al [23] and use the same notations as there tocompute the coordinates describing the center manifold 119862
0
at 120583 = 0 Let 119909119905be the solution of (48) when 120583 = 0 Define
119911 (119905) = ⟨119902lowast 119909119905⟩ 119882 (119905 120579) = 119909
119905(120579) minus 2Re 119911 (119905) 119902 (120579)
(62)
On the center manifold 1198620 we have
119882(119905 120579) = 119882 (119911 (119905) 119911 (119905) 120579)
= 11988220(120579)
1199112
2
+11988211(120579) 119911119911 +119882
02(120579)
1199112
2
+11988230(120579)
1199113
6
+ sdot sdot sdot
(63)
where 119911 and 119911 are local coordinates for center manifold 1198620in
the direction of 119902 and 119902 Note that 119882 is real if 119909119905is real We
consider only real solutions For the solution 119909119905isin 1198620of (48)
since 120583 = 0 we have
= 1198941205961205911119911 + ⟨119902
lowast(120579) 119891 (0119882 (119911 (119905) 119911 (119905) 120579)
+ 2Re 119911 (119905) 119902 (120579))⟩
= 1198941205961205911119911+119902lowast(0) 119891 (0119882 (119911 (119905) 119911 (119905) 0) +2Re 119911 (119905) 119902 (0))
= 1198941205961205911119911 + 119902lowast(0) 1198910(119911 119911) ≜ 119894120596120591
1119911 + 119892 (119911 119911)
(64)
where
119892 (119911 119911) = 119902lowast(0) 1198910(119911 119911) = 119892
20
1199112
2
+ 11989211119911119911
+ 11989202
1199112
2
+ 11989221
1199112119911
2
+ sdot sdot sdot
(65)
By (62) we have 119909119905(120579) = (119909
1119905(120579) 1199092119905(120579) 1199093119905(120579))119879= 119882(119905 120579) +
119911119902(120579) + 119911 119902(120579) and then
1199091119905(0) = 119911 + 119911 +119882
(1)
20(0)
1199112
2
+119882(1)
11(0) 119911119911
+119882(1)
02(0)
1199112
2
+ 119900 (|(119911 119911)|3)
1199092119905(0) = 119911120572 + 119911 120572 +119882
(2)
20(0)
1199112
2
+119882(2)
11(0) 119911119911 +119882
(2)
02(0)
1199112
2
+ 119900 (|(119911 119911)|3)
1199093119905(0) = 119911120573 + 119911120573 +119882
(3)
20(0)
1199112
2
+119882(3)
11(0) 119911119911
+119882(3)
02(0)
1199112
2
+ 119900 (|(119911 119911)|3)
1199091119905(minus1) = 119911119890
minus1198941205961205911+ 1199111198901198941205961205911+119882(1)
20(minus1)
1199112
2
+119882(1)
11(minus1) 119911119911 +119882
(1)
02(minus1)
1199112
2
+ 119900 (|(119911 119911)|3)
1199092119905(minus1) = 119911120572119890
minus1198941205961205911+ 119911 120572119890
1198941205961205911+119882(2)
20(minus1)
1199112
2
+119882(2)
11(minus1) 119911119911 +119882
(2)
02(minus1)
1199112
2
+ 119900 (|(119911 119911)|3)
1199093119905(minus1) = 119911120573119890
minus1198941205961205911+ 119911120573119890
1205961205911+119882(3)
20(minus1)
1199112
2
+119882(3)
11(minus1) 119911119911 +119882
(3)
02(minus1)
1199112
2
+ 119900 (|(119911 119911)|3)
1199091119905(minus
120591lowast
2
12059110
) = 119911119890minus119894120596120591lowast
2+ 119911119890119894120596120591lowast
2+119882(1)
20(minus
120591lowast
2
1205911
)
1199112
2
+119882(1)
11(minus
120591lowast
2
1205911
)119911119911 +119882(1)
02(minus
120591lowast
2
1205911
)
1199112
2
+ 119900 (|(119911 119911)|3)
1199092119905(minus
120591lowast
2
1205911
) = 119911120572119890minus119894120596120591lowast
2+ 119911 120572119890
119894120596120591lowast
2+119882(2)
20(minus
120591lowast
2
1205911
)
1199112
2
+119882(2)
11(minus
120591lowast
2
1205911
)119911119911 +119882(2)
02(minus
120591lowast
2
1205911
)
1199112
2
+ 119900 (|(119911 119911)|3)
1199093119905(minus
120591lowast
2
1205911
) = 119911120573119890minus119894120596120591lowast
2+ 119911120573119890
120596120591lowast
2+119882(3)
20(minus
120591lowast
2
1205911
)
1199112
2
+119882(3)
11(minus
120591lowast
2
1205911
)119911119911 +119882(3)
02(minus
120591lowast
2
1205911
)
1199112
2
+ 119900 (|(119911 119911)|3)
(66)
Abstract and Applied Analysis 9
It follows together with (50) that
119892 (119911 119911)=119902lowast(0) 1198910(119911 119911) =119863120591
1(1 120572lowast 120573
lowast
) (119891(0)
1119891(0)
2119891(0)
3)
119879
= 1198631205911[minus (119886
11+ 1198973) 1206012
1(0) minus 119897
11206011(0) 1206012(0)
minus 11989721206012
1(0) 1206012(0) minus 119897
51206013
1(0) minus 119897
41206013
1(0) 1206012(0)
+ sdot sdot sdot ]
+ 120572lowast[11989761206012(0) 1206013(0) (1198977+ 11988622) 1206012
2(0)
+ 11989711206011(minus1) 120601
2(0)
+ 11989731206012
1(minus1) minus 119897
81206013
2(0) minus 119897
91206012
2(0) 1206013(0)
+ 11989721206012
1(minus1) 120601
2(0) + 119897
31206013
1(minus1) + sdot sdot sdot ]
+ 120573
lowast
[11989761206012(minus
120591lowast
2
1205911
)1206013(0) + 119897
71206012
2(minus
120591lowast
2
1205911
)
minus 119886331206012
3(0) + 119897
81206013
2(minus
120591lowast
2
1205911
) + sdot sdot sdot ]
(67)
Comparing the coefficients with (65) we have
11989220
= 1198631205911[minus2 (119886
11+ 1198973) minus 2120572119897
1]
+ 120572lowast[21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591
1
minus21198976120572120573 minus 2 (119897
7+ 11988622) 1205722]
+ 120573
lowast
[21198976120572120573119890minus119894120596120591lowast
2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591
lowast
2
minus2119886331205732]
11989211
= 1198631205911[minus2 (119886
11+ 1198973) minus 1198971(120572 + 120572)]
+ 120572lowast[1198971(1205721198901198941205961205911+ 120572119890minus1198941205961205911) minus 1198976(120572120573 + 120572120573)
minus2 (1198977+ 11988622) 120572120572 + 2119897
3]
+ 120573
lowast
[1198976(120573120572119890119894120596120591lowast
2+ 120572120573119890
minus119894120596120591lowast
2)
+ 21198977120572120572 minus 119886
33120573120573]
11989202
= 21198631205911[minus2 (119886
11+ 1198973) minus 21198971120572]
+ 120572lowast[minus21198976120572120573 minus 2 (119897
7+ 11988622) 1205722+ 211989711205721198901198941205961205911
+2119897311989021198941205961205911]
+ 120573
lowast
[21198976120572120573119890119894120596120591lowast
2+ 2119897712057221198902119894120596120591lowast
2minus 11988633120573
2
]
11989221
= 1198631205911[minus (119886
11+ 1198973) (2119882
(1)
20(0) + 4119882
(1)
11(0))
minus 1198971(2120572119882
(1)
11(0) + 120572119882
(1)
20(0) + 119882
(2)
20(0)
+ 2119882(2)
11(0))]
+ 120572lowast[minus1198976(2120573119882
(2)
11(0) + 120572119882
(3)
20(0) + 120573119882
(2)
20(0)
+ 2120572119882(3)
11(0)) minus (119897
7+ 11988622)
times (4120572119882(2)
11(0) + 2120572119882
(2)
20(0))
+ 1198971(2120572119882
(1)
11(minus1) + 120572119882
(1)
20(minus1)
+ 119882(2)
20(0) 1198901198941205961205911+ 2119882
(2)
11(0) 119890minus1198941205961205911)
+ 1198973(4119882(1)
11(minus1) 119890
minus1198941205961205911+2119882(1)
20(minus1) 119890
1198941205961205911)]
+ 120573
lowast
[1198976(2120573119882
(2)
11(minus
120591lowast
2
1205911
) + 120573119882(2)
20(minus
120591lowast
2
1205911
)
+ 120572119882(3)
20(0) 119890119894120596120591lowast
2+ 2120572119882
(3)
11(0) 119890minus119894120596120591lowast
2)
+ 1198977(4120572119882
(2)
11(minus
120591lowast
2
1205911
) 119890minus119894120596120591lowast
2
+2120572119882(2)
20(minus
120591lowast
2
1205911
) 119890119894120596120591lowast
2)
minus 11988633(4120573119882
(3)
11(0) + 2120573119882
(3)
20(0))]
(68)
where
11988220(120579) =
11989411989220
1205961205911
119902 (0) 1198901198941205961205911120579+
11989411989202
31205961205911
119902 (0) 119890minus1198941205961205911120579+ 119864111989021198941205961205911120579
11988211(120579) = minus
11989411989211
1205961205911
119902 (0) 1198901198941205961205911120579+
11989411989211
1205961205911
119902 (0) 119890minus1198941205961205911120579+ 1198642
1198641= 2
[[[[[
[
2119894120596 minus 11988711
minus11988712
0
minus11988821119890minus2119894120596120591
12119894120596 minus 119887
22minus11988723
0 minus11988932119890minus2119894120596120591
lowast
22119894120596 minus 119887
33
]]]]]
]
minus1
times
[[[[[
[
minus2 (11988611
+ 1198973) minus 2120572119897
1
21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591
1minus 21198976120572120573 minus 2 (119897
7+ 11988622) 1205722
21198976120572120573119890minus119894120596120591lowast
2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591
lowast
2minus 2119886331205732
]]]]]
]
10 Abstract and Applied Analysis
1198642= 2
[[[[
[
minus11988711
minus11988712
0
minus11988821
minus11988722
minus11988723
0 minus11988932
minus11988733
]]]]
]
minus1
times
[[[[
[
minus2 (11988611
+ 1198973) minus 1198971(120572 + 120572)
1198971(1205721198901198941205961205911+ 120572119890minus1198941205961205911) minus 1198976(120572120573 + 120572120573) minus 2 (119897
7+ 11988622) 120572120572 + 2119897
3
1198976(120573120572119890119894120596120591lowast
2+ 120572120573119890
minus119894120596120591lowast
2) + 21198977120572120572 minus 119886
33120573120573
]]]]
]
(69)
Thus we can determine11988220(120579) and119882
11(120579) Furthermore
we can determine each 119892119894119895by the parameters and delay in (3)
Thus we can compute the following values
1198881(0) =
119894
21205961205911
(1198922011989211
minus 2100381610038161003816100381611989211
1003816100381610038161003816
2
minus
1
3
100381610038161003816100381611989202
1003816100381610038161003816
2
) +
1
2
11989221
1205832= minus
Re 1198881(0)
Re 1205821015840 (1205911)
1198792= minus
Im 1198881(0) + 120583
2Im 120582
1015840(1205911)
1205961205911
1205732= 2Re 119888
1(0)
(70)
which determine the quantities of bifurcating periodic solu-tions in the center manifold at the critical value 120591
1 Suppose
Re1205821015840(1205911) gt 0 120583
2determines the directions of the Hopf
bifurcation if 1205832
gt 0(lt 0) then the Hopf bifurcationis supercritical (subcritical) and the bifurcation exists for120591 gt 120591
1(lt 1205911) 1205732determines the stability of the bifurcation
periodic solutions the bifurcating periodic solutions arestable (unstable) if 120573
2lt 0(gt 0) and119879
2determines the period
of the bifurcating periodic solutions the period increases(decreases) if 119879
2gt 0(lt 0)
4 Numerical Simulation
We consider system (3) by taking the following coefficients1198861
= 03 11988611
= 58889 11988612
= 1 1198981
= 1 1198862
=
01 11988621
= 27 11988622
= 12 11988623
= 12 1198982
= 1 1198863
=
02 11988632
= 25 11988633
= 12 We have the unique positiveequilibrium 119864
lowast= (00451 00357 00551)
By computation we get 11989810
= minus00104 12059611
= 0516411991111
= 02666 ℎ10158401(11991111) = 03402 120591
20
= 32348 FromTheorem 6 we know that when 120591
1= 0 the positive
equilibrium 119864lowastis locally asymptotically stable for 120591
2isin
[0 32348) When 1205912crosses 120591
20
the equilibrium 119864lowastloses its
stability and Hopf bifurcation occurs From the algorithm inSection 3 we have 120583
2= 56646 120573
2= minus31583 119879
2= 5445
whichmeans that the bifurcation is supercritical and periodicsolution is stable The trajectories and the phase graphs areshown in Figures 1 and 2
Regarding 1205911as a parameter and let 120591
2= 29 isin
[0 32348) we can observe that with 1205911increasing the pos-
itive equilibrium 119864lowastloses its stability and Hopf bifurcation
occurs (see Figures 3 and 4)
5 Global Continuationof Local Hopf Bifurcations
In this section we study the global continuation of peri-odic solutions bifurcating from the positive equilibrium(119864lowast 120591(119895)
1119896) (119896 = 1 2 6 119895 = 0 1 ) Throughout this
section we follow closely the notations in [24] and assumethat 120591
2= 120591lowast
2isin [0 120591
210
) regarding 1205911as a parameter For
simplification of notations setting 119911119905(119905) = (119909
1119905 1199092119905 1199093119905)119879 we
may rewrite system (3) as the following functional differentialequation
(119905) = 119865 (119911119905 1205911 119901) (71)
where 119911119905(120579) = (119909
1119905(120579) 1199092119905(120579) 1199093119905(120579))119879
= (1199091(119905 + 120579) 119909
2(119905 +
120579) 1199093(119905 + 120579))
119879 for 119905 ge 0 and 120579 isin [minus1205911 0] Since 119909
1(119905) 1199092(119905)
and 1199093(119905) denote the densities of the prey the predator and
the top predator respectively the positive solution of system(3) is of interest and its periodic solutions only arise in the firstquadrant Thus we consider system (3) only in the domain1198773
+= (1199091 1199092 1199093) isin 1198773 1199091gt 0 119909
2gt 0 119909
3gt 0 It is obvious
that (71) has a unique positive equilibrium 119864lowast(119909lowast
1 119909lowast
2 119909lowast
3) in
1198773
+under the assumption (119867
1)ndash(1198674) Following the work of
[24] we need to define
119883 = 119862([minus1205911 0] 119877
3
+)
Γ = 119862119897 (119911 1205911 119901) isin X times R times R+
119911 is a 119901-periodic solution of system (71)
N = (119911 1205911 119901) 119865 (119911 120591
1 119901) = 0
(72)
Let ℓ(119864lowast120591(119895)
11198962120587120596
(119895)
1119896)denote the connected component pass-
ing through (119864lowast 120591(119895)
1119896 2120587120596
(119895)
1119896) in Γ where 120591
(119895)
1119896is defined by
(43) We know that ℓ(119864lowast120591(119895)
11198962120587120596
(119895)
1119896)through (119864
lowast 120591(119895)
1119896 2120587120596
(119895)
1119896)
is nonemptyFor the benefit of readers we first state the global Hopf
bifurcation theory due to Wu [24] for functional differentialequations
Lemma 9 Assume that (119911lowast 120591 119901) is an isolated center satis-
fying the hypotheses (A1)ndash(A4) in [24] Denote by ℓ(119911lowast120591119901)
theconnected component of (119911
lowast 120591 119901) in Γ Then either
(i) ℓ(119911lowast120591119901)
is unbounded or(ii) ℓ(119911lowast120591119901)
is bounded ℓ(119911lowast120591119901)
cap Γ is finite and
sum
(119911120591119901)isinℓ(119911lowast120591119901)capN
120574119898(119911lowast 120591 119901) = 0 (73)
for all 119898 = 1 2 where 120574119898(119911lowast 120591 119901) is the 119898119905ℎ crossing
number of (119911lowast 120591 119901) if119898 isin 119869(119911
lowast 120591 119901) or it is zero if otherwise
Clearly if (ii) in Lemma 9 is not true then ℓ(119911lowast120591119901)
isunbounded Thus if the projections of ℓ
(119911lowast120591119901)
onto 119911-spaceand onto 119901-space are bounded then the projection of ℓ
(119911lowast120591119901)
onto 120591-space is unbounded Further if we can show that the
Abstract and Applied Analysis 11
0 200 400 60000451
00451
00452
t
x1(t)
t
0 200 400 60000356
00357
00358
x2(t)
t
0 200 400 6000055
00551
00552
x3(t)
00451
00451
00452
00356
00357
00357
003570055
0055
00551
00551
00552
x 1(t)
x2 (t)
x3(t)
Figure 1 The trajectories and the phase graph with 1205911= 0 120591
2= 29 lt 120591
20= 32348 119864
lowastis locally asymptotically stable
0 500 1000004
0045
005
0 500 1000002
004
006
0 500 10000
005
01
t
t
t
x1(t)
x2(t)
x3(t)
0044
0046
0048
002
003
004
005003
004
005
006
007
008
x 1(t)
x2 (t)
x3(t)
Figure 2 The trajectories and the phase graph with 1205911= 0 120591
2= 33 gt 120591
20= 32348 a periodic orbit bifurcate from 119864
lowast
12 Abstract and Applied Analysis
0 500 1000 1500
00451
00452
0 500 1000 15000035
0036
0037
0 500 1000 15000054
0055
0056
t
t
t
x1(t)
x2(t)
x3(t)
0045100451
0045200452
0035
00355
003600546
00548
0055
00552
00554
00556
00558
x1(t)
x2 (t)
x3(t)
Figure 3 The trajectories and the phase graph with 1205911= 09 120591
2= 29 119864
lowastis locally asymptotically stable
0 500 1000 15000044
0045
0046
0 500 1000 1500002
003
004
0 500 1000 1500004
006
008
t
t
t
x1(t)
x2(t)
x3(t)
00445
0045
00455
0032
0034
0036
00380052
0054
0056
0058
006
0062
0064
x 1(t)
x2 (t)
x3(t)
Figure 4 The trajectories and the phase graph with 1205911= 12 120591
2= 29 a periodic orbit bifurcate from 119864
lowast
Abstract and Applied Analysis 13
projection of ℓ(119911lowast120591119901)
onto 120591-space is away from zero thenthe projection of ℓ
(119911lowast120591119901)
onto 120591-space must include interval[120591 infin) Following this ideal we can prove our results on theglobal continuation of local Hopf bifurcation
Lemma 10 If the conditions (1198671)ndash(1198674) hold then all nontriv-
ial periodic solutions of system (71) with initial conditions
1199091(120579) = 120593 (120579) ge 0 119909
2(120579) = 120595 (120579) ge 0
1199093(120579) = 120601 (120579) ge 0 120579 isin [minus120591
1 0)
120593 (0) gt 0 120595 (0) gt 0 120601 (0) gt 0
(74)
are uniformly bounded
Proof Suppose that (1199091(119905) 1199092(119905) 1199093(119905)) are nonconstant peri-
odic solutions of system (3) and define
1199091(1205851) = min 119909
1(119905) 119909
1(1205781) = max 119909
1(119905)
1199092(1205852) = min 119909
2(119905) 119909
2(1205782) = max 119909
2(119905)
1199093(1205853) = min 119909
3(119905) 119909
3(1205783) = max 119909
3(119905)
(75)
It follows from system (3) that
1199091(119905) = 119909
1(0) expint
119905
0
[1198861minus 119886111199091(119904) minus
119886121199092(119904)
1198981+ 1199091(119904)
] 119889119904
1199092(119905) = 119909
2(0) expint
119905
0
[minus1198862+
119886211199091(119904 minus 1205911)
1198981+ 1199091(119904 minus 1205911)
minus 119886221199092(119904) minus
119886231199093(119904)
1198982+ 1199092(119904)
] 119889119904
1199093(119905) = 119909
3(0) expint
119905
0
[minus1198863+
119886321199092(119904 minus 120591
lowast
2)
1198982+ 1199092(119904 minus 120591
lowast
2)
minus 119886331199093(119904)] 119889119904
(76)
which implies that the solutions of system (3) cannot crossthe 119909119894-axis (119894 = 1 2 3) Thus the nonconstant periodic orbits
must be located in the interior of first quadrant It followsfrom initial data of system (3) that 119909
1(119905) gt 0 119909
2(119905) gt
0 1199093(119905) gt 0 for 119905 ge 0
From the first equation of system (3) we can get
0 = 1198861minus 119886111199091(1205781) minus
119886121199092(1205781)
1198981+ 1199091(1205781)
le 1198861minus 119886111199091(1205781) (77)
thus we have
1199091(1205781) le
1198861
11988611
(78)
From the second equation of (3) we obtain
0 = minus 1198862+
119886211199091(1205782minus 1205911)
1198981+ 1199091(1205782minus 1205911)
minus 119886221199092(1205782)
minus
119886231199093(1205782)
1198982+ 1199092(1205782)
le minus1198862+
11988621(119886111988611)
1198981+ (119886111988611)
minus 119886221199092(1205782)
(79)
therefore one can get
1199092(1205782) le
minus1198862(119886111198981+ 1198861) + 119886111988621
11988622(119886111198981+ 1198861)
≜ 1198721 (80)
Applying the third equation of system (3) we know
0 = minus1198863+
119886321199092(1205783minus 120591lowast
2)
1198982+ 1199092(1205783minus 120591lowast
2)
minus 119886331199093(1205783)
le minus1198863+
119886321198721
1198982+1198721
minus 119886331199093(1205783)
(81)
It follows that
1199093(1205783) le
minus1198863(1198982+1198721) + 119886321198721
11988633(1198982+1198721)
≜ 1198722 (82)
This shows that the nontrivial periodic solution of system (3)is uniformly bounded and the proof is complete
Lemma 11 If the conditions (1198671)ndash(1198674) and
(H7) 11988622
minus (119886211198982) gt 0 [(119886
11minus 11988611198981)11988622
minus
(11988612119886211198982
1)]11988633
minus (11988611
minus (11988611198981))[(119886211198982)11988633
+
(11988623119886321198982
2)] gt 0
hold then system (3) has no nontrivial 1205911-periodic solution
Proof Suppose for a contradiction that system (3) has non-trivial periodic solution with period 120591
1 Then the following
system (83) of ordinary differential equations has nontrivialperiodic solution
1198891199091
119889119905
= 1199091(119905) [1198861minus 119886111199091(119905) minus
119886121199092(119905)
1198981+ 1199091(119905)
]
1198891199092
119889119905
= 1199092(119905) [minus119886
2+
119886211199091(119905)
1198981+ 1199091(119905)
minus 119886221199092(119905) minus
119886231199093(119905)
1198982+ 1199092(119905)
]
1198891199093
119889119905
= 1199093(119905) [minus119886
3+
119886321199092(119905 minus 120591
lowast
2)
1198982+ 1199092(119905 minus 120591
lowast
2)
minus 119886331199093(119905)]
(83)
which has the same equilibria to system (3) that is
1198641= (0 0 0) 119864
2= (
1198861
11988611
0 0)
1198643= (1199091 1199092 0) 119864
lowast= (119909lowast
1 119909lowast
2 119909lowast
3)
(84)
Note that 119909119894-axis (119894 = 1 2 3) are the invariable manifold of
system (83) and the orbits of system (83) do not intersect each
14 Abstract and Applied Analysis
other Thus there are no solutions crossing the coordinateaxes On the other hand note the fact that if system (83) hasa periodic solution then there must be the equilibrium in itsinterior and that 119864
1 1198642 1198643are located on the coordinate
axis Thus we conclude that the periodic orbit of system (83)must lie in the first quadrant If (119867
7) holds it is well known
that the positive equilibrium 119864lowastis globally asymptotically
stable in the first quadrant (see [1])Thus there is no periodicorbit in the first quadrant too The above discussion meansthat (83) does not have any nontrivial periodic solution It isa contradiction Therefore the lemma is confirmed
Theorem 12 Suppose the conditions of Theorem 8 and (1198677)
hold let120596119896and 120591(119895)1119896
be defined in Section 2 then when 1205911gt 120591(119895)
1119896
system (3) has at least 119895 minus 1 periodic solutions
Proof It is sufficient to prove that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is [120591
1 +infin) for each 119895 ge 1 where
1205911le 120591(119895)
1119896
In following we prove that the hypotheses (A1)ndash(A4) in[24] hold
(1) From system (3) we know easily that the followingconditions hold
(A1) 119865 isin 1198622(1198773
+times119877+times119877+) where 119865 = 119865|
1198773
+times119877+times119877+
rarr
1198773
+
(A3) 119865(120601 1205911 119901) is differential with respect to 120601
(2) It follows from system (3) that
119863119911119865 (119911 120591
1 119901) =
[[[[[[[[[[[
[
1198861minus 2119886111199091minus
1198861211989811199092
(1198981+ 1199091)2
minus
119886121199091
1198981+ 1199091
0
1198862111989811199092
(1198981+ 1199091)2
minus1198862+
119886211199091
1198981+ 1199091
minus 2119886221199092minus
1198862311989821199093
(1198982+ 1199092)2
minus
119886231199092
1198982+ 1199092
0
1198863211989821199093
(1198982+ 1199092)2
minus1198863+
119886321199092
1198982+ 1199092
minus 2119886331199093
]]]]]]]]]]]
]
(85)
Then under the assumption (1198671)ndash(1198674) we have
det119863119911119865 (119911lowast 1205911 119901)
= det
[[[[[[[[
[
minus11988611119909lowast
1+
11988612119909lowast
1119909lowast
2
(1198981+ 119909lowast
1)2
11988612119909lowast
1
1198981+ 119909lowast
1
0
119886211198981119909lowast
2
(1198981+ 119909lowast
1)2
minus11988622119909lowast
2+
11988623119909lowast
2119909lowast
3
(1198982+ 119909lowast
2)2
minus
11988623119909lowast
2
1198982+ 119909lowast
2
0
119886321198982119909lowast
3
(1198982+ 119909lowast
2)2
minus11988633119909lowast
3
]]]]]]]]
]
= minus
1198862
21199101lowast1199102lowast
(1 + 11988721199101lowast)3[minus119909lowast+
11988611198871119909lowast1199101lowast
(1 + 1198871119909lowast)2] = 0
(86)
From (86) we know that the hypothesis (A2) in [24]is satisfied
(3) The characteristic matrix of (71) at a stationarysolution (119911 120591
0 1199010) where 119911 = (119911
(1) 119911(2) 119911(3)) isin 119877
3
takes the following form
Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863
120601119865 (119911 120591
1 119901) (119890
120582119868) (87)
that is
Δ (119911 1205911 119901) (120582)
=
[[[[[[[[[[[[[[[[[
[
120582 minus 1198861+ 211988611119911(1)
+
119886121198981119911(2)
(1198981+ 119911(1))
2
11988612119911(1)
1198981+ 119911(1)
0
minus
119886211198981119911(2)
(1198981+ 119911(1))
2119890minus1205821205911
120582 + 1198862minus
11988621119911(1)
1198981+ 119911(1)
+ 211988622119911(2)
+
119886231198982119911(3)
(1198982+ 119911(2))
2
11988623119911(2)
1198982+ 119911(2)
0 minus
119886321198982119911(3)
(1198982+ 119911(2))
2119890minus120582120591lowast
2120582 + 1198863minus
11988632119911(2)
1198982+ 119911(2)
+ 211988633119911(3)
]]]]]]]]]]]]]]]]]
]
(88)
Abstract and Applied Analysis 15
From (88) we have
det (Δ (119864lowast 1205911 119901) (120582))
= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902
0+ 1199030)] 119890minus1205821205911
(89)
Note that (89) is the same as (20) from the discussion inSection 2 about the local Hopf bifurcation it is easy to verifythat (119864
lowast 120591(119895)
1119896 2120587120596
119896) is an isolated center and there exist 120598 gt
0 120575 gt 0 and a smooth curve 120582 (120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575) rarr C
such that det(Δ(120582(1205911))) = 0 |120582(120591
1) minus 120596119896| lt 120598 for all 120591
1isin
[120591(119895)
1119896minus 120575 120591(119895)
1119896+ 120575] and
120582 (120591(119895)
1119896) = 120596119896119894
119889Re 120582 (1205911)
1198891205911
1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)
1119896
gt 0 (90)
Let
Ω1205982120587120596
119896
= (120578 119901) 0 lt 120578 lt 120598
10038161003816100381610038161003816100381610038161003816
119901 minus
2120587
120596119896
10038161003816100381610038161003816100381610038161003816
lt 120598 (91)
It is easy to see that on [120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575] times 120597Ω
1205982120587120596119896
det(Δ(119864
lowast 1205911 119901)(120578 + (2120587119901)119894)) = 0 if and only if 120578 = 0
1205911= 120591(119895)
1119896 119901 = 2120587120596
119896 119896 = 1 2 3 119895 = 0 1 2
Therefore the hypothesis (A4) in [24] is satisfiedIf we define
119867plusmn(119864lowast 120591(119895)
1119896
2120587
120596119896
) (120578 119901)
= det(Δ (119864lowast 120591(119895)
1119896plusmn 120575 119901) (120578 +
2120587
119901
119894))
(92)
then we have the crossing number of isolated center(119864lowast 120591(119895)
1119896 2120587120596
119896) as follows
120574(119864lowast 120591(119895)
1119896
2120587
120596119896
) = deg119861(119867minus(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
minus deg119861(119867+(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
= minus1
(93)
Thus we have
sum
(1199111205911119901)isinC
(119864lowast120591(119895)
11198962120587120596119896)
120574 (119911 1205911 119901) lt 0
(94)
where (119911 1205911 119901) has all or parts of the form
(119864lowast 120591(119896)
1119895 2120587120596
119896) (119895 = 0 1 ) It follows from
Lemma 9 that the connected component ℓ(119864lowast120591(119895)
11198962120587120596
119896)
through (119864lowast 120591(119895)
1119896 2120587120596
119896) is unbounded for each center
(119911lowast 1205911 119901) (119895 = 0 1 ) From the discussion in Section 2
we have
120591(119895)
1119896=
1
120596119896
times arccos(((11990121205962
119896minus1199010) (1199030+1199020)
+ 1205962(1205962minus1199011) (1199031+1199021))
times ((1199030+1199020)2
+ (1199031+1199021)2
1205962
119896)
minus1
) +2119895120587
(95)
where 119896 = 1 2 3 119895 = 0 1 Thus one can get 2120587120596119896le 120591(119895)
1119896
for 119895 ge 1Now we prove that the projection of ℓ
(119864lowast120591(119895)
11198962120587120596
119896)onto
1205911-space is [120591
1 +infin) where 120591
1le 120591(119895)
1119896 Clearly it follows
from the proof of Lemma 11 that system (3) with 1205911
= 0
has no nontrivial periodic solution Hence the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is away from zero
For a contradiction we suppose that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is bounded this means that the
projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is included in a
interval (0 120591lowast) Noticing 2120587120596119896lt 120591119895
1119896and applying Lemma 11
we have 119901 lt 120591lowast for (119911(119905) 120591
1 119901) belonging to ℓ
(119864lowast120591(119895)
11198962120587120596
119896)
This implies that the projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 119901-
space is bounded Then applying Lemma 10 we get thatthe connected component ℓ
(119864lowast120591(119895)
11198962120587120596
119896)is bounded This
contradiction completes the proof
6 Conclusion
In this paper we take our attention to the stability and Hopfbifurcation analysis of a predator-prey systemwithMichaelis-Menten type functional response and two unequal delaysWe obtained some conditions for local stability and Hopfbifurcation occurring When 120591
1= 1205912 we derived the explicit
formulas to determine the properties of periodic solutionsby the normal form method and center manifold theoremSpecially the global existence results of periodic solutionsbifurcating from Hopf bifurcations are also established byusing a global Hopf bifurcation result due to Wu [24]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to thank the anonymous referees fortheir careful reading and constructive suggestions which leadto truly significant improvement of the paper This researchis supported by the National Natural Science Foundation ofChina (no 11061016)
16 Abstract and Applied Analysis
References
[1] R Xu and M A J Chaplain ldquoPersistence and global stabilityin a delayed predator-prey system with Michaelis-Menten typefunctional responserdquo Applied Mathematics and Computationvol 130 no 2-3 pp 441ndash455 2002
[2] Y L Song Y H Peng and J JWei ldquoBifurcations for a predator-prey system with two delaysrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 466ndash479 2008
[3] G-P Hu W-T Li and X-P Yan ldquoHopf bifurcations in apredator-prey system with multiple delaysrdquo Chaos Solitons ampFractals vol 42 no 2 pp 1273ndash1285 2009
[4] M Liao X Tang and C Xu ldquoBifurcation analysis for a three-species predator-prey systemwith two delaysrdquoCommunicationsin Nonlinear Science and Numerical Simulation vol 17 no 1 pp183ndash194 2012
[5] C Xu and P Li ldquoDynamical analysis in a delayed predator-prey model with two delaysrdquo Discrete Dynamics in Nature andSociety vol 2012 Article ID 652947 22 pages 2012
[6] J-F Zhang ldquoStability and bifurcation periodic solutions ina Lotka-Volterra competition system with multiple delaysrdquoNonlinear Dynamics vol 70 no 1 pp 849ndash860 2012
[7] J Xia Z Yu and R Yuan ldquoStability and Hopf bifurcation ina symmetric Lotka-Volterra predator-prey system with delaysrdquoElectronic Journal of Differential Equations vol 2013 no 9 pp1ndash16 2013
[8] H Wan and J Cui ldquoA malaria model with two delaysrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 601265 8pages 2013
[9] M Zhao ldquoHopf bifurcation analysis for a semiratio-dependentpredator-prey system with two delaysrdquo Abstract and AppliedAnalysis vol 2013 Article ID 495072 13 pages 2013
[10] S Feyissa and S Banerjee ldquoDelay-induced oscillatory dynamicsin humoral mediated immune response with two time delaysrdquoNonlinear Analysis Real World Applications vol 14 no 1 pp35ndash52 2013
[11] G Zhang Y Shen andB Chen ldquoHopf bifurcation of a predator-prey systemwith predator harvesting and twodelaysrdquoNonlinearDynamics vol 73 no 4 pp 2119ndash2131 2013
[12] X-Y Meng H-F Huo and H Xiang ldquoHopf bifurcation in athree-species systemwith delaysrdquo Journal of AppliedMathemat-ics and Computing vol 35 no 1-2 pp 635ndash661 2011
[13] J-F Zhang ldquoGlobal existence of bifurcated periodic solutions ina commensalism model with delaysrdquo Applied Mathematics andComputation vol 218 no 23 pp 11688ndash11699 2012
[14] Y Wang W Jiang and H Wang ldquoStability and global Hopfbifurcation in toxic phytoplankton-zooplankton model withdelay and selective harvestingrdquoNonlinear Dynamics vol 73 no1-2 pp 881ndash896 2013
[15] Y Ma ldquoGlobal Hopf bifurcation in the Leslie-Gower predator-prey model with two delaysrdquo Nonlinear Analysis Real WorldApplications vol 13 no 1 pp 370ndash375 2012
[16] X-Y Meng H-F Huo and X-B Zhang ldquoStability and globalHopf bifurcation in a delayed food web consisting of a preyand two predatorsrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4335ndash4348 2011
[17] Y Song J Wei andM Han ldquoLocal and global Hopf bifurcationin a delayed hematopoiesis modelrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol14 no 11 pp 3909ndash3919 2004
[18] Y Song and J Wei ldquoLocal Hopf bifurcation and global periodicsolutions in a delayed predator-prey systemrdquo Journal of Mathe-matical Analysis and Applications vol 301 no 1 pp 1ndash21 2005
[19] J Wei and M Y Li ldquoGlobal existence of periodic solutions in atri-neuron network model with delaysrdquo Physica D vol 198 no1-2 pp 106ndash119 2004
[20] T Zhao Y Kuang and H L Smith ldquoGlobal existence of peri-odic solutions in a class of delayed Gause-type predator-preysystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 28 no 8 pp 1373ndash1394 1997
[21] S J Gao L S Chen and Z D Teng ldquoHopf bifurcation andglobal stability for a delayed predator-prey system with stagestructure for predatorrdquo Applied Mathematics and Computationvol 202 no 2 pp 721ndash729 2008
[22] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977
[23] B Hassard D Kazarinoff and Y WanTheory and Applicationsof Hopf Bifurcation Cambridge University Press CambridgeUK 1981
[24] J H Wu ldquoSymmetric functional-differential equations andneural networks with memoryrdquo Transactions of the AmericanMathematical Society vol 350 no 12 pp 4799ndash4838 1998
[25] V Hutson ldquoThe existence of an equilibrium for permanentsystemsrdquo The Rocky Mountain Journal of Mathematics vol 20no 4 pp 1033ndash1040 1990
[26] S G Ruan and J J Wei ldquoOn the zeros of transcendentalfunctions with applications to stability of delay differentialequations with two delaysrdquo Dynamics of Continuous Discreteamp Impulsive Systems A vol 10 no 6 pp 863ndash874 2003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Abstract and Applied Analysis
Thus we can choose
119863 = 1 + 120572120572lowast+ 120573120573
lowast
+ 11988821120572lowast1205911119890minus1198941205961205911+ 11988932120572120573
lowast
120591lowast
2119890minus119894120596120591lowast
2
minus1
(61)
such that ⟨119902lowast(119904) 119902(120579)⟩ = 1 ⟨119902lowast(119904) 119902(120579)⟩ = 0
In the remainder of this section we follow the ideas inHassard et al [23] and use the same notations as there tocompute the coordinates describing the center manifold 119862
0
at 120583 = 0 Let 119909119905be the solution of (48) when 120583 = 0 Define
119911 (119905) = ⟨119902lowast 119909119905⟩ 119882 (119905 120579) = 119909
119905(120579) minus 2Re 119911 (119905) 119902 (120579)
(62)
On the center manifold 1198620 we have
119882(119905 120579) = 119882 (119911 (119905) 119911 (119905) 120579)
= 11988220(120579)
1199112
2
+11988211(120579) 119911119911 +119882
02(120579)
1199112
2
+11988230(120579)
1199113
6
+ sdot sdot sdot
(63)
where 119911 and 119911 are local coordinates for center manifold 1198620in
the direction of 119902 and 119902 Note that 119882 is real if 119909119905is real We
consider only real solutions For the solution 119909119905isin 1198620of (48)
since 120583 = 0 we have
= 1198941205961205911119911 + ⟨119902
lowast(120579) 119891 (0119882 (119911 (119905) 119911 (119905) 120579)
+ 2Re 119911 (119905) 119902 (120579))⟩
= 1198941205961205911119911+119902lowast(0) 119891 (0119882 (119911 (119905) 119911 (119905) 0) +2Re 119911 (119905) 119902 (0))
= 1198941205961205911119911 + 119902lowast(0) 1198910(119911 119911) ≜ 119894120596120591
1119911 + 119892 (119911 119911)
(64)
where
119892 (119911 119911) = 119902lowast(0) 1198910(119911 119911) = 119892
20
1199112
2
+ 11989211119911119911
+ 11989202
1199112
2
+ 11989221
1199112119911
2
+ sdot sdot sdot
(65)
By (62) we have 119909119905(120579) = (119909
1119905(120579) 1199092119905(120579) 1199093119905(120579))119879= 119882(119905 120579) +
119911119902(120579) + 119911 119902(120579) and then
1199091119905(0) = 119911 + 119911 +119882
(1)
20(0)
1199112
2
+119882(1)
11(0) 119911119911
+119882(1)
02(0)
1199112
2
+ 119900 (|(119911 119911)|3)
1199092119905(0) = 119911120572 + 119911 120572 +119882
(2)
20(0)
1199112
2
+119882(2)
11(0) 119911119911 +119882
(2)
02(0)
1199112
2
+ 119900 (|(119911 119911)|3)
1199093119905(0) = 119911120573 + 119911120573 +119882
(3)
20(0)
1199112
2
+119882(3)
11(0) 119911119911
+119882(3)
02(0)
1199112
2
+ 119900 (|(119911 119911)|3)
1199091119905(minus1) = 119911119890
minus1198941205961205911+ 1199111198901198941205961205911+119882(1)
20(minus1)
1199112
2
+119882(1)
11(minus1) 119911119911 +119882
(1)
02(minus1)
1199112
2
+ 119900 (|(119911 119911)|3)
1199092119905(minus1) = 119911120572119890
minus1198941205961205911+ 119911 120572119890
1198941205961205911+119882(2)
20(minus1)
1199112
2
+119882(2)
11(minus1) 119911119911 +119882
(2)
02(minus1)
1199112
2
+ 119900 (|(119911 119911)|3)
1199093119905(minus1) = 119911120573119890
minus1198941205961205911+ 119911120573119890
1205961205911+119882(3)
20(minus1)
1199112
2
+119882(3)
11(minus1) 119911119911 +119882
(3)
02(minus1)
1199112
2
+ 119900 (|(119911 119911)|3)
1199091119905(minus
120591lowast
2
12059110
) = 119911119890minus119894120596120591lowast
2+ 119911119890119894120596120591lowast
2+119882(1)
20(minus
120591lowast
2
1205911
)
1199112
2
+119882(1)
11(minus
120591lowast
2
1205911
)119911119911 +119882(1)
02(minus
120591lowast
2
1205911
)
1199112
2
+ 119900 (|(119911 119911)|3)
1199092119905(minus
120591lowast
2
1205911
) = 119911120572119890minus119894120596120591lowast
2+ 119911 120572119890
119894120596120591lowast
2+119882(2)
20(minus
120591lowast
2
1205911
)
1199112
2
+119882(2)
11(minus
120591lowast
2
1205911
)119911119911 +119882(2)
02(minus
120591lowast
2
1205911
)
1199112
2
+ 119900 (|(119911 119911)|3)
1199093119905(minus
120591lowast
2
1205911
) = 119911120573119890minus119894120596120591lowast
2+ 119911120573119890
120596120591lowast
2+119882(3)
20(minus
120591lowast
2
1205911
)
1199112
2
+119882(3)
11(minus
120591lowast
2
1205911
)119911119911 +119882(3)
02(minus
120591lowast
2
1205911
)
1199112
2
+ 119900 (|(119911 119911)|3)
(66)
Abstract and Applied Analysis 9
It follows together with (50) that
119892 (119911 119911)=119902lowast(0) 1198910(119911 119911) =119863120591
1(1 120572lowast 120573
lowast
) (119891(0)
1119891(0)
2119891(0)
3)
119879
= 1198631205911[minus (119886
11+ 1198973) 1206012
1(0) minus 119897
11206011(0) 1206012(0)
minus 11989721206012
1(0) 1206012(0) minus 119897
51206013
1(0) minus 119897
41206013
1(0) 1206012(0)
+ sdot sdot sdot ]
+ 120572lowast[11989761206012(0) 1206013(0) (1198977+ 11988622) 1206012
2(0)
+ 11989711206011(minus1) 120601
2(0)
+ 11989731206012
1(minus1) minus 119897
81206013
2(0) minus 119897
91206012
2(0) 1206013(0)
+ 11989721206012
1(minus1) 120601
2(0) + 119897
31206013
1(minus1) + sdot sdot sdot ]
+ 120573
lowast
[11989761206012(minus
120591lowast
2
1205911
)1206013(0) + 119897
71206012
2(minus
120591lowast
2
1205911
)
minus 119886331206012
3(0) + 119897
81206013
2(minus
120591lowast
2
1205911
) + sdot sdot sdot ]
(67)
Comparing the coefficients with (65) we have
11989220
= 1198631205911[minus2 (119886
11+ 1198973) minus 2120572119897
1]
+ 120572lowast[21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591
1
minus21198976120572120573 minus 2 (119897
7+ 11988622) 1205722]
+ 120573
lowast
[21198976120572120573119890minus119894120596120591lowast
2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591
lowast
2
minus2119886331205732]
11989211
= 1198631205911[minus2 (119886
11+ 1198973) minus 1198971(120572 + 120572)]
+ 120572lowast[1198971(1205721198901198941205961205911+ 120572119890minus1198941205961205911) minus 1198976(120572120573 + 120572120573)
minus2 (1198977+ 11988622) 120572120572 + 2119897
3]
+ 120573
lowast
[1198976(120573120572119890119894120596120591lowast
2+ 120572120573119890
minus119894120596120591lowast
2)
+ 21198977120572120572 minus 119886
33120573120573]
11989202
= 21198631205911[minus2 (119886
11+ 1198973) minus 21198971120572]
+ 120572lowast[minus21198976120572120573 minus 2 (119897
7+ 11988622) 1205722+ 211989711205721198901198941205961205911
+2119897311989021198941205961205911]
+ 120573
lowast
[21198976120572120573119890119894120596120591lowast
2+ 2119897712057221198902119894120596120591lowast
2minus 11988633120573
2
]
11989221
= 1198631205911[minus (119886
11+ 1198973) (2119882
(1)
20(0) + 4119882
(1)
11(0))
minus 1198971(2120572119882
(1)
11(0) + 120572119882
(1)
20(0) + 119882
(2)
20(0)
+ 2119882(2)
11(0))]
+ 120572lowast[minus1198976(2120573119882
(2)
11(0) + 120572119882
(3)
20(0) + 120573119882
(2)
20(0)
+ 2120572119882(3)
11(0)) minus (119897
7+ 11988622)
times (4120572119882(2)
11(0) + 2120572119882
(2)
20(0))
+ 1198971(2120572119882
(1)
11(minus1) + 120572119882
(1)
20(minus1)
+ 119882(2)
20(0) 1198901198941205961205911+ 2119882
(2)
11(0) 119890minus1198941205961205911)
+ 1198973(4119882(1)
11(minus1) 119890
minus1198941205961205911+2119882(1)
20(minus1) 119890
1198941205961205911)]
+ 120573
lowast
[1198976(2120573119882
(2)
11(minus
120591lowast
2
1205911
) + 120573119882(2)
20(minus
120591lowast
2
1205911
)
+ 120572119882(3)
20(0) 119890119894120596120591lowast
2+ 2120572119882
(3)
11(0) 119890minus119894120596120591lowast
2)
+ 1198977(4120572119882
(2)
11(minus
120591lowast
2
1205911
) 119890minus119894120596120591lowast
2
+2120572119882(2)
20(minus
120591lowast
2
1205911
) 119890119894120596120591lowast
2)
minus 11988633(4120573119882
(3)
11(0) + 2120573119882
(3)
20(0))]
(68)
where
11988220(120579) =
11989411989220
1205961205911
119902 (0) 1198901198941205961205911120579+
11989411989202
31205961205911
119902 (0) 119890minus1198941205961205911120579+ 119864111989021198941205961205911120579
11988211(120579) = minus
11989411989211
1205961205911
119902 (0) 1198901198941205961205911120579+
11989411989211
1205961205911
119902 (0) 119890minus1198941205961205911120579+ 1198642
1198641= 2
[[[[[
[
2119894120596 minus 11988711
minus11988712
0
minus11988821119890minus2119894120596120591
12119894120596 minus 119887
22minus11988723
0 minus11988932119890minus2119894120596120591
lowast
22119894120596 minus 119887
33
]]]]]
]
minus1
times
[[[[[
[
minus2 (11988611
+ 1198973) minus 2120572119897
1
21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591
1minus 21198976120572120573 minus 2 (119897
7+ 11988622) 1205722
21198976120572120573119890minus119894120596120591lowast
2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591
lowast
2minus 2119886331205732
]]]]]
]
10 Abstract and Applied Analysis
1198642= 2
[[[[
[
minus11988711
minus11988712
0
minus11988821
minus11988722
minus11988723
0 minus11988932
minus11988733
]]]]
]
minus1
times
[[[[
[
minus2 (11988611
+ 1198973) minus 1198971(120572 + 120572)
1198971(1205721198901198941205961205911+ 120572119890minus1198941205961205911) minus 1198976(120572120573 + 120572120573) minus 2 (119897
7+ 11988622) 120572120572 + 2119897
3
1198976(120573120572119890119894120596120591lowast
2+ 120572120573119890
minus119894120596120591lowast
2) + 21198977120572120572 minus 119886
33120573120573
]]]]
]
(69)
Thus we can determine11988220(120579) and119882
11(120579) Furthermore
we can determine each 119892119894119895by the parameters and delay in (3)
Thus we can compute the following values
1198881(0) =
119894
21205961205911
(1198922011989211
minus 2100381610038161003816100381611989211
1003816100381610038161003816
2
minus
1
3
100381610038161003816100381611989202
1003816100381610038161003816
2
) +
1
2
11989221
1205832= minus
Re 1198881(0)
Re 1205821015840 (1205911)
1198792= minus
Im 1198881(0) + 120583
2Im 120582
1015840(1205911)
1205961205911
1205732= 2Re 119888
1(0)
(70)
which determine the quantities of bifurcating periodic solu-tions in the center manifold at the critical value 120591
1 Suppose
Re1205821015840(1205911) gt 0 120583
2determines the directions of the Hopf
bifurcation if 1205832
gt 0(lt 0) then the Hopf bifurcationis supercritical (subcritical) and the bifurcation exists for120591 gt 120591
1(lt 1205911) 1205732determines the stability of the bifurcation
periodic solutions the bifurcating periodic solutions arestable (unstable) if 120573
2lt 0(gt 0) and119879
2determines the period
of the bifurcating periodic solutions the period increases(decreases) if 119879
2gt 0(lt 0)
4 Numerical Simulation
We consider system (3) by taking the following coefficients1198861
= 03 11988611
= 58889 11988612
= 1 1198981
= 1 1198862
=
01 11988621
= 27 11988622
= 12 11988623
= 12 1198982
= 1 1198863
=
02 11988632
= 25 11988633
= 12 We have the unique positiveequilibrium 119864
lowast= (00451 00357 00551)
By computation we get 11989810
= minus00104 12059611
= 0516411991111
= 02666 ℎ10158401(11991111) = 03402 120591
20
= 32348 FromTheorem 6 we know that when 120591
1= 0 the positive
equilibrium 119864lowastis locally asymptotically stable for 120591
2isin
[0 32348) When 1205912crosses 120591
20
the equilibrium 119864lowastloses its
stability and Hopf bifurcation occurs From the algorithm inSection 3 we have 120583
2= 56646 120573
2= minus31583 119879
2= 5445
whichmeans that the bifurcation is supercritical and periodicsolution is stable The trajectories and the phase graphs areshown in Figures 1 and 2
Regarding 1205911as a parameter and let 120591
2= 29 isin
[0 32348) we can observe that with 1205911increasing the pos-
itive equilibrium 119864lowastloses its stability and Hopf bifurcation
occurs (see Figures 3 and 4)
5 Global Continuationof Local Hopf Bifurcations
In this section we study the global continuation of peri-odic solutions bifurcating from the positive equilibrium(119864lowast 120591(119895)
1119896) (119896 = 1 2 6 119895 = 0 1 ) Throughout this
section we follow closely the notations in [24] and assumethat 120591
2= 120591lowast
2isin [0 120591
210
) regarding 1205911as a parameter For
simplification of notations setting 119911119905(119905) = (119909
1119905 1199092119905 1199093119905)119879 we
may rewrite system (3) as the following functional differentialequation
(119905) = 119865 (119911119905 1205911 119901) (71)
where 119911119905(120579) = (119909
1119905(120579) 1199092119905(120579) 1199093119905(120579))119879
= (1199091(119905 + 120579) 119909
2(119905 +
120579) 1199093(119905 + 120579))
119879 for 119905 ge 0 and 120579 isin [minus1205911 0] Since 119909
1(119905) 1199092(119905)
and 1199093(119905) denote the densities of the prey the predator and
the top predator respectively the positive solution of system(3) is of interest and its periodic solutions only arise in the firstquadrant Thus we consider system (3) only in the domain1198773
+= (1199091 1199092 1199093) isin 1198773 1199091gt 0 119909
2gt 0 119909
3gt 0 It is obvious
that (71) has a unique positive equilibrium 119864lowast(119909lowast
1 119909lowast
2 119909lowast
3) in
1198773
+under the assumption (119867
1)ndash(1198674) Following the work of
[24] we need to define
119883 = 119862([minus1205911 0] 119877
3
+)
Γ = 119862119897 (119911 1205911 119901) isin X times R times R+
119911 is a 119901-periodic solution of system (71)
N = (119911 1205911 119901) 119865 (119911 120591
1 119901) = 0
(72)
Let ℓ(119864lowast120591(119895)
11198962120587120596
(119895)
1119896)denote the connected component pass-
ing through (119864lowast 120591(119895)
1119896 2120587120596
(119895)
1119896) in Γ where 120591
(119895)
1119896is defined by
(43) We know that ℓ(119864lowast120591(119895)
11198962120587120596
(119895)
1119896)through (119864
lowast 120591(119895)
1119896 2120587120596
(119895)
1119896)
is nonemptyFor the benefit of readers we first state the global Hopf
bifurcation theory due to Wu [24] for functional differentialequations
Lemma 9 Assume that (119911lowast 120591 119901) is an isolated center satis-
fying the hypotheses (A1)ndash(A4) in [24] Denote by ℓ(119911lowast120591119901)
theconnected component of (119911
lowast 120591 119901) in Γ Then either
(i) ℓ(119911lowast120591119901)
is unbounded or(ii) ℓ(119911lowast120591119901)
is bounded ℓ(119911lowast120591119901)
cap Γ is finite and
sum
(119911120591119901)isinℓ(119911lowast120591119901)capN
120574119898(119911lowast 120591 119901) = 0 (73)
for all 119898 = 1 2 where 120574119898(119911lowast 120591 119901) is the 119898119905ℎ crossing
number of (119911lowast 120591 119901) if119898 isin 119869(119911
lowast 120591 119901) or it is zero if otherwise
Clearly if (ii) in Lemma 9 is not true then ℓ(119911lowast120591119901)
isunbounded Thus if the projections of ℓ
(119911lowast120591119901)
onto 119911-spaceand onto 119901-space are bounded then the projection of ℓ
(119911lowast120591119901)
onto 120591-space is unbounded Further if we can show that the
Abstract and Applied Analysis 11
0 200 400 60000451
00451
00452
t
x1(t)
t
0 200 400 60000356
00357
00358
x2(t)
t
0 200 400 6000055
00551
00552
x3(t)
00451
00451
00452
00356
00357
00357
003570055
0055
00551
00551
00552
x 1(t)
x2 (t)
x3(t)
Figure 1 The trajectories and the phase graph with 1205911= 0 120591
2= 29 lt 120591
20= 32348 119864
lowastis locally asymptotically stable
0 500 1000004
0045
005
0 500 1000002
004
006
0 500 10000
005
01
t
t
t
x1(t)
x2(t)
x3(t)
0044
0046
0048
002
003
004
005003
004
005
006
007
008
x 1(t)
x2 (t)
x3(t)
Figure 2 The trajectories and the phase graph with 1205911= 0 120591
2= 33 gt 120591
20= 32348 a periodic orbit bifurcate from 119864
lowast
12 Abstract and Applied Analysis
0 500 1000 1500
00451
00452
0 500 1000 15000035
0036
0037
0 500 1000 15000054
0055
0056
t
t
t
x1(t)
x2(t)
x3(t)
0045100451
0045200452
0035
00355
003600546
00548
0055
00552
00554
00556
00558
x1(t)
x2 (t)
x3(t)
Figure 3 The trajectories and the phase graph with 1205911= 09 120591
2= 29 119864
lowastis locally asymptotically stable
0 500 1000 15000044
0045
0046
0 500 1000 1500002
003
004
0 500 1000 1500004
006
008
t
t
t
x1(t)
x2(t)
x3(t)
00445
0045
00455
0032
0034
0036
00380052
0054
0056
0058
006
0062
0064
x 1(t)
x2 (t)
x3(t)
Figure 4 The trajectories and the phase graph with 1205911= 12 120591
2= 29 a periodic orbit bifurcate from 119864
lowast
Abstract and Applied Analysis 13
projection of ℓ(119911lowast120591119901)
onto 120591-space is away from zero thenthe projection of ℓ
(119911lowast120591119901)
onto 120591-space must include interval[120591 infin) Following this ideal we can prove our results on theglobal continuation of local Hopf bifurcation
Lemma 10 If the conditions (1198671)ndash(1198674) hold then all nontriv-
ial periodic solutions of system (71) with initial conditions
1199091(120579) = 120593 (120579) ge 0 119909
2(120579) = 120595 (120579) ge 0
1199093(120579) = 120601 (120579) ge 0 120579 isin [minus120591
1 0)
120593 (0) gt 0 120595 (0) gt 0 120601 (0) gt 0
(74)
are uniformly bounded
Proof Suppose that (1199091(119905) 1199092(119905) 1199093(119905)) are nonconstant peri-
odic solutions of system (3) and define
1199091(1205851) = min 119909
1(119905) 119909
1(1205781) = max 119909
1(119905)
1199092(1205852) = min 119909
2(119905) 119909
2(1205782) = max 119909
2(119905)
1199093(1205853) = min 119909
3(119905) 119909
3(1205783) = max 119909
3(119905)
(75)
It follows from system (3) that
1199091(119905) = 119909
1(0) expint
119905
0
[1198861minus 119886111199091(119904) minus
119886121199092(119904)
1198981+ 1199091(119904)
] 119889119904
1199092(119905) = 119909
2(0) expint
119905
0
[minus1198862+
119886211199091(119904 minus 1205911)
1198981+ 1199091(119904 minus 1205911)
minus 119886221199092(119904) minus
119886231199093(119904)
1198982+ 1199092(119904)
] 119889119904
1199093(119905) = 119909
3(0) expint
119905
0
[minus1198863+
119886321199092(119904 minus 120591
lowast
2)
1198982+ 1199092(119904 minus 120591
lowast
2)
minus 119886331199093(119904)] 119889119904
(76)
which implies that the solutions of system (3) cannot crossthe 119909119894-axis (119894 = 1 2 3) Thus the nonconstant periodic orbits
must be located in the interior of first quadrant It followsfrom initial data of system (3) that 119909
1(119905) gt 0 119909
2(119905) gt
0 1199093(119905) gt 0 for 119905 ge 0
From the first equation of system (3) we can get
0 = 1198861minus 119886111199091(1205781) minus
119886121199092(1205781)
1198981+ 1199091(1205781)
le 1198861minus 119886111199091(1205781) (77)
thus we have
1199091(1205781) le
1198861
11988611
(78)
From the second equation of (3) we obtain
0 = minus 1198862+
119886211199091(1205782minus 1205911)
1198981+ 1199091(1205782minus 1205911)
minus 119886221199092(1205782)
minus
119886231199093(1205782)
1198982+ 1199092(1205782)
le minus1198862+
11988621(119886111988611)
1198981+ (119886111988611)
minus 119886221199092(1205782)
(79)
therefore one can get
1199092(1205782) le
minus1198862(119886111198981+ 1198861) + 119886111988621
11988622(119886111198981+ 1198861)
≜ 1198721 (80)
Applying the third equation of system (3) we know
0 = minus1198863+
119886321199092(1205783minus 120591lowast
2)
1198982+ 1199092(1205783minus 120591lowast
2)
minus 119886331199093(1205783)
le minus1198863+
119886321198721
1198982+1198721
minus 119886331199093(1205783)
(81)
It follows that
1199093(1205783) le
minus1198863(1198982+1198721) + 119886321198721
11988633(1198982+1198721)
≜ 1198722 (82)
This shows that the nontrivial periodic solution of system (3)is uniformly bounded and the proof is complete
Lemma 11 If the conditions (1198671)ndash(1198674) and
(H7) 11988622
minus (119886211198982) gt 0 [(119886
11minus 11988611198981)11988622
minus
(11988612119886211198982
1)]11988633
minus (11988611
minus (11988611198981))[(119886211198982)11988633
+
(11988623119886321198982
2)] gt 0
hold then system (3) has no nontrivial 1205911-periodic solution
Proof Suppose for a contradiction that system (3) has non-trivial periodic solution with period 120591
1 Then the following
system (83) of ordinary differential equations has nontrivialperiodic solution
1198891199091
119889119905
= 1199091(119905) [1198861minus 119886111199091(119905) minus
119886121199092(119905)
1198981+ 1199091(119905)
]
1198891199092
119889119905
= 1199092(119905) [minus119886
2+
119886211199091(119905)
1198981+ 1199091(119905)
minus 119886221199092(119905) minus
119886231199093(119905)
1198982+ 1199092(119905)
]
1198891199093
119889119905
= 1199093(119905) [minus119886
3+
119886321199092(119905 minus 120591
lowast
2)
1198982+ 1199092(119905 minus 120591
lowast
2)
minus 119886331199093(119905)]
(83)
which has the same equilibria to system (3) that is
1198641= (0 0 0) 119864
2= (
1198861
11988611
0 0)
1198643= (1199091 1199092 0) 119864
lowast= (119909lowast
1 119909lowast
2 119909lowast
3)
(84)
Note that 119909119894-axis (119894 = 1 2 3) are the invariable manifold of
system (83) and the orbits of system (83) do not intersect each
14 Abstract and Applied Analysis
other Thus there are no solutions crossing the coordinateaxes On the other hand note the fact that if system (83) hasa periodic solution then there must be the equilibrium in itsinterior and that 119864
1 1198642 1198643are located on the coordinate
axis Thus we conclude that the periodic orbit of system (83)must lie in the first quadrant If (119867
7) holds it is well known
that the positive equilibrium 119864lowastis globally asymptotically
stable in the first quadrant (see [1])Thus there is no periodicorbit in the first quadrant too The above discussion meansthat (83) does not have any nontrivial periodic solution It isa contradiction Therefore the lemma is confirmed
Theorem 12 Suppose the conditions of Theorem 8 and (1198677)
hold let120596119896and 120591(119895)1119896
be defined in Section 2 then when 1205911gt 120591(119895)
1119896
system (3) has at least 119895 minus 1 periodic solutions
Proof It is sufficient to prove that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is [120591
1 +infin) for each 119895 ge 1 where
1205911le 120591(119895)
1119896
In following we prove that the hypotheses (A1)ndash(A4) in[24] hold
(1) From system (3) we know easily that the followingconditions hold
(A1) 119865 isin 1198622(1198773
+times119877+times119877+) where 119865 = 119865|
1198773
+times119877+times119877+
rarr
1198773
+
(A3) 119865(120601 1205911 119901) is differential with respect to 120601
(2) It follows from system (3) that
119863119911119865 (119911 120591
1 119901) =
[[[[[[[[[[[
[
1198861minus 2119886111199091minus
1198861211989811199092
(1198981+ 1199091)2
minus
119886121199091
1198981+ 1199091
0
1198862111989811199092
(1198981+ 1199091)2
minus1198862+
119886211199091
1198981+ 1199091
minus 2119886221199092minus
1198862311989821199093
(1198982+ 1199092)2
minus
119886231199092
1198982+ 1199092
0
1198863211989821199093
(1198982+ 1199092)2
minus1198863+
119886321199092
1198982+ 1199092
minus 2119886331199093
]]]]]]]]]]]
]
(85)
Then under the assumption (1198671)ndash(1198674) we have
det119863119911119865 (119911lowast 1205911 119901)
= det
[[[[[[[[
[
minus11988611119909lowast
1+
11988612119909lowast
1119909lowast
2
(1198981+ 119909lowast
1)2
11988612119909lowast
1
1198981+ 119909lowast
1
0
119886211198981119909lowast
2
(1198981+ 119909lowast
1)2
minus11988622119909lowast
2+
11988623119909lowast
2119909lowast
3
(1198982+ 119909lowast
2)2
minus
11988623119909lowast
2
1198982+ 119909lowast
2
0
119886321198982119909lowast
3
(1198982+ 119909lowast
2)2
minus11988633119909lowast
3
]]]]]]]]
]
= minus
1198862
21199101lowast1199102lowast
(1 + 11988721199101lowast)3[minus119909lowast+
11988611198871119909lowast1199101lowast
(1 + 1198871119909lowast)2] = 0
(86)
From (86) we know that the hypothesis (A2) in [24]is satisfied
(3) The characteristic matrix of (71) at a stationarysolution (119911 120591
0 1199010) where 119911 = (119911
(1) 119911(2) 119911(3)) isin 119877
3
takes the following form
Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863
120601119865 (119911 120591
1 119901) (119890
120582119868) (87)
that is
Δ (119911 1205911 119901) (120582)
=
[[[[[[[[[[[[[[[[[
[
120582 minus 1198861+ 211988611119911(1)
+
119886121198981119911(2)
(1198981+ 119911(1))
2
11988612119911(1)
1198981+ 119911(1)
0
minus
119886211198981119911(2)
(1198981+ 119911(1))
2119890minus1205821205911
120582 + 1198862minus
11988621119911(1)
1198981+ 119911(1)
+ 211988622119911(2)
+
119886231198982119911(3)
(1198982+ 119911(2))
2
11988623119911(2)
1198982+ 119911(2)
0 minus
119886321198982119911(3)
(1198982+ 119911(2))
2119890minus120582120591lowast
2120582 + 1198863minus
11988632119911(2)
1198982+ 119911(2)
+ 211988633119911(3)
]]]]]]]]]]]]]]]]]
]
(88)
Abstract and Applied Analysis 15
From (88) we have
det (Δ (119864lowast 1205911 119901) (120582))
= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902
0+ 1199030)] 119890minus1205821205911
(89)
Note that (89) is the same as (20) from the discussion inSection 2 about the local Hopf bifurcation it is easy to verifythat (119864
lowast 120591(119895)
1119896 2120587120596
119896) is an isolated center and there exist 120598 gt
0 120575 gt 0 and a smooth curve 120582 (120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575) rarr C
such that det(Δ(120582(1205911))) = 0 |120582(120591
1) minus 120596119896| lt 120598 for all 120591
1isin
[120591(119895)
1119896minus 120575 120591(119895)
1119896+ 120575] and
120582 (120591(119895)
1119896) = 120596119896119894
119889Re 120582 (1205911)
1198891205911
1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)
1119896
gt 0 (90)
Let
Ω1205982120587120596
119896
= (120578 119901) 0 lt 120578 lt 120598
10038161003816100381610038161003816100381610038161003816
119901 minus
2120587
120596119896
10038161003816100381610038161003816100381610038161003816
lt 120598 (91)
It is easy to see that on [120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575] times 120597Ω
1205982120587120596119896
det(Δ(119864
lowast 1205911 119901)(120578 + (2120587119901)119894)) = 0 if and only if 120578 = 0
1205911= 120591(119895)
1119896 119901 = 2120587120596
119896 119896 = 1 2 3 119895 = 0 1 2
Therefore the hypothesis (A4) in [24] is satisfiedIf we define
119867plusmn(119864lowast 120591(119895)
1119896
2120587
120596119896
) (120578 119901)
= det(Δ (119864lowast 120591(119895)
1119896plusmn 120575 119901) (120578 +
2120587
119901
119894))
(92)
then we have the crossing number of isolated center(119864lowast 120591(119895)
1119896 2120587120596
119896) as follows
120574(119864lowast 120591(119895)
1119896
2120587
120596119896
) = deg119861(119867minus(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
minus deg119861(119867+(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
= minus1
(93)
Thus we have
sum
(1199111205911119901)isinC
(119864lowast120591(119895)
11198962120587120596119896)
120574 (119911 1205911 119901) lt 0
(94)
where (119911 1205911 119901) has all or parts of the form
(119864lowast 120591(119896)
1119895 2120587120596
119896) (119895 = 0 1 ) It follows from
Lemma 9 that the connected component ℓ(119864lowast120591(119895)
11198962120587120596
119896)
through (119864lowast 120591(119895)
1119896 2120587120596
119896) is unbounded for each center
(119911lowast 1205911 119901) (119895 = 0 1 ) From the discussion in Section 2
we have
120591(119895)
1119896=
1
120596119896
times arccos(((11990121205962
119896minus1199010) (1199030+1199020)
+ 1205962(1205962minus1199011) (1199031+1199021))
times ((1199030+1199020)2
+ (1199031+1199021)2
1205962
119896)
minus1
) +2119895120587
(95)
where 119896 = 1 2 3 119895 = 0 1 Thus one can get 2120587120596119896le 120591(119895)
1119896
for 119895 ge 1Now we prove that the projection of ℓ
(119864lowast120591(119895)
11198962120587120596
119896)onto
1205911-space is [120591
1 +infin) where 120591
1le 120591(119895)
1119896 Clearly it follows
from the proof of Lemma 11 that system (3) with 1205911
= 0
has no nontrivial periodic solution Hence the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is away from zero
For a contradiction we suppose that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is bounded this means that the
projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is included in a
interval (0 120591lowast) Noticing 2120587120596119896lt 120591119895
1119896and applying Lemma 11
we have 119901 lt 120591lowast for (119911(119905) 120591
1 119901) belonging to ℓ
(119864lowast120591(119895)
11198962120587120596
119896)
This implies that the projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 119901-
space is bounded Then applying Lemma 10 we get thatthe connected component ℓ
(119864lowast120591(119895)
11198962120587120596
119896)is bounded This
contradiction completes the proof
6 Conclusion
In this paper we take our attention to the stability and Hopfbifurcation analysis of a predator-prey systemwithMichaelis-Menten type functional response and two unequal delaysWe obtained some conditions for local stability and Hopfbifurcation occurring When 120591
1= 1205912 we derived the explicit
formulas to determine the properties of periodic solutionsby the normal form method and center manifold theoremSpecially the global existence results of periodic solutionsbifurcating from Hopf bifurcations are also established byusing a global Hopf bifurcation result due to Wu [24]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to thank the anonymous referees fortheir careful reading and constructive suggestions which leadto truly significant improvement of the paper This researchis supported by the National Natural Science Foundation ofChina (no 11061016)
16 Abstract and Applied Analysis
References
[1] R Xu and M A J Chaplain ldquoPersistence and global stabilityin a delayed predator-prey system with Michaelis-Menten typefunctional responserdquo Applied Mathematics and Computationvol 130 no 2-3 pp 441ndash455 2002
[2] Y L Song Y H Peng and J JWei ldquoBifurcations for a predator-prey system with two delaysrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 466ndash479 2008
[3] G-P Hu W-T Li and X-P Yan ldquoHopf bifurcations in apredator-prey system with multiple delaysrdquo Chaos Solitons ampFractals vol 42 no 2 pp 1273ndash1285 2009
[4] M Liao X Tang and C Xu ldquoBifurcation analysis for a three-species predator-prey systemwith two delaysrdquoCommunicationsin Nonlinear Science and Numerical Simulation vol 17 no 1 pp183ndash194 2012
[5] C Xu and P Li ldquoDynamical analysis in a delayed predator-prey model with two delaysrdquo Discrete Dynamics in Nature andSociety vol 2012 Article ID 652947 22 pages 2012
[6] J-F Zhang ldquoStability and bifurcation periodic solutions ina Lotka-Volterra competition system with multiple delaysrdquoNonlinear Dynamics vol 70 no 1 pp 849ndash860 2012
[7] J Xia Z Yu and R Yuan ldquoStability and Hopf bifurcation ina symmetric Lotka-Volterra predator-prey system with delaysrdquoElectronic Journal of Differential Equations vol 2013 no 9 pp1ndash16 2013
[8] H Wan and J Cui ldquoA malaria model with two delaysrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 601265 8pages 2013
[9] M Zhao ldquoHopf bifurcation analysis for a semiratio-dependentpredator-prey system with two delaysrdquo Abstract and AppliedAnalysis vol 2013 Article ID 495072 13 pages 2013
[10] S Feyissa and S Banerjee ldquoDelay-induced oscillatory dynamicsin humoral mediated immune response with two time delaysrdquoNonlinear Analysis Real World Applications vol 14 no 1 pp35ndash52 2013
[11] G Zhang Y Shen andB Chen ldquoHopf bifurcation of a predator-prey systemwith predator harvesting and twodelaysrdquoNonlinearDynamics vol 73 no 4 pp 2119ndash2131 2013
[12] X-Y Meng H-F Huo and H Xiang ldquoHopf bifurcation in athree-species systemwith delaysrdquo Journal of AppliedMathemat-ics and Computing vol 35 no 1-2 pp 635ndash661 2011
[13] J-F Zhang ldquoGlobal existence of bifurcated periodic solutions ina commensalism model with delaysrdquo Applied Mathematics andComputation vol 218 no 23 pp 11688ndash11699 2012
[14] Y Wang W Jiang and H Wang ldquoStability and global Hopfbifurcation in toxic phytoplankton-zooplankton model withdelay and selective harvestingrdquoNonlinear Dynamics vol 73 no1-2 pp 881ndash896 2013
[15] Y Ma ldquoGlobal Hopf bifurcation in the Leslie-Gower predator-prey model with two delaysrdquo Nonlinear Analysis Real WorldApplications vol 13 no 1 pp 370ndash375 2012
[16] X-Y Meng H-F Huo and X-B Zhang ldquoStability and globalHopf bifurcation in a delayed food web consisting of a preyand two predatorsrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4335ndash4348 2011
[17] Y Song J Wei andM Han ldquoLocal and global Hopf bifurcationin a delayed hematopoiesis modelrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol14 no 11 pp 3909ndash3919 2004
[18] Y Song and J Wei ldquoLocal Hopf bifurcation and global periodicsolutions in a delayed predator-prey systemrdquo Journal of Mathe-matical Analysis and Applications vol 301 no 1 pp 1ndash21 2005
[19] J Wei and M Y Li ldquoGlobal existence of periodic solutions in atri-neuron network model with delaysrdquo Physica D vol 198 no1-2 pp 106ndash119 2004
[20] T Zhao Y Kuang and H L Smith ldquoGlobal existence of peri-odic solutions in a class of delayed Gause-type predator-preysystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 28 no 8 pp 1373ndash1394 1997
[21] S J Gao L S Chen and Z D Teng ldquoHopf bifurcation andglobal stability for a delayed predator-prey system with stagestructure for predatorrdquo Applied Mathematics and Computationvol 202 no 2 pp 721ndash729 2008
[22] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977
[23] B Hassard D Kazarinoff and Y WanTheory and Applicationsof Hopf Bifurcation Cambridge University Press CambridgeUK 1981
[24] J H Wu ldquoSymmetric functional-differential equations andneural networks with memoryrdquo Transactions of the AmericanMathematical Society vol 350 no 12 pp 4799ndash4838 1998
[25] V Hutson ldquoThe existence of an equilibrium for permanentsystemsrdquo The Rocky Mountain Journal of Mathematics vol 20no 4 pp 1033ndash1040 1990
[26] S G Ruan and J J Wei ldquoOn the zeros of transcendentalfunctions with applications to stability of delay differentialequations with two delaysrdquo Dynamics of Continuous Discreteamp Impulsive Systems A vol 10 no 6 pp 863ndash874 2003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 9
It follows together with (50) that
119892 (119911 119911)=119902lowast(0) 1198910(119911 119911) =119863120591
1(1 120572lowast 120573
lowast
) (119891(0)
1119891(0)
2119891(0)
3)
119879
= 1198631205911[minus (119886
11+ 1198973) 1206012
1(0) minus 119897
11206011(0) 1206012(0)
minus 11989721206012
1(0) 1206012(0) minus 119897
51206013
1(0) minus 119897
41206013
1(0) 1206012(0)
+ sdot sdot sdot ]
+ 120572lowast[11989761206012(0) 1206013(0) (1198977+ 11988622) 1206012
2(0)
+ 11989711206011(minus1) 120601
2(0)
+ 11989731206012
1(minus1) minus 119897
81206013
2(0) minus 119897
91206012
2(0) 1206013(0)
+ 11989721206012
1(minus1) 120601
2(0) + 119897
31206013
1(minus1) + sdot sdot sdot ]
+ 120573
lowast
[11989761206012(minus
120591lowast
2
1205911
)1206013(0) + 119897
71206012
2(minus
120591lowast
2
1205911
)
minus 119886331206012
3(0) + 119897
81206013
2(minus
120591lowast
2
1205911
) + sdot sdot sdot ]
(67)
Comparing the coefficients with (65) we have
11989220
= 1198631205911[minus2 (119886
11+ 1198973) minus 2120572119897
1]
+ 120572lowast[21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591
1
minus21198976120572120573 minus 2 (119897
7+ 11988622) 1205722]
+ 120573
lowast
[21198976120572120573119890minus119894120596120591lowast
2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591
lowast
2
minus2119886331205732]
11989211
= 1198631205911[minus2 (119886
11+ 1198973) minus 1198971(120572 + 120572)]
+ 120572lowast[1198971(1205721198901198941205961205911+ 120572119890minus1198941205961205911) minus 1198976(120572120573 + 120572120573)
minus2 (1198977+ 11988622) 120572120572 + 2119897
3]
+ 120573
lowast
[1198976(120573120572119890119894120596120591lowast
2+ 120572120573119890
minus119894120596120591lowast
2)
+ 21198977120572120572 minus 119886
33120573120573]
11989202
= 21198631205911[minus2 (119886
11+ 1198973) minus 21198971120572]
+ 120572lowast[minus21198976120572120573 minus 2 (119897
7+ 11988622) 1205722+ 211989711205721198901198941205961205911
+2119897311989021198941205961205911]
+ 120573
lowast
[21198976120572120573119890119894120596120591lowast
2+ 2119897712057221198902119894120596120591lowast
2minus 11988633120573
2
]
11989221
= 1198631205911[minus (119886
11+ 1198973) (2119882
(1)
20(0) + 4119882
(1)
11(0))
minus 1198971(2120572119882
(1)
11(0) + 120572119882
(1)
20(0) + 119882
(2)
20(0)
+ 2119882(2)
11(0))]
+ 120572lowast[minus1198976(2120573119882
(2)
11(0) + 120572119882
(3)
20(0) + 120573119882
(2)
20(0)
+ 2120572119882(3)
11(0)) minus (119897
7+ 11988622)
times (4120572119882(2)
11(0) + 2120572119882
(2)
20(0))
+ 1198971(2120572119882
(1)
11(minus1) + 120572119882
(1)
20(minus1)
+ 119882(2)
20(0) 1198901198941205961205911+ 2119882
(2)
11(0) 119890minus1198941205961205911)
+ 1198973(4119882(1)
11(minus1) 119890
minus1198941205961205911+2119882(1)
20(minus1) 119890
1198941205961205911)]
+ 120573
lowast
[1198976(2120573119882
(2)
11(minus
120591lowast
2
1205911
) + 120573119882(2)
20(minus
120591lowast
2
1205911
)
+ 120572119882(3)
20(0) 119890119894120596120591lowast
2+ 2120572119882
(3)
11(0) 119890minus119894120596120591lowast
2)
+ 1198977(4120572119882
(2)
11(minus
120591lowast
2
1205911
) 119890minus119894120596120591lowast
2
+2120572119882(2)
20(minus
120591lowast
2
1205911
) 119890119894120596120591lowast
2)
minus 11988633(4120573119882
(3)
11(0) + 2120573119882
(3)
20(0))]
(68)
where
11988220(120579) =
11989411989220
1205961205911
119902 (0) 1198901198941205961205911120579+
11989411989202
31205961205911
119902 (0) 119890minus1198941205961205911120579+ 119864111989021198941205961205911120579
11988211(120579) = minus
11989411989211
1205961205911
119902 (0) 1198901198941205961205911120579+
11989411989211
1205961205911
119902 (0) 119890minus1198941205961205911120579+ 1198642
1198641= 2
[[[[[
[
2119894120596 minus 11988711
minus11988712
0
minus11988821119890minus2119894120596120591
12119894120596 minus 119887
22minus11988723
0 minus11988932119890minus2119894120596120591
lowast
22119894120596 minus 119887
33
]]]]]
]
minus1
times
[[[[[
[
minus2 (11988611
+ 1198973) minus 2120572119897
1
21198971120572119890minus1198941205961205911+ 21198973119890minus2119894120596120591
1minus 21198976120572120573 minus 2 (119897
7+ 11988622) 1205722
21198976120572120573119890minus119894120596120591lowast
2+ 2 (1198977+ 11988622) 1205722119890minus2119894120596120591
lowast
2minus 2119886331205732
]]]]]
]
10 Abstract and Applied Analysis
1198642= 2
[[[[
[
minus11988711
minus11988712
0
minus11988821
minus11988722
minus11988723
0 minus11988932
minus11988733
]]]]
]
minus1
times
[[[[
[
minus2 (11988611
+ 1198973) minus 1198971(120572 + 120572)
1198971(1205721198901198941205961205911+ 120572119890minus1198941205961205911) minus 1198976(120572120573 + 120572120573) minus 2 (119897
7+ 11988622) 120572120572 + 2119897
3
1198976(120573120572119890119894120596120591lowast
2+ 120572120573119890
minus119894120596120591lowast
2) + 21198977120572120572 minus 119886
33120573120573
]]]]
]
(69)
Thus we can determine11988220(120579) and119882
11(120579) Furthermore
we can determine each 119892119894119895by the parameters and delay in (3)
Thus we can compute the following values
1198881(0) =
119894
21205961205911
(1198922011989211
minus 2100381610038161003816100381611989211
1003816100381610038161003816
2
minus
1
3
100381610038161003816100381611989202
1003816100381610038161003816
2
) +
1
2
11989221
1205832= minus
Re 1198881(0)
Re 1205821015840 (1205911)
1198792= minus
Im 1198881(0) + 120583
2Im 120582
1015840(1205911)
1205961205911
1205732= 2Re 119888
1(0)
(70)
which determine the quantities of bifurcating periodic solu-tions in the center manifold at the critical value 120591
1 Suppose
Re1205821015840(1205911) gt 0 120583
2determines the directions of the Hopf
bifurcation if 1205832
gt 0(lt 0) then the Hopf bifurcationis supercritical (subcritical) and the bifurcation exists for120591 gt 120591
1(lt 1205911) 1205732determines the stability of the bifurcation
periodic solutions the bifurcating periodic solutions arestable (unstable) if 120573
2lt 0(gt 0) and119879
2determines the period
of the bifurcating periodic solutions the period increases(decreases) if 119879
2gt 0(lt 0)
4 Numerical Simulation
We consider system (3) by taking the following coefficients1198861
= 03 11988611
= 58889 11988612
= 1 1198981
= 1 1198862
=
01 11988621
= 27 11988622
= 12 11988623
= 12 1198982
= 1 1198863
=
02 11988632
= 25 11988633
= 12 We have the unique positiveequilibrium 119864
lowast= (00451 00357 00551)
By computation we get 11989810
= minus00104 12059611
= 0516411991111
= 02666 ℎ10158401(11991111) = 03402 120591
20
= 32348 FromTheorem 6 we know that when 120591
1= 0 the positive
equilibrium 119864lowastis locally asymptotically stable for 120591
2isin
[0 32348) When 1205912crosses 120591
20
the equilibrium 119864lowastloses its
stability and Hopf bifurcation occurs From the algorithm inSection 3 we have 120583
2= 56646 120573
2= minus31583 119879
2= 5445
whichmeans that the bifurcation is supercritical and periodicsolution is stable The trajectories and the phase graphs areshown in Figures 1 and 2
Regarding 1205911as a parameter and let 120591
2= 29 isin
[0 32348) we can observe that with 1205911increasing the pos-
itive equilibrium 119864lowastloses its stability and Hopf bifurcation
occurs (see Figures 3 and 4)
5 Global Continuationof Local Hopf Bifurcations
In this section we study the global continuation of peri-odic solutions bifurcating from the positive equilibrium(119864lowast 120591(119895)
1119896) (119896 = 1 2 6 119895 = 0 1 ) Throughout this
section we follow closely the notations in [24] and assumethat 120591
2= 120591lowast
2isin [0 120591
210
) regarding 1205911as a parameter For
simplification of notations setting 119911119905(119905) = (119909
1119905 1199092119905 1199093119905)119879 we
may rewrite system (3) as the following functional differentialequation
(119905) = 119865 (119911119905 1205911 119901) (71)
where 119911119905(120579) = (119909
1119905(120579) 1199092119905(120579) 1199093119905(120579))119879
= (1199091(119905 + 120579) 119909
2(119905 +
120579) 1199093(119905 + 120579))
119879 for 119905 ge 0 and 120579 isin [minus1205911 0] Since 119909
1(119905) 1199092(119905)
and 1199093(119905) denote the densities of the prey the predator and
the top predator respectively the positive solution of system(3) is of interest and its periodic solutions only arise in the firstquadrant Thus we consider system (3) only in the domain1198773
+= (1199091 1199092 1199093) isin 1198773 1199091gt 0 119909
2gt 0 119909
3gt 0 It is obvious
that (71) has a unique positive equilibrium 119864lowast(119909lowast
1 119909lowast
2 119909lowast
3) in
1198773
+under the assumption (119867
1)ndash(1198674) Following the work of
[24] we need to define
119883 = 119862([minus1205911 0] 119877
3
+)
Γ = 119862119897 (119911 1205911 119901) isin X times R times R+
119911 is a 119901-periodic solution of system (71)
N = (119911 1205911 119901) 119865 (119911 120591
1 119901) = 0
(72)
Let ℓ(119864lowast120591(119895)
11198962120587120596
(119895)
1119896)denote the connected component pass-
ing through (119864lowast 120591(119895)
1119896 2120587120596
(119895)
1119896) in Γ where 120591
(119895)
1119896is defined by
(43) We know that ℓ(119864lowast120591(119895)
11198962120587120596
(119895)
1119896)through (119864
lowast 120591(119895)
1119896 2120587120596
(119895)
1119896)
is nonemptyFor the benefit of readers we first state the global Hopf
bifurcation theory due to Wu [24] for functional differentialequations
Lemma 9 Assume that (119911lowast 120591 119901) is an isolated center satis-
fying the hypotheses (A1)ndash(A4) in [24] Denote by ℓ(119911lowast120591119901)
theconnected component of (119911
lowast 120591 119901) in Γ Then either
(i) ℓ(119911lowast120591119901)
is unbounded or(ii) ℓ(119911lowast120591119901)
is bounded ℓ(119911lowast120591119901)
cap Γ is finite and
sum
(119911120591119901)isinℓ(119911lowast120591119901)capN
120574119898(119911lowast 120591 119901) = 0 (73)
for all 119898 = 1 2 where 120574119898(119911lowast 120591 119901) is the 119898119905ℎ crossing
number of (119911lowast 120591 119901) if119898 isin 119869(119911
lowast 120591 119901) or it is zero if otherwise
Clearly if (ii) in Lemma 9 is not true then ℓ(119911lowast120591119901)
isunbounded Thus if the projections of ℓ
(119911lowast120591119901)
onto 119911-spaceand onto 119901-space are bounded then the projection of ℓ
(119911lowast120591119901)
onto 120591-space is unbounded Further if we can show that the
Abstract and Applied Analysis 11
0 200 400 60000451
00451
00452
t
x1(t)
t
0 200 400 60000356
00357
00358
x2(t)
t
0 200 400 6000055
00551
00552
x3(t)
00451
00451
00452
00356
00357
00357
003570055
0055
00551
00551
00552
x 1(t)
x2 (t)
x3(t)
Figure 1 The trajectories and the phase graph with 1205911= 0 120591
2= 29 lt 120591
20= 32348 119864
lowastis locally asymptotically stable
0 500 1000004
0045
005
0 500 1000002
004
006
0 500 10000
005
01
t
t
t
x1(t)
x2(t)
x3(t)
0044
0046
0048
002
003
004
005003
004
005
006
007
008
x 1(t)
x2 (t)
x3(t)
Figure 2 The trajectories and the phase graph with 1205911= 0 120591
2= 33 gt 120591
20= 32348 a periodic orbit bifurcate from 119864
lowast
12 Abstract and Applied Analysis
0 500 1000 1500
00451
00452
0 500 1000 15000035
0036
0037
0 500 1000 15000054
0055
0056
t
t
t
x1(t)
x2(t)
x3(t)
0045100451
0045200452
0035
00355
003600546
00548
0055
00552
00554
00556
00558
x1(t)
x2 (t)
x3(t)
Figure 3 The trajectories and the phase graph with 1205911= 09 120591
2= 29 119864
lowastis locally asymptotically stable
0 500 1000 15000044
0045
0046
0 500 1000 1500002
003
004
0 500 1000 1500004
006
008
t
t
t
x1(t)
x2(t)
x3(t)
00445
0045
00455
0032
0034
0036
00380052
0054
0056
0058
006
0062
0064
x 1(t)
x2 (t)
x3(t)
Figure 4 The trajectories and the phase graph with 1205911= 12 120591
2= 29 a periodic orbit bifurcate from 119864
lowast
Abstract and Applied Analysis 13
projection of ℓ(119911lowast120591119901)
onto 120591-space is away from zero thenthe projection of ℓ
(119911lowast120591119901)
onto 120591-space must include interval[120591 infin) Following this ideal we can prove our results on theglobal continuation of local Hopf bifurcation
Lemma 10 If the conditions (1198671)ndash(1198674) hold then all nontriv-
ial periodic solutions of system (71) with initial conditions
1199091(120579) = 120593 (120579) ge 0 119909
2(120579) = 120595 (120579) ge 0
1199093(120579) = 120601 (120579) ge 0 120579 isin [minus120591
1 0)
120593 (0) gt 0 120595 (0) gt 0 120601 (0) gt 0
(74)
are uniformly bounded
Proof Suppose that (1199091(119905) 1199092(119905) 1199093(119905)) are nonconstant peri-
odic solutions of system (3) and define
1199091(1205851) = min 119909
1(119905) 119909
1(1205781) = max 119909
1(119905)
1199092(1205852) = min 119909
2(119905) 119909
2(1205782) = max 119909
2(119905)
1199093(1205853) = min 119909
3(119905) 119909
3(1205783) = max 119909
3(119905)
(75)
It follows from system (3) that
1199091(119905) = 119909
1(0) expint
119905
0
[1198861minus 119886111199091(119904) minus
119886121199092(119904)
1198981+ 1199091(119904)
] 119889119904
1199092(119905) = 119909
2(0) expint
119905
0
[minus1198862+
119886211199091(119904 minus 1205911)
1198981+ 1199091(119904 minus 1205911)
minus 119886221199092(119904) minus
119886231199093(119904)
1198982+ 1199092(119904)
] 119889119904
1199093(119905) = 119909
3(0) expint
119905
0
[minus1198863+
119886321199092(119904 minus 120591
lowast
2)
1198982+ 1199092(119904 minus 120591
lowast
2)
minus 119886331199093(119904)] 119889119904
(76)
which implies that the solutions of system (3) cannot crossthe 119909119894-axis (119894 = 1 2 3) Thus the nonconstant periodic orbits
must be located in the interior of first quadrant It followsfrom initial data of system (3) that 119909
1(119905) gt 0 119909
2(119905) gt
0 1199093(119905) gt 0 for 119905 ge 0
From the first equation of system (3) we can get
0 = 1198861minus 119886111199091(1205781) minus
119886121199092(1205781)
1198981+ 1199091(1205781)
le 1198861minus 119886111199091(1205781) (77)
thus we have
1199091(1205781) le
1198861
11988611
(78)
From the second equation of (3) we obtain
0 = minus 1198862+
119886211199091(1205782minus 1205911)
1198981+ 1199091(1205782minus 1205911)
minus 119886221199092(1205782)
minus
119886231199093(1205782)
1198982+ 1199092(1205782)
le minus1198862+
11988621(119886111988611)
1198981+ (119886111988611)
minus 119886221199092(1205782)
(79)
therefore one can get
1199092(1205782) le
minus1198862(119886111198981+ 1198861) + 119886111988621
11988622(119886111198981+ 1198861)
≜ 1198721 (80)
Applying the third equation of system (3) we know
0 = minus1198863+
119886321199092(1205783minus 120591lowast
2)
1198982+ 1199092(1205783minus 120591lowast
2)
minus 119886331199093(1205783)
le minus1198863+
119886321198721
1198982+1198721
minus 119886331199093(1205783)
(81)
It follows that
1199093(1205783) le
minus1198863(1198982+1198721) + 119886321198721
11988633(1198982+1198721)
≜ 1198722 (82)
This shows that the nontrivial periodic solution of system (3)is uniformly bounded and the proof is complete
Lemma 11 If the conditions (1198671)ndash(1198674) and
(H7) 11988622
minus (119886211198982) gt 0 [(119886
11minus 11988611198981)11988622
minus
(11988612119886211198982
1)]11988633
minus (11988611
minus (11988611198981))[(119886211198982)11988633
+
(11988623119886321198982
2)] gt 0
hold then system (3) has no nontrivial 1205911-periodic solution
Proof Suppose for a contradiction that system (3) has non-trivial periodic solution with period 120591
1 Then the following
system (83) of ordinary differential equations has nontrivialperiodic solution
1198891199091
119889119905
= 1199091(119905) [1198861minus 119886111199091(119905) minus
119886121199092(119905)
1198981+ 1199091(119905)
]
1198891199092
119889119905
= 1199092(119905) [minus119886
2+
119886211199091(119905)
1198981+ 1199091(119905)
minus 119886221199092(119905) minus
119886231199093(119905)
1198982+ 1199092(119905)
]
1198891199093
119889119905
= 1199093(119905) [minus119886
3+
119886321199092(119905 minus 120591
lowast
2)
1198982+ 1199092(119905 minus 120591
lowast
2)
minus 119886331199093(119905)]
(83)
which has the same equilibria to system (3) that is
1198641= (0 0 0) 119864
2= (
1198861
11988611
0 0)
1198643= (1199091 1199092 0) 119864
lowast= (119909lowast
1 119909lowast
2 119909lowast
3)
(84)
Note that 119909119894-axis (119894 = 1 2 3) are the invariable manifold of
system (83) and the orbits of system (83) do not intersect each
14 Abstract and Applied Analysis
other Thus there are no solutions crossing the coordinateaxes On the other hand note the fact that if system (83) hasa periodic solution then there must be the equilibrium in itsinterior and that 119864
1 1198642 1198643are located on the coordinate
axis Thus we conclude that the periodic orbit of system (83)must lie in the first quadrant If (119867
7) holds it is well known
that the positive equilibrium 119864lowastis globally asymptotically
stable in the first quadrant (see [1])Thus there is no periodicorbit in the first quadrant too The above discussion meansthat (83) does not have any nontrivial periodic solution It isa contradiction Therefore the lemma is confirmed
Theorem 12 Suppose the conditions of Theorem 8 and (1198677)
hold let120596119896and 120591(119895)1119896
be defined in Section 2 then when 1205911gt 120591(119895)
1119896
system (3) has at least 119895 minus 1 periodic solutions
Proof It is sufficient to prove that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is [120591
1 +infin) for each 119895 ge 1 where
1205911le 120591(119895)
1119896
In following we prove that the hypotheses (A1)ndash(A4) in[24] hold
(1) From system (3) we know easily that the followingconditions hold
(A1) 119865 isin 1198622(1198773
+times119877+times119877+) where 119865 = 119865|
1198773
+times119877+times119877+
rarr
1198773
+
(A3) 119865(120601 1205911 119901) is differential with respect to 120601
(2) It follows from system (3) that
119863119911119865 (119911 120591
1 119901) =
[[[[[[[[[[[
[
1198861minus 2119886111199091minus
1198861211989811199092
(1198981+ 1199091)2
minus
119886121199091
1198981+ 1199091
0
1198862111989811199092
(1198981+ 1199091)2
minus1198862+
119886211199091
1198981+ 1199091
minus 2119886221199092minus
1198862311989821199093
(1198982+ 1199092)2
minus
119886231199092
1198982+ 1199092
0
1198863211989821199093
(1198982+ 1199092)2
minus1198863+
119886321199092
1198982+ 1199092
minus 2119886331199093
]]]]]]]]]]]
]
(85)
Then under the assumption (1198671)ndash(1198674) we have
det119863119911119865 (119911lowast 1205911 119901)
= det
[[[[[[[[
[
minus11988611119909lowast
1+
11988612119909lowast
1119909lowast
2
(1198981+ 119909lowast
1)2
11988612119909lowast
1
1198981+ 119909lowast
1
0
119886211198981119909lowast
2
(1198981+ 119909lowast
1)2
minus11988622119909lowast
2+
11988623119909lowast
2119909lowast
3
(1198982+ 119909lowast
2)2
minus
11988623119909lowast
2
1198982+ 119909lowast
2
0
119886321198982119909lowast
3
(1198982+ 119909lowast
2)2
minus11988633119909lowast
3
]]]]]]]]
]
= minus
1198862
21199101lowast1199102lowast
(1 + 11988721199101lowast)3[minus119909lowast+
11988611198871119909lowast1199101lowast
(1 + 1198871119909lowast)2] = 0
(86)
From (86) we know that the hypothesis (A2) in [24]is satisfied
(3) The characteristic matrix of (71) at a stationarysolution (119911 120591
0 1199010) where 119911 = (119911
(1) 119911(2) 119911(3)) isin 119877
3
takes the following form
Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863
120601119865 (119911 120591
1 119901) (119890
120582119868) (87)
that is
Δ (119911 1205911 119901) (120582)
=
[[[[[[[[[[[[[[[[[
[
120582 minus 1198861+ 211988611119911(1)
+
119886121198981119911(2)
(1198981+ 119911(1))
2
11988612119911(1)
1198981+ 119911(1)
0
minus
119886211198981119911(2)
(1198981+ 119911(1))
2119890minus1205821205911
120582 + 1198862minus
11988621119911(1)
1198981+ 119911(1)
+ 211988622119911(2)
+
119886231198982119911(3)
(1198982+ 119911(2))
2
11988623119911(2)
1198982+ 119911(2)
0 minus
119886321198982119911(3)
(1198982+ 119911(2))
2119890minus120582120591lowast
2120582 + 1198863minus
11988632119911(2)
1198982+ 119911(2)
+ 211988633119911(3)
]]]]]]]]]]]]]]]]]
]
(88)
Abstract and Applied Analysis 15
From (88) we have
det (Δ (119864lowast 1205911 119901) (120582))
= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902
0+ 1199030)] 119890minus1205821205911
(89)
Note that (89) is the same as (20) from the discussion inSection 2 about the local Hopf bifurcation it is easy to verifythat (119864
lowast 120591(119895)
1119896 2120587120596
119896) is an isolated center and there exist 120598 gt
0 120575 gt 0 and a smooth curve 120582 (120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575) rarr C
such that det(Δ(120582(1205911))) = 0 |120582(120591
1) minus 120596119896| lt 120598 for all 120591
1isin
[120591(119895)
1119896minus 120575 120591(119895)
1119896+ 120575] and
120582 (120591(119895)
1119896) = 120596119896119894
119889Re 120582 (1205911)
1198891205911
1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)
1119896
gt 0 (90)
Let
Ω1205982120587120596
119896
= (120578 119901) 0 lt 120578 lt 120598
10038161003816100381610038161003816100381610038161003816
119901 minus
2120587
120596119896
10038161003816100381610038161003816100381610038161003816
lt 120598 (91)
It is easy to see that on [120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575] times 120597Ω
1205982120587120596119896
det(Δ(119864
lowast 1205911 119901)(120578 + (2120587119901)119894)) = 0 if and only if 120578 = 0
1205911= 120591(119895)
1119896 119901 = 2120587120596
119896 119896 = 1 2 3 119895 = 0 1 2
Therefore the hypothesis (A4) in [24] is satisfiedIf we define
119867plusmn(119864lowast 120591(119895)
1119896
2120587
120596119896
) (120578 119901)
= det(Δ (119864lowast 120591(119895)
1119896plusmn 120575 119901) (120578 +
2120587
119901
119894))
(92)
then we have the crossing number of isolated center(119864lowast 120591(119895)
1119896 2120587120596
119896) as follows
120574(119864lowast 120591(119895)
1119896
2120587
120596119896
) = deg119861(119867minus(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
minus deg119861(119867+(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
= minus1
(93)
Thus we have
sum
(1199111205911119901)isinC
(119864lowast120591(119895)
11198962120587120596119896)
120574 (119911 1205911 119901) lt 0
(94)
where (119911 1205911 119901) has all or parts of the form
(119864lowast 120591(119896)
1119895 2120587120596
119896) (119895 = 0 1 ) It follows from
Lemma 9 that the connected component ℓ(119864lowast120591(119895)
11198962120587120596
119896)
through (119864lowast 120591(119895)
1119896 2120587120596
119896) is unbounded for each center
(119911lowast 1205911 119901) (119895 = 0 1 ) From the discussion in Section 2
we have
120591(119895)
1119896=
1
120596119896
times arccos(((11990121205962
119896minus1199010) (1199030+1199020)
+ 1205962(1205962minus1199011) (1199031+1199021))
times ((1199030+1199020)2
+ (1199031+1199021)2
1205962
119896)
minus1
) +2119895120587
(95)
where 119896 = 1 2 3 119895 = 0 1 Thus one can get 2120587120596119896le 120591(119895)
1119896
for 119895 ge 1Now we prove that the projection of ℓ
(119864lowast120591(119895)
11198962120587120596
119896)onto
1205911-space is [120591
1 +infin) where 120591
1le 120591(119895)
1119896 Clearly it follows
from the proof of Lemma 11 that system (3) with 1205911
= 0
has no nontrivial periodic solution Hence the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is away from zero
For a contradiction we suppose that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is bounded this means that the
projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is included in a
interval (0 120591lowast) Noticing 2120587120596119896lt 120591119895
1119896and applying Lemma 11
we have 119901 lt 120591lowast for (119911(119905) 120591
1 119901) belonging to ℓ
(119864lowast120591(119895)
11198962120587120596
119896)
This implies that the projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 119901-
space is bounded Then applying Lemma 10 we get thatthe connected component ℓ
(119864lowast120591(119895)
11198962120587120596
119896)is bounded This
contradiction completes the proof
6 Conclusion
In this paper we take our attention to the stability and Hopfbifurcation analysis of a predator-prey systemwithMichaelis-Menten type functional response and two unequal delaysWe obtained some conditions for local stability and Hopfbifurcation occurring When 120591
1= 1205912 we derived the explicit
formulas to determine the properties of periodic solutionsby the normal form method and center manifold theoremSpecially the global existence results of periodic solutionsbifurcating from Hopf bifurcations are also established byusing a global Hopf bifurcation result due to Wu [24]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to thank the anonymous referees fortheir careful reading and constructive suggestions which leadto truly significant improvement of the paper This researchis supported by the National Natural Science Foundation ofChina (no 11061016)
16 Abstract and Applied Analysis
References
[1] R Xu and M A J Chaplain ldquoPersistence and global stabilityin a delayed predator-prey system with Michaelis-Menten typefunctional responserdquo Applied Mathematics and Computationvol 130 no 2-3 pp 441ndash455 2002
[2] Y L Song Y H Peng and J JWei ldquoBifurcations for a predator-prey system with two delaysrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 466ndash479 2008
[3] G-P Hu W-T Li and X-P Yan ldquoHopf bifurcations in apredator-prey system with multiple delaysrdquo Chaos Solitons ampFractals vol 42 no 2 pp 1273ndash1285 2009
[4] M Liao X Tang and C Xu ldquoBifurcation analysis for a three-species predator-prey systemwith two delaysrdquoCommunicationsin Nonlinear Science and Numerical Simulation vol 17 no 1 pp183ndash194 2012
[5] C Xu and P Li ldquoDynamical analysis in a delayed predator-prey model with two delaysrdquo Discrete Dynamics in Nature andSociety vol 2012 Article ID 652947 22 pages 2012
[6] J-F Zhang ldquoStability and bifurcation periodic solutions ina Lotka-Volterra competition system with multiple delaysrdquoNonlinear Dynamics vol 70 no 1 pp 849ndash860 2012
[7] J Xia Z Yu and R Yuan ldquoStability and Hopf bifurcation ina symmetric Lotka-Volterra predator-prey system with delaysrdquoElectronic Journal of Differential Equations vol 2013 no 9 pp1ndash16 2013
[8] H Wan and J Cui ldquoA malaria model with two delaysrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 601265 8pages 2013
[9] M Zhao ldquoHopf bifurcation analysis for a semiratio-dependentpredator-prey system with two delaysrdquo Abstract and AppliedAnalysis vol 2013 Article ID 495072 13 pages 2013
[10] S Feyissa and S Banerjee ldquoDelay-induced oscillatory dynamicsin humoral mediated immune response with two time delaysrdquoNonlinear Analysis Real World Applications vol 14 no 1 pp35ndash52 2013
[11] G Zhang Y Shen andB Chen ldquoHopf bifurcation of a predator-prey systemwith predator harvesting and twodelaysrdquoNonlinearDynamics vol 73 no 4 pp 2119ndash2131 2013
[12] X-Y Meng H-F Huo and H Xiang ldquoHopf bifurcation in athree-species systemwith delaysrdquo Journal of AppliedMathemat-ics and Computing vol 35 no 1-2 pp 635ndash661 2011
[13] J-F Zhang ldquoGlobal existence of bifurcated periodic solutions ina commensalism model with delaysrdquo Applied Mathematics andComputation vol 218 no 23 pp 11688ndash11699 2012
[14] Y Wang W Jiang and H Wang ldquoStability and global Hopfbifurcation in toxic phytoplankton-zooplankton model withdelay and selective harvestingrdquoNonlinear Dynamics vol 73 no1-2 pp 881ndash896 2013
[15] Y Ma ldquoGlobal Hopf bifurcation in the Leslie-Gower predator-prey model with two delaysrdquo Nonlinear Analysis Real WorldApplications vol 13 no 1 pp 370ndash375 2012
[16] X-Y Meng H-F Huo and X-B Zhang ldquoStability and globalHopf bifurcation in a delayed food web consisting of a preyand two predatorsrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4335ndash4348 2011
[17] Y Song J Wei andM Han ldquoLocal and global Hopf bifurcationin a delayed hematopoiesis modelrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol14 no 11 pp 3909ndash3919 2004
[18] Y Song and J Wei ldquoLocal Hopf bifurcation and global periodicsolutions in a delayed predator-prey systemrdquo Journal of Mathe-matical Analysis and Applications vol 301 no 1 pp 1ndash21 2005
[19] J Wei and M Y Li ldquoGlobal existence of periodic solutions in atri-neuron network model with delaysrdquo Physica D vol 198 no1-2 pp 106ndash119 2004
[20] T Zhao Y Kuang and H L Smith ldquoGlobal existence of peri-odic solutions in a class of delayed Gause-type predator-preysystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 28 no 8 pp 1373ndash1394 1997
[21] S J Gao L S Chen and Z D Teng ldquoHopf bifurcation andglobal stability for a delayed predator-prey system with stagestructure for predatorrdquo Applied Mathematics and Computationvol 202 no 2 pp 721ndash729 2008
[22] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977
[23] B Hassard D Kazarinoff and Y WanTheory and Applicationsof Hopf Bifurcation Cambridge University Press CambridgeUK 1981
[24] J H Wu ldquoSymmetric functional-differential equations andneural networks with memoryrdquo Transactions of the AmericanMathematical Society vol 350 no 12 pp 4799ndash4838 1998
[25] V Hutson ldquoThe existence of an equilibrium for permanentsystemsrdquo The Rocky Mountain Journal of Mathematics vol 20no 4 pp 1033ndash1040 1990
[26] S G Ruan and J J Wei ldquoOn the zeros of transcendentalfunctions with applications to stability of delay differentialequations with two delaysrdquo Dynamics of Continuous Discreteamp Impulsive Systems A vol 10 no 6 pp 863ndash874 2003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Abstract and Applied Analysis
1198642= 2
[[[[
[
minus11988711
minus11988712
0
minus11988821
minus11988722
minus11988723
0 minus11988932
minus11988733
]]]]
]
minus1
times
[[[[
[
minus2 (11988611
+ 1198973) minus 1198971(120572 + 120572)
1198971(1205721198901198941205961205911+ 120572119890minus1198941205961205911) minus 1198976(120572120573 + 120572120573) minus 2 (119897
7+ 11988622) 120572120572 + 2119897
3
1198976(120573120572119890119894120596120591lowast
2+ 120572120573119890
minus119894120596120591lowast
2) + 21198977120572120572 minus 119886
33120573120573
]]]]
]
(69)
Thus we can determine11988220(120579) and119882
11(120579) Furthermore
we can determine each 119892119894119895by the parameters and delay in (3)
Thus we can compute the following values
1198881(0) =
119894
21205961205911
(1198922011989211
minus 2100381610038161003816100381611989211
1003816100381610038161003816
2
minus
1
3
100381610038161003816100381611989202
1003816100381610038161003816
2
) +
1
2
11989221
1205832= minus
Re 1198881(0)
Re 1205821015840 (1205911)
1198792= minus
Im 1198881(0) + 120583
2Im 120582
1015840(1205911)
1205961205911
1205732= 2Re 119888
1(0)
(70)
which determine the quantities of bifurcating periodic solu-tions in the center manifold at the critical value 120591
1 Suppose
Re1205821015840(1205911) gt 0 120583
2determines the directions of the Hopf
bifurcation if 1205832
gt 0(lt 0) then the Hopf bifurcationis supercritical (subcritical) and the bifurcation exists for120591 gt 120591
1(lt 1205911) 1205732determines the stability of the bifurcation
periodic solutions the bifurcating periodic solutions arestable (unstable) if 120573
2lt 0(gt 0) and119879
2determines the period
of the bifurcating periodic solutions the period increases(decreases) if 119879
2gt 0(lt 0)
4 Numerical Simulation
We consider system (3) by taking the following coefficients1198861
= 03 11988611
= 58889 11988612
= 1 1198981
= 1 1198862
=
01 11988621
= 27 11988622
= 12 11988623
= 12 1198982
= 1 1198863
=
02 11988632
= 25 11988633
= 12 We have the unique positiveequilibrium 119864
lowast= (00451 00357 00551)
By computation we get 11989810
= minus00104 12059611
= 0516411991111
= 02666 ℎ10158401(11991111) = 03402 120591
20
= 32348 FromTheorem 6 we know that when 120591
1= 0 the positive
equilibrium 119864lowastis locally asymptotically stable for 120591
2isin
[0 32348) When 1205912crosses 120591
20
the equilibrium 119864lowastloses its
stability and Hopf bifurcation occurs From the algorithm inSection 3 we have 120583
2= 56646 120573
2= minus31583 119879
2= 5445
whichmeans that the bifurcation is supercritical and periodicsolution is stable The trajectories and the phase graphs areshown in Figures 1 and 2
Regarding 1205911as a parameter and let 120591
2= 29 isin
[0 32348) we can observe that with 1205911increasing the pos-
itive equilibrium 119864lowastloses its stability and Hopf bifurcation
occurs (see Figures 3 and 4)
5 Global Continuationof Local Hopf Bifurcations
In this section we study the global continuation of peri-odic solutions bifurcating from the positive equilibrium(119864lowast 120591(119895)
1119896) (119896 = 1 2 6 119895 = 0 1 ) Throughout this
section we follow closely the notations in [24] and assumethat 120591
2= 120591lowast
2isin [0 120591
210
) regarding 1205911as a parameter For
simplification of notations setting 119911119905(119905) = (119909
1119905 1199092119905 1199093119905)119879 we
may rewrite system (3) as the following functional differentialequation
(119905) = 119865 (119911119905 1205911 119901) (71)
where 119911119905(120579) = (119909
1119905(120579) 1199092119905(120579) 1199093119905(120579))119879
= (1199091(119905 + 120579) 119909
2(119905 +
120579) 1199093(119905 + 120579))
119879 for 119905 ge 0 and 120579 isin [minus1205911 0] Since 119909
1(119905) 1199092(119905)
and 1199093(119905) denote the densities of the prey the predator and
the top predator respectively the positive solution of system(3) is of interest and its periodic solutions only arise in the firstquadrant Thus we consider system (3) only in the domain1198773
+= (1199091 1199092 1199093) isin 1198773 1199091gt 0 119909
2gt 0 119909
3gt 0 It is obvious
that (71) has a unique positive equilibrium 119864lowast(119909lowast
1 119909lowast
2 119909lowast
3) in
1198773
+under the assumption (119867
1)ndash(1198674) Following the work of
[24] we need to define
119883 = 119862([minus1205911 0] 119877
3
+)
Γ = 119862119897 (119911 1205911 119901) isin X times R times R+
119911 is a 119901-periodic solution of system (71)
N = (119911 1205911 119901) 119865 (119911 120591
1 119901) = 0
(72)
Let ℓ(119864lowast120591(119895)
11198962120587120596
(119895)
1119896)denote the connected component pass-
ing through (119864lowast 120591(119895)
1119896 2120587120596
(119895)
1119896) in Γ where 120591
(119895)
1119896is defined by
(43) We know that ℓ(119864lowast120591(119895)
11198962120587120596
(119895)
1119896)through (119864
lowast 120591(119895)
1119896 2120587120596
(119895)
1119896)
is nonemptyFor the benefit of readers we first state the global Hopf
bifurcation theory due to Wu [24] for functional differentialequations
Lemma 9 Assume that (119911lowast 120591 119901) is an isolated center satis-
fying the hypotheses (A1)ndash(A4) in [24] Denote by ℓ(119911lowast120591119901)
theconnected component of (119911
lowast 120591 119901) in Γ Then either
(i) ℓ(119911lowast120591119901)
is unbounded or(ii) ℓ(119911lowast120591119901)
is bounded ℓ(119911lowast120591119901)
cap Γ is finite and
sum
(119911120591119901)isinℓ(119911lowast120591119901)capN
120574119898(119911lowast 120591 119901) = 0 (73)
for all 119898 = 1 2 where 120574119898(119911lowast 120591 119901) is the 119898119905ℎ crossing
number of (119911lowast 120591 119901) if119898 isin 119869(119911
lowast 120591 119901) or it is zero if otherwise
Clearly if (ii) in Lemma 9 is not true then ℓ(119911lowast120591119901)
isunbounded Thus if the projections of ℓ
(119911lowast120591119901)
onto 119911-spaceand onto 119901-space are bounded then the projection of ℓ
(119911lowast120591119901)
onto 120591-space is unbounded Further if we can show that the
Abstract and Applied Analysis 11
0 200 400 60000451
00451
00452
t
x1(t)
t
0 200 400 60000356
00357
00358
x2(t)
t
0 200 400 6000055
00551
00552
x3(t)
00451
00451
00452
00356
00357
00357
003570055
0055
00551
00551
00552
x 1(t)
x2 (t)
x3(t)
Figure 1 The trajectories and the phase graph with 1205911= 0 120591
2= 29 lt 120591
20= 32348 119864
lowastis locally asymptotically stable
0 500 1000004
0045
005
0 500 1000002
004
006
0 500 10000
005
01
t
t
t
x1(t)
x2(t)
x3(t)
0044
0046
0048
002
003
004
005003
004
005
006
007
008
x 1(t)
x2 (t)
x3(t)
Figure 2 The trajectories and the phase graph with 1205911= 0 120591
2= 33 gt 120591
20= 32348 a periodic orbit bifurcate from 119864
lowast
12 Abstract and Applied Analysis
0 500 1000 1500
00451
00452
0 500 1000 15000035
0036
0037
0 500 1000 15000054
0055
0056
t
t
t
x1(t)
x2(t)
x3(t)
0045100451
0045200452
0035
00355
003600546
00548
0055
00552
00554
00556
00558
x1(t)
x2 (t)
x3(t)
Figure 3 The trajectories and the phase graph with 1205911= 09 120591
2= 29 119864
lowastis locally asymptotically stable
0 500 1000 15000044
0045
0046
0 500 1000 1500002
003
004
0 500 1000 1500004
006
008
t
t
t
x1(t)
x2(t)
x3(t)
00445
0045
00455
0032
0034
0036
00380052
0054
0056
0058
006
0062
0064
x 1(t)
x2 (t)
x3(t)
Figure 4 The trajectories and the phase graph with 1205911= 12 120591
2= 29 a periodic orbit bifurcate from 119864
lowast
Abstract and Applied Analysis 13
projection of ℓ(119911lowast120591119901)
onto 120591-space is away from zero thenthe projection of ℓ
(119911lowast120591119901)
onto 120591-space must include interval[120591 infin) Following this ideal we can prove our results on theglobal continuation of local Hopf bifurcation
Lemma 10 If the conditions (1198671)ndash(1198674) hold then all nontriv-
ial periodic solutions of system (71) with initial conditions
1199091(120579) = 120593 (120579) ge 0 119909
2(120579) = 120595 (120579) ge 0
1199093(120579) = 120601 (120579) ge 0 120579 isin [minus120591
1 0)
120593 (0) gt 0 120595 (0) gt 0 120601 (0) gt 0
(74)
are uniformly bounded
Proof Suppose that (1199091(119905) 1199092(119905) 1199093(119905)) are nonconstant peri-
odic solutions of system (3) and define
1199091(1205851) = min 119909
1(119905) 119909
1(1205781) = max 119909
1(119905)
1199092(1205852) = min 119909
2(119905) 119909
2(1205782) = max 119909
2(119905)
1199093(1205853) = min 119909
3(119905) 119909
3(1205783) = max 119909
3(119905)
(75)
It follows from system (3) that
1199091(119905) = 119909
1(0) expint
119905
0
[1198861minus 119886111199091(119904) minus
119886121199092(119904)
1198981+ 1199091(119904)
] 119889119904
1199092(119905) = 119909
2(0) expint
119905
0
[minus1198862+
119886211199091(119904 minus 1205911)
1198981+ 1199091(119904 minus 1205911)
minus 119886221199092(119904) minus
119886231199093(119904)
1198982+ 1199092(119904)
] 119889119904
1199093(119905) = 119909
3(0) expint
119905
0
[minus1198863+
119886321199092(119904 minus 120591
lowast
2)
1198982+ 1199092(119904 minus 120591
lowast
2)
minus 119886331199093(119904)] 119889119904
(76)
which implies that the solutions of system (3) cannot crossthe 119909119894-axis (119894 = 1 2 3) Thus the nonconstant periodic orbits
must be located in the interior of first quadrant It followsfrom initial data of system (3) that 119909
1(119905) gt 0 119909
2(119905) gt
0 1199093(119905) gt 0 for 119905 ge 0
From the first equation of system (3) we can get
0 = 1198861minus 119886111199091(1205781) minus
119886121199092(1205781)
1198981+ 1199091(1205781)
le 1198861minus 119886111199091(1205781) (77)
thus we have
1199091(1205781) le
1198861
11988611
(78)
From the second equation of (3) we obtain
0 = minus 1198862+
119886211199091(1205782minus 1205911)
1198981+ 1199091(1205782minus 1205911)
minus 119886221199092(1205782)
minus
119886231199093(1205782)
1198982+ 1199092(1205782)
le minus1198862+
11988621(119886111988611)
1198981+ (119886111988611)
minus 119886221199092(1205782)
(79)
therefore one can get
1199092(1205782) le
minus1198862(119886111198981+ 1198861) + 119886111988621
11988622(119886111198981+ 1198861)
≜ 1198721 (80)
Applying the third equation of system (3) we know
0 = minus1198863+
119886321199092(1205783minus 120591lowast
2)
1198982+ 1199092(1205783minus 120591lowast
2)
minus 119886331199093(1205783)
le minus1198863+
119886321198721
1198982+1198721
minus 119886331199093(1205783)
(81)
It follows that
1199093(1205783) le
minus1198863(1198982+1198721) + 119886321198721
11988633(1198982+1198721)
≜ 1198722 (82)
This shows that the nontrivial periodic solution of system (3)is uniformly bounded and the proof is complete
Lemma 11 If the conditions (1198671)ndash(1198674) and
(H7) 11988622
minus (119886211198982) gt 0 [(119886
11minus 11988611198981)11988622
minus
(11988612119886211198982
1)]11988633
minus (11988611
minus (11988611198981))[(119886211198982)11988633
+
(11988623119886321198982
2)] gt 0
hold then system (3) has no nontrivial 1205911-periodic solution
Proof Suppose for a contradiction that system (3) has non-trivial periodic solution with period 120591
1 Then the following
system (83) of ordinary differential equations has nontrivialperiodic solution
1198891199091
119889119905
= 1199091(119905) [1198861minus 119886111199091(119905) minus
119886121199092(119905)
1198981+ 1199091(119905)
]
1198891199092
119889119905
= 1199092(119905) [minus119886
2+
119886211199091(119905)
1198981+ 1199091(119905)
minus 119886221199092(119905) minus
119886231199093(119905)
1198982+ 1199092(119905)
]
1198891199093
119889119905
= 1199093(119905) [minus119886
3+
119886321199092(119905 minus 120591
lowast
2)
1198982+ 1199092(119905 minus 120591
lowast
2)
minus 119886331199093(119905)]
(83)
which has the same equilibria to system (3) that is
1198641= (0 0 0) 119864
2= (
1198861
11988611
0 0)
1198643= (1199091 1199092 0) 119864
lowast= (119909lowast
1 119909lowast
2 119909lowast
3)
(84)
Note that 119909119894-axis (119894 = 1 2 3) are the invariable manifold of
system (83) and the orbits of system (83) do not intersect each
14 Abstract and Applied Analysis
other Thus there are no solutions crossing the coordinateaxes On the other hand note the fact that if system (83) hasa periodic solution then there must be the equilibrium in itsinterior and that 119864
1 1198642 1198643are located on the coordinate
axis Thus we conclude that the periodic orbit of system (83)must lie in the first quadrant If (119867
7) holds it is well known
that the positive equilibrium 119864lowastis globally asymptotically
stable in the first quadrant (see [1])Thus there is no periodicorbit in the first quadrant too The above discussion meansthat (83) does not have any nontrivial periodic solution It isa contradiction Therefore the lemma is confirmed
Theorem 12 Suppose the conditions of Theorem 8 and (1198677)
hold let120596119896and 120591(119895)1119896
be defined in Section 2 then when 1205911gt 120591(119895)
1119896
system (3) has at least 119895 minus 1 periodic solutions
Proof It is sufficient to prove that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is [120591
1 +infin) for each 119895 ge 1 where
1205911le 120591(119895)
1119896
In following we prove that the hypotheses (A1)ndash(A4) in[24] hold
(1) From system (3) we know easily that the followingconditions hold
(A1) 119865 isin 1198622(1198773
+times119877+times119877+) where 119865 = 119865|
1198773
+times119877+times119877+
rarr
1198773
+
(A3) 119865(120601 1205911 119901) is differential with respect to 120601
(2) It follows from system (3) that
119863119911119865 (119911 120591
1 119901) =
[[[[[[[[[[[
[
1198861minus 2119886111199091minus
1198861211989811199092
(1198981+ 1199091)2
minus
119886121199091
1198981+ 1199091
0
1198862111989811199092
(1198981+ 1199091)2
minus1198862+
119886211199091
1198981+ 1199091
minus 2119886221199092minus
1198862311989821199093
(1198982+ 1199092)2
minus
119886231199092
1198982+ 1199092
0
1198863211989821199093
(1198982+ 1199092)2
minus1198863+
119886321199092
1198982+ 1199092
minus 2119886331199093
]]]]]]]]]]]
]
(85)
Then under the assumption (1198671)ndash(1198674) we have
det119863119911119865 (119911lowast 1205911 119901)
= det
[[[[[[[[
[
minus11988611119909lowast
1+
11988612119909lowast
1119909lowast
2
(1198981+ 119909lowast
1)2
11988612119909lowast
1
1198981+ 119909lowast
1
0
119886211198981119909lowast
2
(1198981+ 119909lowast
1)2
minus11988622119909lowast
2+
11988623119909lowast
2119909lowast
3
(1198982+ 119909lowast
2)2
minus
11988623119909lowast
2
1198982+ 119909lowast
2
0
119886321198982119909lowast
3
(1198982+ 119909lowast
2)2
minus11988633119909lowast
3
]]]]]]]]
]
= minus
1198862
21199101lowast1199102lowast
(1 + 11988721199101lowast)3[minus119909lowast+
11988611198871119909lowast1199101lowast
(1 + 1198871119909lowast)2] = 0
(86)
From (86) we know that the hypothesis (A2) in [24]is satisfied
(3) The characteristic matrix of (71) at a stationarysolution (119911 120591
0 1199010) where 119911 = (119911
(1) 119911(2) 119911(3)) isin 119877
3
takes the following form
Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863
120601119865 (119911 120591
1 119901) (119890
120582119868) (87)
that is
Δ (119911 1205911 119901) (120582)
=
[[[[[[[[[[[[[[[[[
[
120582 minus 1198861+ 211988611119911(1)
+
119886121198981119911(2)
(1198981+ 119911(1))
2
11988612119911(1)
1198981+ 119911(1)
0
minus
119886211198981119911(2)
(1198981+ 119911(1))
2119890minus1205821205911
120582 + 1198862minus
11988621119911(1)
1198981+ 119911(1)
+ 211988622119911(2)
+
119886231198982119911(3)
(1198982+ 119911(2))
2
11988623119911(2)
1198982+ 119911(2)
0 minus
119886321198982119911(3)
(1198982+ 119911(2))
2119890minus120582120591lowast
2120582 + 1198863minus
11988632119911(2)
1198982+ 119911(2)
+ 211988633119911(3)
]]]]]]]]]]]]]]]]]
]
(88)
Abstract and Applied Analysis 15
From (88) we have
det (Δ (119864lowast 1205911 119901) (120582))
= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902
0+ 1199030)] 119890minus1205821205911
(89)
Note that (89) is the same as (20) from the discussion inSection 2 about the local Hopf bifurcation it is easy to verifythat (119864
lowast 120591(119895)
1119896 2120587120596
119896) is an isolated center and there exist 120598 gt
0 120575 gt 0 and a smooth curve 120582 (120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575) rarr C
such that det(Δ(120582(1205911))) = 0 |120582(120591
1) minus 120596119896| lt 120598 for all 120591
1isin
[120591(119895)
1119896minus 120575 120591(119895)
1119896+ 120575] and
120582 (120591(119895)
1119896) = 120596119896119894
119889Re 120582 (1205911)
1198891205911
1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)
1119896
gt 0 (90)
Let
Ω1205982120587120596
119896
= (120578 119901) 0 lt 120578 lt 120598
10038161003816100381610038161003816100381610038161003816
119901 minus
2120587
120596119896
10038161003816100381610038161003816100381610038161003816
lt 120598 (91)
It is easy to see that on [120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575] times 120597Ω
1205982120587120596119896
det(Δ(119864
lowast 1205911 119901)(120578 + (2120587119901)119894)) = 0 if and only if 120578 = 0
1205911= 120591(119895)
1119896 119901 = 2120587120596
119896 119896 = 1 2 3 119895 = 0 1 2
Therefore the hypothesis (A4) in [24] is satisfiedIf we define
119867plusmn(119864lowast 120591(119895)
1119896
2120587
120596119896
) (120578 119901)
= det(Δ (119864lowast 120591(119895)
1119896plusmn 120575 119901) (120578 +
2120587
119901
119894))
(92)
then we have the crossing number of isolated center(119864lowast 120591(119895)
1119896 2120587120596
119896) as follows
120574(119864lowast 120591(119895)
1119896
2120587
120596119896
) = deg119861(119867minus(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
minus deg119861(119867+(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
= minus1
(93)
Thus we have
sum
(1199111205911119901)isinC
(119864lowast120591(119895)
11198962120587120596119896)
120574 (119911 1205911 119901) lt 0
(94)
where (119911 1205911 119901) has all or parts of the form
(119864lowast 120591(119896)
1119895 2120587120596
119896) (119895 = 0 1 ) It follows from
Lemma 9 that the connected component ℓ(119864lowast120591(119895)
11198962120587120596
119896)
through (119864lowast 120591(119895)
1119896 2120587120596
119896) is unbounded for each center
(119911lowast 1205911 119901) (119895 = 0 1 ) From the discussion in Section 2
we have
120591(119895)
1119896=
1
120596119896
times arccos(((11990121205962
119896minus1199010) (1199030+1199020)
+ 1205962(1205962minus1199011) (1199031+1199021))
times ((1199030+1199020)2
+ (1199031+1199021)2
1205962
119896)
minus1
) +2119895120587
(95)
where 119896 = 1 2 3 119895 = 0 1 Thus one can get 2120587120596119896le 120591(119895)
1119896
for 119895 ge 1Now we prove that the projection of ℓ
(119864lowast120591(119895)
11198962120587120596
119896)onto
1205911-space is [120591
1 +infin) where 120591
1le 120591(119895)
1119896 Clearly it follows
from the proof of Lemma 11 that system (3) with 1205911
= 0
has no nontrivial periodic solution Hence the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is away from zero
For a contradiction we suppose that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is bounded this means that the
projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is included in a
interval (0 120591lowast) Noticing 2120587120596119896lt 120591119895
1119896and applying Lemma 11
we have 119901 lt 120591lowast for (119911(119905) 120591
1 119901) belonging to ℓ
(119864lowast120591(119895)
11198962120587120596
119896)
This implies that the projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 119901-
space is bounded Then applying Lemma 10 we get thatthe connected component ℓ
(119864lowast120591(119895)
11198962120587120596
119896)is bounded This
contradiction completes the proof
6 Conclusion
In this paper we take our attention to the stability and Hopfbifurcation analysis of a predator-prey systemwithMichaelis-Menten type functional response and two unequal delaysWe obtained some conditions for local stability and Hopfbifurcation occurring When 120591
1= 1205912 we derived the explicit
formulas to determine the properties of periodic solutionsby the normal form method and center manifold theoremSpecially the global existence results of periodic solutionsbifurcating from Hopf bifurcations are also established byusing a global Hopf bifurcation result due to Wu [24]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to thank the anonymous referees fortheir careful reading and constructive suggestions which leadto truly significant improvement of the paper This researchis supported by the National Natural Science Foundation ofChina (no 11061016)
16 Abstract and Applied Analysis
References
[1] R Xu and M A J Chaplain ldquoPersistence and global stabilityin a delayed predator-prey system with Michaelis-Menten typefunctional responserdquo Applied Mathematics and Computationvol 130 no 2-3 pp 441ndash455 2002
[2] Y L Song Y H Peng and J JWei ldquoBifurcations for a predator-prey system with two delaysrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 466ndash479 2008
[3] G-P Hu W-T Li and X-P Yan ldquoHopf bifurcations in apredator-prey system with multiple delaysrdquo Chaos Solitons ampFractals vol 42 no 2 pp 1273ndash1285 2009
[4] M Liao X Tang and C Xu ldquoBifurcation analysis for a three-species predator-prey systemwith two delaysrdquoCommunicationsin Nonlinear Science and Numerical Simulation vol 17 no 1 pp183ndash194 2012
[5] C Xu and P Li ldquoDynamical analysis in a delayed predator-prey model with two delaysrdquo Discrete Dynamics in Nature andSociety vol 2012 Article ID 652947 22 pages 2012
[6] J-F Zhang ldquoStability and bifurcation periodic solutions ina Lotka-Volterra competition system with multiple delaysrdquoNonlinear Dynamics vol 70 no 1 pp 849ndash860 2012
[7] J Xia Z Yu and R Yuan ldquoStability and Hopf bifurcation ina symmetric Lotka-Volterra predator-prey system with delaysrdquoElectronic Journal of Differential Equations vol 2013 no 9 pp1ndash16 2013
[8] H Wan and J Cui ldquoA malaria model with two delaysrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 601265 8pages 2013
[9] M Zhao ldquoHopf bifurcation analysis for a semiratio-dependentpredator-prey system with two delaysrdquo Abstract and AppliedAnalysis vol 2013 Article ID 495072 13 pages 2013
[10] S Feyissa and S Banerjee ldquoDelay-induced oscillatory dynamicsin humoral mediated immune response with two time delaysrdquoNonlinear Analysis Real World Applications vol 14 no 1 pp35ndash52 2013
[11] G Zhang Y Shen andB Chen ldquoHopf bifurcation of a predator-prey systemwith predator harvesting and twodelaysrdquoNonlinearDynamics vol 73 no 4 pp 2119ndash2131 2013
[12] X-Y Meng H-F Huo and H Xiang ldquoHopf bifurcation in athree-species systemwith delaysrdquo Journal of AppliedMathemat-ics and Computing vol 35 no 1-2 pp 635ndash661 2011
[13] J-F Zhang ldquoGlobal existence of bifurcated periodic solutions ina commensalism model with delaysrdquo Applied Mathematics andComputation vol 218 no 23 pp 11688ndash11699 2012
[14] Y Wang W Jiang and H Wang ldquoStability and global Hopfbifurcation in toxic phytoplankton-zooplankton model withdelay and selective harvestingrdquoNonlinear Dynamics vol 73 no1-2 pp 881ndash896 2013
[15] Y Ma ldquoGlobal Hopf bifurcation in the Leslie-Gower predator-prey model with two delaysrdquo Nonlinear Analysis Real WorldApplications vol 13 no 1 pp 370ndash375 2012
[16] X-Y Meng H-F Huo and X-B Zhang ldquoStability and globalHopf bifurcation in a delayed food web consisting of a preyand two predatorsrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4335ndash4348 2011
[17] Y Song J Wei andM Han ldquoLocal and global Hopf bifurcationin a delayed hematopoiesis modelrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol14 no 11 pp 3909ndash3919 2004
[18] Y Song and J Wei ldquoLocal Hopf bifurcation and global periodicsolutions in a delayed predator-prey systemrdquo Journal of Mathe-matical Analysis and Applications vol 301 no 1 pp 1ndash21 2005
[19] J Wei and M Y Li ldquoGlobal existence of periodic solutions in atri-neuron network model with delaysrdquo Physica D vol 198 no1-2 pp 106ndash119 2004
[20] T Zhao Y Kuang and H L Smith ldquoGlobal existence of peri-odic solutions in a class of delayed Gause-type predator-preysystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 28 no 8 pp 1373ndash1394 1997
[21] S J Gao L S Chen and Z D Teng ldquoHopf bifurcation andglobal stability for a delayed predator-prey system with stagestructure for predatorrdquo Applied Mathematics and Computationvol 202 no 2 pp 721ndash729 2008
[22] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977
[23] B Hassard D Kazarinoff and Y WanTheory and Applicationsof Hopf Bifurcation Cambridge University Press CambridgeUK 1981
[24] J H Wu ldquoSymmetric functional-differential equations andneural networks with memoryrdquo Transactions of the AmericanMathematical Society vol 350 no 12 pp 4799ndash4838 1998
[25] V Hutson ldquoThe existence of an equilibrium for permanentsystemsrdquo The Rocky Mountain Journal of Mathematics vol 20no 4 pp 1033ndash1040 1990
[26] S G Ruan and J J Wei ldquoOn the zeros of transcendentalfunctions with applications to stability of delay differentialequations with two delaysrdquo Dynamics of Continuous Discreteamp Impulsive Systems A vol 10 no 6 pp 863ndash874 2003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 11
0 200 400 60000451
00451
00452
t
x1(t)
t
0 200 400 60000356
00357
00358
x2(t)
t
0 200 400 6000055
00551
00552
x3(t)
00451
00451
00452
00356
00357
00357
003570055
0055
00551
00551
00552
x 1(t)
x2 (t)
x3(t)
Figure 1 The trajectories and the phase graph with 1205911= 0 120591
2= 29 lt 120591
20= 32348 119864
lowastis locally asymptotically stable
0 500 1000004
0045
005
0 500 1000002
004
006
0 500 10000
005
01
t
t
t
x1(t)
x2(t)
x3(t)
0044
0046
0048
002
003
004
005003
004
005
006
007
008
x 1(t)
x2 (t)
x3(t)
Figure 2 The trajectories and the phase graph with 1205911= 0 120591
2= 33 gt 120591
20= 32348 a periodic orbit bifurcate from 119864
lowast
12 Abstract and Applied Analysis
0 500 1000 1500
00451
00452
0 500 1000 15000035
0036
0037
0 500 1000 15000054
0055
0056
t
t
t
x1(t)
x2(t)
x3(t)
0045100451
0045200452
0035
00355
003600546
00548
0055
00552
00554
00556
00558
x1(t)
x2 (t)
x3(t)
Figure 3 The trajectories and the phase graph with 1205911= 09 120591
2= 29 119864
lowastis locally asymptotically stable
0 500 1000 15000044
0045
0046
0 500 1000 1500002
003
004
0 500 1000 1500004
006
008
t
t
t
x1(t)
x2(t)
x3(t)
00445
0045
00455
0032
0034
0036
00380052
0054
0056
0058
006
0062
0064
x 1(t)
x2 (t)
x3(t)
Figure 4 The trajectories and the phase graph with 1205911= 12 120591
2= 29 a periodic orbit bifurcate from 119864
lowast
Abstract and Applied Analysis 13
projection of ℓ(119911lowast120591119901)
onto 120591-space is away from zero thenthe projection of ℓ
(119911lowast120591119901)
onto 120591-space must include interval[120591 infin) Following this ideal we can prove our results on theglobal continuation of local Hopf bifurcation
Lemma 10 If the conditions (1198671)ndash(1198674) hold then all nontriv-
ial periodic solutions of system (71) with initial conditions
1199091(120579) = 120593 (120579) ge 0 119909
2(120579) = 120595 (120579) ge 0
1199093(120579) = 120601 (120579) ge 0 120579 isin [minus120591
1 0)
120593 (0) gt 0 120595 (0) gt 0 120601 (0) gt 0
(74)
are uniformly bounded
Proof Suppose that (1199091(119905) 1199092(119905) 1199093(119905)) are nonconstant peri-
odic solutions of system (3) and define
1199091(1205851) = min 119909
1(119905) 119909
1(1205781) = max 119909
1(119905)
1199092(1205852) = min 119909
2(119905) 119909
2(1205782) = max 119909
2(119905)
1199093(1205853) = min 119909
3(119905) 119909
3(1205783) = max 119909
3(119905)
(75)
It follows from system (3) that
1199091(119905) = 119909
1(0) expint
119905
0
[1198861minus 119886111199091(119904) minus
119886121199092(119904)
1198981+ 1199091(119904)
] 119889119904
1199092(119905) = 119909
2(0) expint
119905
0
[minus1198862+
119886211199091(119904 minus 1205911)
1198981+ 1199091(119904 minus 1205911)
minus 119886221199092(119904) minus
119886231199093(119904)
1198982+ 1199092(119904)
] 119889119904
1199093(119905) = 119909
3(0) expint
119905
0
[minus1198863+
119886321199092(119904 minus 120591
lowast
2)
1198982+ 1199092(119904 minus 120591
lowast
2)
minus 119886331199093(119904)] 119889119904
(76)
which implies that the solutions of system (3) cannot crossthe 119909119894-axis (119894 = 1 2 3) Thus the nonconstant periodic orbits
must be located in the interior of first quadrant It followsfrom initial data of system (3) that 119909
1(119905) gt 0 119909
2(119905) gt
0 1199093(119905) gt 0 for 119905 ge 0
From the first equation of system (3) we can get
0 = 1198861minus 119886111199091(1205781) minus
119886121199092(1205781)
1198981+ 1199091(1205781)
le 1198861minus 119886111199091(1205781) (77)
thus we have
1199091(1205781) le
1198861
11988611
(78)
From the second equation of (3) we obtain
0 = minus 1198862+
119886211199091(1205782minus 1205911)
1198981+ 1199091(1205782minus 1205911)
minus 119886221199092(1205782)
minus
119886231199093(1205782)
1198982+ 1199092(1205782)
le minus1198862+
11988621(119886111988611)
1198981+ (119886111988611)
minus 119886221199092(1205782)
(79)
therefore one can get
1199092(1205782) le
minus1198862(119886111198981+ 1198861) + 119886111988621
11988622(119886111198981+ 1198861)
≜ 1198721 (80)
Applying the third equation of system (3) we know
0 = minus1198863+
119886321199092(1205783minus 120591lowast
2)
1198982+ 1199092(1205783minus 120591lowast
2)
minus 119886331199093(1205783)
le minus1198863+
119886321198721
1198982+1198721
minus 119886331199093(1205783)
(81)
It follows that
1199093(1205783) le
minus1198863(1198982+1198721) + 119886321198721
11988633(1198982+1198721)
≜ 1198722 (82)
This shows that the nontrivial periodic solution of system (3)is uniformly bounded and the proof is complete
Lemma 11 If the conditions (1198671)ndash(1198674) and
(H7) 11988622
minus (119886211198982) gt 0 [(119886
11minus 11988611198981)11988622
minus
(11988612119886211198982
1)]11988633
minus (11988611
minus (11988611198981))[(119886211198982)11988633
+
(11988623119886321198982
2)] gt 0
hold then system (3) has no nontrivial 1205911-periodic solution
Proof Suppose for a contradiction that system (3) has non-trivial periodic solution with period 120591
1 Then the following
system (83) of ordinary differential equations has nontrivialperiodic solution
1198891199091
119889119905
= 1199091(119905) [1198861minus 119886111199091(119905) minus
119886121199092(119905)
1198981+ 1199091(119905)
]
1198891199092
119889119905
= 1199092(119905) [minus119886
2+
119886211199091(119905)
1198981+ 1199091(119905)
minus 119886221199092(119905) minus
119886231199093(119905)
1198982+ 1199092(119905)
]
1198891199093
119889119905
= 1199093(119905) [minus119886
3+
119886321199092(119905 minus 120591
lowast
2)
1198982+ 1199092(119905 minus 120591
lowast
2)
minus 119886331199093(119905)]
(83)
which has the same equilibria to system (3) that is
1198641= (0 0 0) 119864
2= (
1198861
11988611
0 0)
1198643= (1199091 1199092 0) 119864
lowast= (119909lowast
1 119909lowast
2 119909lowast
3)
(84)
Note that 119909119894-axis (119894 = 1 2 3) are the invariable manifold of
system (83) and the orbits of system (83) do not intersect each
14 Abstract and Applied Analysis
other Thus there are no solutions crossing the coordinateaxes On the other hand note the fact that if system (83) hasa periodic solution then there must be the equilibrium in itsinterior and that 119864
1 1198642 1198643are located on the coordinate
axis Thus we conclude that the periodic orbit of system (83)must lie in the first quadrant If (119867
7) holds it is well known
that the positive equilibrium 119864lowastis globally asymptotically
stable in the first quadrant (see [1])Thus there is no periodicorbit in the first quadrant too The above discussion meansthat (83) does not have any nontrivial periodic solution It isa contradiction Therefore the lemma is confirmed
Theorem 12 Suppose the conditions of Theorem 8 and (1198677)
hold let120596119896and 120591(119895)1119896
be defined in Section 2 then when 1205911gt 120591(119895)
1119896
system (3) has at least 119895 minus 1 periodic solutions
Proof It is sufficient to prove that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is [120591
1 +infin) for each 119895 ge 1 where
1205911le 120591(119895)
1119896
In following we prove that the hypotheses (A1)ndash(A4) in[24] hold
(1) From system (3) we know easily that the followingconditions hold
(A1) 119865 isin 1198622(1198773
+times119877+times119877+) where 119865 = 119865|
1198773
+times119877+times119877+
rarr
1198773
+
(A3) 119865(120601 1205911 119901) is differential with respect to 120601
(2) It follows from system (3) that
119863119911119865 (119911 120591
1 119901) =
[[[[[[[[[[[
[
1198861minus 2119886111199091minus
1198861211989811199092
(1198981+ 1199091)2
minus
119886121199091
1198981+ 1199091
0
1198862111989811199092
(1198981+ 1199091)2
minus1198862+
119886211199091
1198981+ 1199091
minus 2119886221199092minus
1198862311989821199093
(1198982+ 1199092)2
minus
119886231199092
1198982+ 1199092
0
1198863211989821199093
(1198982+ 1199092)2
minus1198863+
119886321199092
1198982+ 1199092
minus 2119886331199093
]]]]]]]]]]]
]
(85)
Then under the assumption (1198671)ndash(1198674) we have
det119863119911119865 (119911lowast 1205911 119901)
= det
[[[[[[[[
[
minus11988611119909lowast
1+
11988612119909lowast
1119909lowast
2
(1198981+ 119909lowast
1)2
11988612119909lowast
1
1198981+ 119909lowast
1
0
119886211198981119909lowast
2
(1198981+ 119909lowast
1)2
minus11988622119909lowast
2+
11988623119909lowast
2119909lowast
3
(1198982+ 119909lowast
2)2
minus
11988623119909lowast
2
1198982+ 119909lowast
2
0
119886321198982119909lowast
3
(1198982+ 119909lowast
2)2
minus11988633119909lowast
3
]]]]]]]]
]
= minus
1198862
21199101lowast1199102lowast
(1 + 11988721199101lowast)3[minus119909lowast+
11988611198871119909lowast1199101lowast
(1 + 1198871119909lowast)2] = 0
(86)
From (86) we know that the hypothesis (A2) in [24]is satisfied
(3) The characteristic matrix of (71) at a stationarysolution (119911 120591
0 1199010) where 119911 = (119911
(1) 119911(2) 119911(3)) isin 119877
3
takes the following form
Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863
120601119865 (119911 120591
1 119901) (119890
120582119868) (87)
that is
Δ (119911 1205911 119901) (120582)
=
[[[[[[[[[[[[[[[[[
[
120582 minus 1198861+ 211988611119911(1)
+
119886121198981119911(2)
(1198981+ 119911(1))
2
11988612119911(1)
1198981+ 119911(1)
0
minus
119886211198981119911(2)
(1198981+ 119911(1))
2119890minus1205821205911
120582 + 1198862minus
11988621119911(1)
1198981+ 119911(1)
+ 211988622119911(2)
+
119886231198982119911(3)
(1198982+ 119911(2))
2
11988623119911(2)
1198982+ 119911(2)
0 minus
119886321198982119911(3)
(1198982+ 119911(2))
2119890minus120582120591lowast
2120582 + 1198863minus
11988632119911(2)
1198982+ 119911(2)
+ 211988633119911(3)
]]]]]]]]]]]]]]]]]
]
(88)
Abstract and Applied Analysis 15
From (88) we have
det (Δ (119864lowast 1205911 119901) (120582))
= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902
0+ 1199030)] 119890minus1205821205911
(89)
Note that (89) is the same as (20) from the discussion inSection 2 about the local Hopf bifurcation it is easy to verifythat (119864
lowast 120591(119895)
1119896 2120587120596
119896) is an isolated center and there exist 120598 gt
0 120575 gt 0 and a smooth curve 120582 (120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575) rarr C
such that det(Δ(120582(1205911))) = 0 |120582(120591
1) minus 120596119896| lt 120598 for all 120591
1isin
[120591(119895)
1119896minus 120575 120591(119895)
1119896+ 120575] and
120582 (120591(119895)
1119896) = 120596119896119894
119889Re 120582 (1205911)
1198891205911
1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)
1119896
gt 0 (90)
Let
Ω1205982120587120596
119896
= (120578 119901) 0 lt 120578 lt 120598
10038161003816100381610038161003816100381610038161003816
119901 minus
2120587
120596119896
10038161003816100381610038161003816100381610038161003816
lt 120598 (91)
It is easy to see that on [120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575] times 120597Ω
1205982120587120596119896
det(Δ(119864
lowast 1205911 119901)(120578 + (2120587119901)119894)) = 0 if and only if 120578 = 0
1205911= 120591(119895)
1119896 119901 = 2120587120596
119896 119896 = 1 2 3 119895 = 0 1 2
Therefore the hypothesis (A4) in [24] is satisfiedIf we define
119867plusmn(119864lowast 120591(119895)
1119896
2120587
120596119896
) (120578 119901)
= det(Δ (119864lowast 120591(119895)
1119896plusmn 120575 119901) (120578 +
2120587
119901
119894))
(92)
then we have the crossing number of isolated center(119864lowast 120591(119895)
1119896 2120587120596
119896) as follows
120574(119864lowast 120591(119895)
1119896
2120587
120596119896
) = deg119861(119867minus(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
minus deg119861(119867+(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
= minus1
(93)
Thus we have
sum
(1199111205911119901)isinC
(119864lowast120591(119895)
11198962120587120596119896)
120574 (119911 1205911 119901) lt 0
(94)
where (119911 1205911 119901) has all or parts of the form
(119864lowast 120591(119896)
1119895 2120587120596
119896) (119895 = 0 1 ) It follows from
Lemma 9 that the connected component ℓ(119864lowast120591(119895)
11198962120587120596
119896)
through (119864lowast 120591(119895)
1119896 2120587120596
119896) is unbounded for each center
(119911lowast 1205911 119901) (119895 = 0 1 ) From the discussion in Section 2
we have
120591(119895)
1119896=
1
120596119896
times arccos(((11990121205962
119896minus1199010) (1199030+1199020)
+ 1205962(1205962minus1199011) (1199031+1199021))
times ((1199030+1199020)2
+ (1199031+1199021)2
1205962
119896)
minus1
) +2119895120587
(95)
where 119896 = 1 2 3 119895 = 0 1 Thus one can get 2120587120596119896le 120591(119895)
1119896
for 119895 ge 1Now we prove that the projection of ℓ
(119864lowast120591(119895)
11198962120587120596
119896)onto
1205911-space is [120591
1 +infin) where 120591
1le 120591(119895)
1119896 Clearly it follows
from the proof of Lemma 11 that system (3) with 1205911
= 0
has no nontrivial periodic solution Hence the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is away from zero
For a contradiction we suppose that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is bounded this means that the
projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is included in a
interval (0 120591lowast) Noticing 2120587120596119896lt 120591119895
1119896and applying Lemma 11
we have 119901 lt 120591lowast for (119911(119905) 120591
1 119901) belonging to ℓ
(119864lowast120591(119895)
11198962120587120596
119896)
This implies that the projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 119901-
space is bounded Then applying Lemma 10 we get thatthe connected component ℓ
(119864lowast120591(119895)
11198962120587120596
119896)is bounded This
contradiction completes the proof
6 Conclusion
In this paper we take our attention to the stability and Hopfbifurcation analysis of a predator-prey systemwithMichaelis-Menten type functional response and two unequal delaysWe obtained some conditions for local stability and Hopfbifurcation occurring When 120591
1= 1205912 we derived the explicit
formulas to determine the properties of periodic solutionsby the normal form method and center manifold theoremSpecially the global existence results of periodic solutionsbifurcating from Hopf bifurcations are also established byusing a global Hopf bifurcation result due to Wu [24]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to thank the anonymous referees fortheir careful reading and constructive suggestions which leadto truly significant improvement of the paper This researchis supported by the National Natural Science Foundation ofChina (no 11061016)
16 Abstract and Applied Analysis
References
[1] R Xu and M A J Chaplain ldquoPersistence and global stabilityin a delayed predator-prey system with Michaelis-Menten typefunctional responserdquo Applied Mathematics and Computationvol 130 no 2-3 pp 441ndash455 2002
[2] Y L Song Y H Peng and J JWei ldquoBifurcations for a predator-prey system with two delaysrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 466ndash479 2008
[3] G-P Hu W-T Li and X-P Yan ldquoHopf bifurcations in apredator-prey system with multiple delaysrdquo Chaos Solitons ampFractals vol 42 no 2 pp 1273ndash1285 2009
[4] M Liao X Tang and C Xu ldquoBifurcation analysis for a three-species predator-prey systemwith two delaysrdquoCommunicationsin Nonlinear Science and Numerical Simulation vol 17 no 1 pp183ndash194 2012
[5] C Xu and P Li ldquoDynamical analysis in a delayed predator-prey model with two delaysrdquo Discrete Dynamics in Nature andSociety vol 2012 Article ID 652947 22 pages 2012
[6] J-F Zhang ldquoStability and bifurcation periodic solutions ina Lotka-Volterra competition system with multiple delaysrdquoNonlinear Dynamics vol 70 no 1 pp 849ndash860 2012
[7] J Xia Z Yu and R Yuan ldquoStability and Hopf bifurcation ina symmetric Lotka-Volterra predator-prey system with delaysrdquoElectronic Journal of Differential Equations vol 2013 no 9 pp1ndash16 2013
[8] H Wan and J Cui ldquoA malaria model with two delaysrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 601265 8pages 2013
[9] M Zhao ldquoHopf bifurcation analysis for a semiratio-dependentpredator-prey system with two delaysrdquo Abstract and AppliedAnalysis vol 2013 Article ID 495072 13 pages 2013
[10] S Feyissa and S Banerjee ldquoDelay-induced oscillatory dynamicsin humoral mediated immune response with two time delaysrdquoNonlinear Analysis Real World Applications vol 14 no 1 pp35ndash52 2013
[11] G Zhang Y Shen andB Chen ldquoHopf bifurcation of a predator-prey systemwith predator harvesting and twodelaysrdquoNonlinearDynamics vol 73 no 4 pp 2119ndash2131 2013
[12] X-Y Meng H-F Huo and H Xiang ldquoHopf bifurcation in athree-species systemwith delaysrdquo Journal of AppliedMathemat-ics and Computing vol 35 no 1-2 pp 635ndash661 2011
[13] J-F Zhang ldquoGlobal existence of bifurcated periodic solutions ina commensalism model with delaysrdquo Applied Mathematics andComputation vol 218 no 23 pp 11688ndash11699 2012
[14] Y Wang W Jiang and H Wang ldquoStability and global Hopfbifurcation in toxic phytoplankton-zooplankton model withdelay and selective harvestingrdquoNonlinear Dynamics vol 73 no1-2 pp 881ndash896 2013
[15] Y Ma ldquoGlobal Hopf bifurcation in the Leslie-Gower predator-prey model with two delaysrdquo Nonlinear Analysis Real WorldApplications vol 13 no 1 pp 370ndash375 2012
[16] X-Y Meng H-F Huo and X-B Zhang ldquoStability and globalHopf bifurcation in a delayed food web consisting of a preyand two predatorsrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4335ndash4348 2011
[17] Y Song J Wei andM Han ldquoLocal and global Hopf bifurcationin a delayed hematopoiesis modelrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol14 no 11 pp 3909ndash3919 2004
[18] Y Song and J Wei ldquoLocal Hopf bifurcation and global periodicsolutions in a delayed predator-prey systemrdquo Journal of Mathe-matical Analysis and Applications vol 301 no 1 pp 1ndash21 2005
[19] J Wei and M Y Li ldquoGlobal existence of periodic solutions in atri-neuron network model with delaysrdquo Physica D vol 198 no1-2 pp 106ndash119 2004
[20] T Zhao Y Kuang and H L Smith ldquoGlobal existence of peri-odic solutions in a class of delayed Gause-type predator-preysystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 28 no 8 pp 1373ndash1394 1997
[21] S J Gao L S Chen and Z D Teng ldquoHopf bifurcation andglobal stability for a delayed predator-prey system with stagestructure for predatorrdquo Applied Mathematics and Computationvol 202 no 2 pp 721ndash729 2008
[22] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977
[23] B Hassard D Kazarinoff and Y WanTheory and Applicationsof Hopf Bifurcation Cambridge University Press CambridgeUK 1981
[24] J H Wu ldquoSymmetric functional-differential equations andneural networks with memoryrdquo Transactions of the AmericanMathematical Society vol 350 no 12 pp 4799ndash4838 1998
[25] V Hutson ldquoThe existence of an equilibrium for permanentsystemsrdquo The Rocky Mountain Journal of Mathematics vol 20no 4 pp 1033ndash1040 1990
[26] S G Ruan and J J Wei ldquoOn the zeros of transcendentalfunctions with applications to stability of delay differentialequations with two delaysrdquo Dynamics of Continuous Discreteamp Impulsive Systems A vol 10 no 6 pp 863ndash874 2003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Abstract and Applied Analysis
0 500 1000 1500
00451
00452
0 500 1000 15000035
0036
0037
0 500 1000 15000054
0055
0056
t
t
t
x1(t)
x2(t)
x3(t)
0045100451
0045200452
0035
00355
003600546
00548
0055
00552
00554
00556
00558
x1(t)
x2 (t)
x3(t)
Figure 3 The trajectories and the phase graph with 1205911= 09 120591
2= 29 119864
lowastis locally asymptotically stable
0 500 1000 15000044
0045
0046
0 500 1000 1500002
003
004
0 500 1000 1500004
006
008
t
t
t
x1(t)
x2(t)
x3(t)
00445
0045
00455
0032
0034
0036
00380052
0054
0056
0058
006
0062
0064
x 1(t)
x2 (t)
x3(t)
Figure 4 The trajectories and the phase graph with 1205911= 12 120591
2= 29 a periodic orbit bifurcate from 119864
lowast
Abstract and Applied Analysis 13
projection of ℓ(119911lowast120591119901)
onto 120591-space is away from zero thenthe projection of ℓ
(119911lowast120591119901)
onto 120591-space must include interval[120591 infin) Following this ideal we can prove our results on theglobal continuation of local Hopf bifurcation
Lemma 10 If the conditions (1198671)ndash(1198674) hold then all nontriv-
ial periodic solutions of system (71) with initial conditions
1199091(120579) = 120593 (120579) ge 0 119909
2(120579) = 120595 (120579) ge 0
1199093(120579) = 120601 (120579) ge 0 120579 isin [minus120591
1 0)
120593 (0) gt 0 120595 (0) gt 0 120601 (0) gt 0
(74)
are uniformly bounded
Proof Suppose that (1199091(119905) 1199092(119905) 1199093(119905)) are nonconstant peri-
odic solutions of system (3) and define
1199091(1205851) = min 119909
1(119905) 119909
1(1205781) = max 119909
1(119905)
1199092(1205852) = min 119909
2(119905) 119909
2(1205782) = max 119909
2(119905)
1199093(1205853) = min 119909
3(119905) 119909
3(1205783) = max 119909
3(119905)
(75)
It follows from system (3) that
1199091(119905) = 119909
1(0) expint
119905
0
[1198861minus 119886111199091(119904) minus
119886121199092(119904)
1198981+ 1199091(119904)
] 119889119904
1199092(119905) = 119909
2(0) expint
119905
0
[minus1198862+
119886211199091(119904 minus 1205911)
1198981+ 1199091(119904 minus 1205911)
minus 119886221199092(119904) minus
119886231199093(119904)
1198982+ 1199092(119904)
] 119889119904
1199093(119905) = 119909
3(0) expint
119905
0
[minus1198863+
119886321199092(119904 minus 120591
lowast
2)
1198982+ 1199092(119904 minus 120591
lowast
2)
minus 119886331199093(119904)] 119889119904
(76)
which implies that the solutions of system (3) cannot crossthe 119909119894-axis (119894 = 1 2 3) Thus the nonconstant periodic orbits
must be located in the interior of first quadrant It followsfrom initial data of system (3) that 119909
1(119905) gt 0 119909
2(119905) gt
0 1199093(119905) gt 0 for 119905 ge 0
From the first equation of system (3) we can get
0 = 1198861minus 119886111199091(1205781) minus
119886121199092(1205781)
1198981+ 1199091(1205781)
le 1198861minus 119886111199091(1205781) (77)
thus we have
1199091(1205781) le
1198861
11988611
(78)
From the second equation of (3) we obtain
0 = minus 1198862+
119886211199091(1205782minus 1205911)
1198981+ 1199091(1205782minus 1205911)
minus 119886221199092(1205782)
minus
119886231199093(1205782)
1198982+ 1199092(1205782)
le minus1198862+
11988621(119886111988611)
1198981+ (119886111988611)
minus 119886221199092(1205782)
(79)
therefore one can get
1199092(1205782) le
minus1198862(119886111198981+ 1198861) + 119886111988621
11988622(119886111198981+ 1198861)
≜ 1198721 (80)
Applying the third equation of system (3) we know
0 = minus1198863+
119886321199092(1205783minus 120591lowast
2)
1198982+ 1199092(1205783minus 120591lowast
2)
minus 119886331199093(1205783)
le minus1198863+
119886321198721
1198982+1198721
minus 119886331199093(1205783)
(81)
It follows that
1199093(1205783) le
minus1198863(1198982+1198721) + 119886321198721
11988633(1198982+1198721)
≜ 1198722 (82)
This shows that the nontrivial periodic solution of system (3)is uniformly bounded and the proof is complete
Lemma 11 If the conditions (1198671)ndash(1198674) and
(H7) 11988622
minus (119886211198982) gt 0 [(119886
11minus 11988611198981)11988622
minus
(11988612119886211198982
1)]11988633
minus (11988611
minus (11988611198981))[(119886211198982)11988633
+
(11988623119886321198982
2)] gt 0
hold then system (3) has no nontrivial 1205911-periodic solution
Proof Suppose for a contradiction that system (3) has non-trivial periodic solution with period 120591
1 Then the following
system (83) of ordinary differential equations has nontrivialperiodic solution
1198891199091
119889119905
= 1199091(119905) [1198861minus 119886111199091(119905) minus
119886121199092(119905)
1198981+ 1199091(119905)
]
1198891199092
119889119905
= 1199092(119905) [minus119886
2+
119886211199091(119905)
1198981+ 1199091(119905)
minus 119886221199092(119905) minus
119886231199093(119905)
1198982+ 1199092(119905)
]
1198891199093
119889119905
= 1199093(119905) [minus119886
3+
119886321199092(119905 minus 120591
lowast
2)
1198982+ 1199092(119905 minus 120591
lowast
2)
minus 119886331199093(119905)]
(83)
which has the same equilibria to system (3) that is
1198641= (0 0 0) 119864
2= (
1198861
11988611
0 0)
1198643= (1199091 1199092 0) 119864
lowast= (119909lowast
1 119909lowast
2 119909lowast
3)
(84)
Note that 119909119894-axis (119894 = 1 2 3) are the invariable manifold of
system (83) and the orbits of system (83) do not intersect each
14 Abstract and Applied Analysis
other Thus there are no solutions crossing the coordinateaxes On the other hand note the fact that if system (83) hasa periodic solution then there must be the equilibrium in itsinterior and that 119864
1 1198642 1198643are located on the coordinate
axis Thus we conclude that the periodic orbit of system (83)must lie in the first quadrant If (119867
7) holds it is well known
that the positive equilibrium 119864lowastis globally asymptotically
stable in the first quadrant (see [1])Thus there is no periodicorbit in the first quadrant too The above discussion meansthat (83) does not have any nontrivial periodic solution It isa contradiction Therefore the lemma is confirmed
Theorem 12 Suppose the conditions of Theorem 8 and (1198677)
hold let120596119896and 120591(119895)1119896
be defined in Section 2 then when 1205911gt 120591(119895)
1119896
system (3) has at least 119895 minus 1 periodic solutions
Proof It is sufficient to prove that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is [120591
1 +infin) for each 119895 ge 1 where
1205911le 120591(119895)
1119896
In following we prove that the hypotheses (A1)ndash(A4) in[24] hold
(1) From system (3) we know easily that the followingconditions hold
(A1) 119865 isin 1198622(1198773
+times119877+times119877+) where 119865 = 119865|
1198773
+times119877+times119877+
rarr
1198773
+
(A3) 119865(120601 1205911 119901) is differential with respect to 120601
(2) It follows from system (3) that
119863119911119865 (119911 120591
1 119901) =
[[[[[[[[[[[
[
1198861minus 2119886111199091minus
1198861211989811199092
(1198981+ 1199091)2
minus
119886121199091
1198981+ 1199091
0
1198862111989811199092
(1198981+ 1199091)2
minus1198862+
119886211199091
1198981+ 1199091
minus 2119886221199092minus
1198862311989821199093
(1198982+ 1199092)2
minus
119886231199092
1198982+ 1199092
0
1198863211989821199093
(1198982+ 1199092)2
minus1198863+
119886321199092
1198982+ 1199092
minus 2119886331199093
]]]]]]]]]]]
]
(85)
Then under the assumption (1198671)ndash(1198674) we have
det119863119911119865 (119911lowast 1205911 119901)
= det
[[[[[[[[
[
minus11988611119909lowast
1+
11988612119909lowast
1119909lowast
2
(1198981+ 119909lowast
1)2
11988612119909lowast
1
1198981+ 119909lowast
1
0
119886211198981119909lowast
2
(1198981+ 119909lowast
1)2
minus11988622119909lowast
2+
11988623119909lowast
2119909lowast
3
(1198982+ 119909lowast
2)2
minus
11988623119909lowast
2
1198982+ 119909lowast
2
0
119886321198982119909lowast
3
(1198982+ 119909lowast
2)2
minus11988633119909lowast
3
]]]]]]]]
]
= minus
1198862
21199101lowast1199102lowast
(1 + 11988721199101lowast)3[minus119909lowast+
11988611198871119909lowast1199101lowast
(1 + 1198871119909lowast)2] = 0
(86)
From (86) we know that the hypothesis (A2) in [24]is satisfied
(3) The characteristic matrix of (71) at a stationarysolution (119911 120591
0 1199010) where 119911 = (119911
(1) 119911(2) 119911(3)) isin 119877
3
takes the following form
Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863
120601119865 (119911 120591
1 119901) (119890
120582119868) (87)
that is
Δ (119911 1205911 119901) (120582)
=
[[[[[[[[[[[[[[[[[
[
120582 minus 1198861+ 211988611119911(1)
+
119886121198981119911(2)
(1198981+ 119911(1))
2
11988612119911(1)
1198981+ 119911(1)
0
minus
119886211198981119911(2)
(1198981+ 119911(1))
2119890minus1205821205911
120582 + 1198862minus
11988621119911(1)
1198981+ 119911(1)
+ 211988622119911(2)
+
119886231198982119911(3)
(1198982+ 119911(2))
2
11988623119911(2)
1198982+ 119911(2)
0 minus
119886321198982119911(3)
(1198982+ 119911(2))
2119890minus120582120591lowast
2120582 + 1198863minus
11988632119911(2)
1198982+ 119911(2)
+ 211988633119911(3)
]]]]]]]]]]]]]]]]]
]
(88)
Abstract and Applied Analysis 15
From (88) we have
det (Δ (119864lowast 1205911 119901) (120582))
= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902
0+ 1199030)] 119890minus1205821205911
(89)
Note that (89) is the same as (20) from the discussion inSection 2 about the local Hopf bifurcation it is easy to verifythat (119864
lowast 120591(119895)
1119896 2120587120596
119896) is an isolated center and there exist 120598 gt
0 120575 gt 0 and a smooth curve 120582 (120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575) rarr C
such that det(Δ(120582(1205911))) = 0 |120582(120591
1) minus 120596119896| lt 120598 for all 120591
1isin
[120591(119895)
1119896minus 120575 120591(119895)
1119896+ 120575] and
120582 (120591(119895)
1119896) = 120596119896119894
119889Re 120582 (1205911)
1198891205911
1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)
1119896
gt 0 (90)
Let
Ω1205982120587120596
119896
= (120578 119901) 0 lt 120578 lt 120598
10038161003816100381610038161003816100381610038161003816
119901 minus
2120587
120596119896
10038161003816100381610038161003816100381610038161003816
lt 120598 (91)
It is easy to see that on [120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575] times 120597Ω
1205982120587120596119896
det(Δ(119864
lowast 1205911 119901)(120578 + (2120587119901)119894)) = 0 if and only if 120578 = 0
1205911= 120591(119895)
1119896 119901 = 2120587120596
119896 119896 = 1 2 3 119895 = 0 1 2
Therefore the hypothesis (A4) in [24] is satisfiedIf we define
119867plusmn(119864lowast 120591(119895)
1119896
2120587
120596119896
) (120578 119901)
= det(Δ (119864lowast 120591(119895)
1119896plusmn 120575 119901) (120578 +
2120587
119901
119894))
(92)
then we have the crossing number of isolated center(119864lowast 120591(119895)
1119896 2120587120596
119896) as follows
120574(119864lowast 120591(119895)
1119896
2120587
120596119896
) = deg119861(119867minus(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
minus deg119861(119867+(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
= minus1
(93)
Thus we have
sum
(1199111205911119901)isinC
(119864lowast120591(119895)
11198962120587120596119896)
120574 (119911 1205911 119901) lt 0
(94)
where (119911 1205911 119901) has all or parts of the form
(119864lowast 120591(119896)
1119895 2120587120596
119896) (119895 = 0 1 ) It follows from
Lemma 9 that the connected component ℓ(119864lowast120591(119895)
11198962120587120596
119896)
through (119864lowast 120591(119895)
1119896 2120587120596
119896) is unbounded for each center
(119911lowast 1205911 119901) (119895 = 0 1 ) From the discussion in Section 2
we have
120591(119895)
1119896=
1
120596119896
times arccos(((11990121205962
119896minus1199010) (1199030+1199020)
+ 1205962(1205962minus1199011) (1199031+1199021))
times ((1199030+1199020)2
+ (1199031+1199021)2
1205962
119896)
minus1
) +2119895120587
(95)
where 119896 = 1 2 3 119895 = 0 1 Thus one can get 2120587120596119896le 120591(119895)
1119896
for 119895 ge 1Now we prove that the projection of ℓ
(119864lowast120591(119895)
11198962120587120596
119896)onto
1205911-space is [120591
1 +infin) where 120591
1le 120591(119895)
1119896 Clearly it follows
from the proof of Lemma 11 that system (3) with 1205911
= 0
has no nontrivial periodic solution Hence the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is away from zero
For a contradiction we suppose that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is bounded this means that the
projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is included in a
interval (0 120591lowast) Noticing 2120587120596119896lt 120591119895
1119896and applying Lemma 11
we have 119901 lt 120591lowast for (119911(119905) 120591
1 119901) belonging to ℓ
(119864lowast120591(119895)
11198962120587120596
119896)
This implies that the projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 119901-
space is bounded Then applying Lemma 10 we get thatthe connected component ℓ
(119864lowast120591(119895)
11198962120587120596
119896)is bounded This
contradiction completes the proof
6 Conclusion
In this paper we take our attention to the stability and Hopfbifurcation analysis of a predator-prey systemwithMichaelis-Menten type functional response and two unequal delaysWe obtained some conditions for local stability and Hopfbifurcation occurring When 120591
1= 1205912 we derived the explicit
formulas to determine the properties of periodic solutionsby the normal form method and center manifold theoremSpecially the global existence results of periodic solutionsbifurcating from Hopf bifurcations are also established byusing a global Hopf bifurcation result due to Wu [24]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to thank the anonymous referees fortheir careful reading and constructive suggestions which leadto truly significant improvement of the paper This researchis supported by the National Natural Science Foundation ofChina (no 11061016)
16 Abstract and Applied Analysis
References
[1] R Xu and M A J Chaplain ldquoPersistence and global stabilityin a delayed predator-prey system with Michaelis-Menten typefunctional responserdquo Applied Mathematics and Computationvol 130 no 2-3 pp 441ndash455 2002
[2] Y L Song Y H Peng and J JWei ldquoBifurcations for a predator-prey system with two delaysrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 466ndash479 2008
[3] G-P Hu W-T Li and X-P Yan ldquoHopf bifurcations in apredator-prey system with multiple delaysrdquo Chaos Solitons ampFractals vol 42 no 2 pp 1273ndash1285 2009
[4] M Liao X Tang and C Xu ldquoBifurcation analysis for a three-species predator-prey systemwith two delaysrdquoCommunicationsin Nonlinear Science and Numerical Simulation vol 17 no 1 pp183ndash194 2012
[5] C Xu and P Li ldquoDynamical analysis in a delayed predator-prey model with two delaysrdquo Discrete Dynamics in Nature andSociety vol 2012 Article ID 652947 22 pages 2012
[6] J-F Zhang ldquoStability and bifurcation periodic solutions ina Lotka-Volterra competition system with multiple delaysrdquoNonlinear Dynamics vol 70 no 1 pp 849ndash860 2012
[7] J Xia Z Yu and R Yuan ldquoStability and Hopf bifurcation ina symmetric Lotka-Volterra predator-prey system with delaysrdquoElectronic Journal of Differential Equations vol 2013 no 9 pp1ndash16 2013
[8] H Wan and J Cui ldquoA malaria model with two delaysrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 601265 8pages 2013
[9] M Zhao ldquoHopf bifurcation analysis for a semiratio-dependentpredator-prey system with two delaysrdquo Abstract and AppliedAnalysis vol 2013 Article ID 495072 13 pages 2013
[10] S Feyissa and S Banerjee ldquoDelay-induced oscillatory dynamicsin humoral mediated immune response with two time delaysrdquoNonlinear Analysis Real World Applications vol 14 no 1 pp35ndash52 2013
[11] G Zhang Y Shen andB Chen ldquoHopf bifurcation of a predator-prey systemwith predator harvesting and twodelaysrdquoNonlinearDynamics vol 73 no 4 pp 2119ndash2131 2013
[12] X-Y Meng H-F Huo and H Xiang ldquoHopf bifurcation in athree-species systemwith delaysrdquo Journal of AppliedMathemat-ics and Computing vol 35 no 1-2 pp 635ndash661 2011
[13] J-F Zhang ldquoGlobal existence of bifurcated periodic solutions ina commensalism model with delaysrdquo Applied Mathematics andComputation vol 218 no 23 pp 11688ndash11699 2012
[14] Y Wang W Jiang and H Wang ldquoStability and global Hopfbifurcation in toxic phytoplankton-zooplankton model withdelay and selective harvestingrdquoNonlinear Dynamics vol 73 no1-2 pp 881ndash896 2013
[15] Y Ma ldquoGlobal Hopf bifurcation in the Leslie-Gower predator-prey model with two delaysrdquo Nonlinear Analysis Real WorldApplications vol 13 no 1 pp 370ndash375 2012
[16] X-Y Meng H-F Huo and X-B Zhang ldquoStability and globalHopf bifurcation in a delayed food web consisting of a preyand two predatorsrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4335ndash4348 2011
[17] Y Song J Wei andM Han ldquoLocal and global Hopf bifurcationin a delayed hematopoiesis modelrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol14 no 11 pp 3909ndash3919 2004
[18] Y Song and J Wei ldquoLocal Hopf bifurcation and global periodicsolutions in a delayed predator-prey systemrdquo Journal of Mathe-matical Analysis and Applications vol 301 no 1 pp 1ndash21 2005
[19] J Wei and M Y Li ldquoGlobal existence of periodic solutions in atri-neuron network model with delaysrdquo Physica D vol 198 no1-2 pp 106ndash119 2004
[20] T Zhao Y Kuang and H L Smith ldquoGlobal existence of peri-odic solutions in a class of delayed Gause-type predator-preysystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 28 no 8 pp 1373ndash1394 1997
[21] S J Gao L S Chen and Z D Teng ldquoHopf bifurcation andglobal stability for a delayed predator-prey system with stagestructure for predatorrdquo Applied Mathematics and Computationvol 202 no 2 pp 721ndash729 2008
[22] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977
[23] B Hassard D Kazarinoff and Y WanTheory and Applicationsof Hopf Bifurcation Cambridge University Press CambridgeUK 1981
[24] J H Wu ldquoSymmetric functional-differential equations andneural networks with memoryrdquo Transactions of the AmericanMathematical Society vol 350 no 12 pp 4799ndash4838 1998
[25] V Hutson ldquoThe existence of an equilibrium for permanentsystemsrdquo The Rocky Mountain Journal of Mathematics vol 20no 4 pp 1033ndash1040 1990
[26] S G Ruan and J J Wei ldquoOn the zeros of transcendentalfunctions with applications to stability of delay differentialequations with two delaysrdquo Dynamics of Continuous Discreteamp Impulsive Systems A vol 10 no 6 pp 863ndash874 2003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 13
projection of ℓ(119911lowast120591119901)
onto 120591-space is away from zero thenthe projection of ℓ
(119911lowast120591119901)
onto 120591-space must include interval[120591 infin) Following this ideal we can prove our results on theglobal continuation of local Hopf bifurcation
Lemma 10 If the conditions (1198671)ndash(1198674) hold then all nontriv-
ial periodic solutions of system (71) with initial conditions
1199091(120579) = 120593 (120579) ge 0 119909
2(120579) = 120595 (120579) ge 0
1199093(120579) = 120601 (120579) ge 0 120579 isin [minus120591
1 0)
120593 (0) gt 0 120595 (0) gt 0 120601 (0) gt 0
(74)
are uniformly bounded
Proof Suppose that (1199091(119905) 1199092(119905) 1199093(119905)) are nonconstant peri-
odic solutions of system (3) and define
1199091(1205851) = min 119909
1(119905) 119909
1(1205781) = max 119909
1(119905)
1199092(1205852) = min 119909
2(119905) 119909
2(1205782) = max 119909
2(119905)
1199093(1205853) = min 119909
3(119905) 119909
3(1205783) = max 119909
3(119905)
(75)
It follows from system (3) that
1199091(119905) = 119909
1(0) expint
119905
0
[1198861minus 119886111199091(119904) minus
119886121199092(119904)
1198981+ 1199091(119904)
] 119889119904
1199092(119905) = 119909
2(0) expint
119905
0
[minus1198862+
119886211199091(119904 minus 1205911)
1198981+ 1199091(119904 minus 1205911)
minus 119886221199092(119904) minus
119886231199093(119904)
1198982+ 1199092(119904)
] 119889119904
1199093(119905) = 119909
3(0) expint
119905
0
[minus1198863+
119886321199092(119904 minus 120591
lowast
2)
1198982+ 1199092(119904 minus 120591
lowast
2)
minus 119886331199093(119904)] 119889119904
(76)
which implies that the solutions of system (3) cannot crossthe 119909119894-axis (119894 = 1 2 3) Thus the nonconstant periodic orbits
must be located in the interior of first quadrant It followsfrom initial data of system (3) that 119909
1(119905) gt 0 119909
2(119905) gt
0 1199093(119905) gt 0 for 119905 ge 0
From the first equation of system (3) we can get
0 = 1198861minus 119886111199091(1205781) minus
119886121199092(1205781)
1198981+ 1199091(1205781)
le 1198861minus 119886111199091(1205781) (77)
thus we have
1199091(1205781) le
1198861
11988611
(78)
From the second equation of (3) we obtain
0 = minus 1198862+
119886211199091(1205782minus 1205911)
1198981+ 1199091(1205782minus 1205911)
minus 119886221199092(1205782)
minus
119886231199093(1205782)
1198982+ 1199092(1205782)
le minus1198862+
11988621(119886111988611)
1198981+ (119886111988611)
minus 119886221199092(1205782)
(79)
therefore one can get
1199092(1205782) le
minus1198862(119886111198981+ 1198861) + 119886111988621
11988622(119886111198981+ 1198861)
≜ 1198721 (80)
Applying the third equation of system (3) we know
0 = minus1198863+
119886321199092(1205783minus 120591lowast
2)
1198982+ 1199092(1205783minus 120591lowast
2)
minus 119886331199093(1205783)
le minus1198863+
119886321198721
1198982+1198721
minus 119886331199093(1205783)
(81)
It follows that
1199093(1205783) le
minus1198863(1198982+1198721) + 119886321198721
11988633(1198982+1198721)
≜ 1198722 (82)
This shows that the nontrivial periodic solution of system (3)is uniformly bounded and the proof is complete
Lemma 11 If the conditions (1198671)ndash(1198674) and
(H7) 11988622
minus (119886211198982) gt 0 [(119886
11minus 11988611198981)11988622
minus
(11988612119886211198982
1)]11988633
minus (11988611
minus (11988611198981))[(119886211198982)11988633
+
(11988623119886321198982
2)] gt 0
hold then system (3) has no nontrivial 1205911-periodic solution
Proof Suppose for a contradiction that system (3) has non-trivial periodic solution with period 120591
1 Then the following
system (83) of ordinary differential equations has nontrivialperiodic solution
1198891199091
119889119905
= 1199091(119905) [1198861minus 119886111199091(119905) minus
119886121199092(119905)
1198981+ 1199091(119905)
]
1198891199092
119889119905
= 1199092(119905) [minus119886
2+
119886211199091(119905)
1198981+ 1199091(119905)
minus 119886221199092(119905) minus
119886231199093(119905)
1198982+ 1199092(119905)
]
1198891199093
119889119905
= 1199093(119905) [minus119886
3+
119886321199092(119905 minus 120591
lowast
2)
1198982+ 1199092(119905 minus 120591
lowast
2)
minus 119886331199093(119905)]
(83)
which has the same equilibria to system (3) that is
1198641= (0 0 0) 119864
2= (
1198861
11988611
0 0)
1198643= (1199091 1199092 0) 119864
lowast= (119909lowast
1 119909lowast
2 119909lowast
3)
(84)
Note that 119909119894-axis (119894 = 1 2 3) are the invariable manifold of
system (83) and the orbits of system (83) do not intersect each
14 Abstract and Applied Analysis
other Thus there are no solutions crossing the coordinateaxes On the other hand note the fact that if system (83) hasa periodic solution then there must be the equilibrium in itsinterior and that 119864
1 1198642 1198643are located on the coordinate
axis Thus we conclude that the periodic orbit of system (83)must lie in the first quadrant If (119867
7) holds it is well known
that the positive equilibrium 119864lowastis globally asymptotically
stable in the first quadrant (see [1])Thus there is no periodicorbit in the first quadrant too The above discussion meansthat (83) does not have any nontrivial periodic solution It isa contradiction Therefore the lemma is confirmed
Theorem 12 Suppose the conditions of Theorem 8 and (1198677)
hold let120596119896and 120591(119895)1119896
be defined in Section 2 then when 1205911gt 120591(119895)
1119896
system (3) has at least 119895 minus 1 periodic solutions
Proof It is sufficient to prove that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is [120591
1 +infin) for each 119895 ge 1 where
1205911le 120591(119895)
1119896
In following we prove that the hypotheses (A1)ndash(A4) in[24] hold
(1) From system (3) we know easily that the followingconditions hold
(A1) 119865 isin 1198622(1198773
+times119877+times119877+) where 119865 = 119865|
1198773
+times119877+times119877+
rarr
1198773
+
(A3) 119865(120601 1205911 119901) is differential with respect to 120601
(2) It follows from system (3) that
119863119911119865 (119911 120591
1 119901) =
[[[[[[[[[[[
[
1198861minus 2119886111199091minus
1198861211989811199092
(1198981+ 1199091)2
minus
119886121199091
1198981+ 1199091
0
1198862111989811199092
(1198981+ 1199091)2
minus1198862+
119886211199091
1198981+ 1199091
minus 2119886221199092minus
1198862311989821199093
(1198982+ 1199092)2
minus
119886231199092
1198982+ 1199092
0
1198863211989821199093
(1198982+ 1199092)2
minus1198863+
119886321199092
1198982+ 1199092
minus 2119886331199093
]]]]]]]]]]]
]
(85)
Then under the assumption (1198671)ndash(1198674) we have
det119863119911119865 (119911lowast 1205911 119901)
= det
[[[[[[[[
[
minus11988611119909lowast
1+
11988612119909lowast
1119909lowast
2
(1198981+ 119909lowast
1)2
11988612119909lowast
1
1198981+ 119909lowast
1
0
119886211198981119909lowast
2
(1198981+ 119909lowast
1)2
minus11988622119909lowast
2+
11988623119909lowast
2119909lowast
3
(1198982+ 119909lowast
2)2
minus
11988623119909lowast
2
1198982+ 119909lowast
2
0
119886321198982119909lowast
3
(1198982+ 119909lowast
2)2
minus11988633119909lowast
3
]]]]]]]]
]
= minus
1198862
21199101lowast1199102lowast
(1 + 11988721199101lowast)3[minus119909lowast+
11988611198871119909lowast1199101lowast
(1 + 1198871119909lowast)2] = 0
(86)
From (86) we know that the hypothesis (A2) in [24]is satisfied
(3) The characteristic matrix of (71) at a stationarysolution (119911 120591
0 1199010) where 119911 = (119911
(1) 119911(2) 119911(3)) isin 119877
3
takes the following form
Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863
120601119865 (119911 120591
1 119901) (119890
120582119868) (87)
that is
Δ (119911 1205911 119901) (120582)
=
[[[[[[[[[[[[[[[[[
[
120582 minus 1198861+ 211988611119911(1)
+
119886121198981119911(2)
(1198981+ 119911(1))
2
11988612119911(1)
1198981+ 119911(1)
0
minus
119886211198981119911(2)
(1198981+ 119911(1))
2119890minus1205821205911
120582 + 1198862minus
11988621119911(1)
1198981+ 119911(1)
+ 211988622119911(2)
+
119886231198982119911(3)
(1198982+ 119911(2))
2
11988623119911(2)
1198982+ 119911(2)
0 minus
119886321198982119911(3)
(1198982+ 119911(2))
2119890minus120582120591lowast
2120582 + 1198863minus
11988632119911(2)
1198982+ 119911(2)
+ 211988633119911(3)
]]]]]]]]]]]]]]]]]
]
(88)
Abstract and Applied Analysis 15
From (88) we have
det (Δ (119864lowast 1205911 119901) (120582))
= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902
0+ 1199030)] 119890minus1205821205911
(89)
Note that (89) is the same as (20) from the discussion inSection 2 about the local Hopf bifurcation it is easy to verifythat (119864
lowast 120591(119895)
1119896 2120587120596
119896) is an isolated center and there exist 120598 gt
0 120575 gt 0 and a smooth curve 120582 (120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575) rarr C
such that det(Δ(120582(1205911))) = 0 |120582(120591
1) minus 120596119896| lt 120598 for all 120591
1isin
[120591(119895)
1119896minus 120575 120591(119895)
1119896+ 120575] and
120582 (120591(119895)
1119896) = 120596119896119894
119889Re 120582 (1205911)
1198891205911
1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)
1119896
gt 0 (90)
Let
Ω1205982120587120596
119896
= (120578 119901) 0 lt 120578 lt 120598
10038161003816100381610038161003816100381610038161003816
119901 minus
2120587
120596119896
10038161003816100381610038161003816100381610038161003816
lt 120598 (91)
It is easy to see that on [120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575] times 120597Ω
1205982120587120596119896
det(Δ(119864
lowast 1205911 119901)(120578 + (2120587119901)119894)) = 0 if and only if 120578 = 0
1205911= 120591(119895)
1119896 119901 = 2120587120596
119896 119896 = 1 2 3 119895 = 0 1 2
Therefore the hypothesis (A4) in [24] is satisfiedIf we define
119867plusmn(119864lowast 120591(119895)
1119896
2120587
120596119896
) (120578 119901)
= det(Δ (119864lowast 120591(119895)
1119896plusmn 120575 119901) (120578 +
2120587
119901
119894))
(92)
then we have the crossing number of isolated center(119864lowast 120591(119895)
1119896 2120587120596
119896) as follows
120574(119864lowast 120591(119895)
1119896
2120587
120596119896
) = deg119861(119867minus(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
minus deg119861(119867+(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
= minus1
(93)
Thus we have
sum
(1199111205911119901)isinC
(119864lowast120591(119895)
11198962120587120596119896)
120574 (119911 1205911 119901) lt 0
(94)
where (119911 1205911 119901) has all or parts of the form
(119864lowast 120591(119896)
1119895 2120587120596
119896) (119895 = 0 1 ) It follows from
Lemma 9 that the connected component ℓ(119864lowast120591(119895)
11198962120587120596
119896)
through (119864lowast 120591(119895)
1119896 2120587120596
119896) is unbounded for each center
(119911lowast 1205911 119901) (119895 = 0 1 ) From the discussion in Section 2
we have
120591(119895)
1119896=
1
120596119896
times arccos(((11990121205962
119896minus1199010) (1199030+1199020)
+ 1205962(1205962minus1199011) (1199031+1199021))
times ((1199030+1199020)2
+ (1199031+1199021)2
1205962
119896)
minus1
) +2119895120587
(95)
where 119896 = 1 2 3 119895 = 0 1 Thus one can get 2120587120596119896le 120591(119895)
1119896
for 119895 ge 1Now we prove that the projection of ℓ
(119864lowast120591(119895)
11198962120587120596
119896)onto
1205911-space is [120591
1 +infin) where 120591
1le 120591(119895)
1119896 Clearly it follows
from the proof of Lemma 11 that system (3) with 1205911
= 0
has no nontrivial periodic solution Hence the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is away from zero
For a contradiction we suppose that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is bounded this means that the
projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is included in a
interval (0 120591lowast) Noticing 2120587120596119896lt 120591119895
1119896and applying Lemma 11
we have 119901 lt 120591lowast for (119911(119905) 120591
1 119901) belonging to ℓ
(119864lowast120591(119895)
11198962120587120596
119896)
This implies that the projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 119901-
space is bounded Then applying Lemma 10 we get thatthe connected component ℓ
(119864lowast120591(119895)
11198962120587120596
119896)is bounded This
contradiction completes the proof
6 Conclusion
In this paper we take our attention to the stability and Hopfbifurcation analysis of a predator-prey systemwithMichaelis-Menten type functional response and two unequal delaysWe obtained some conditions for local stability and Hopfbifurcation occurring When 120591
1= 1205912 we derived the explicit
formulas to determine the properties of periodic solutionsby the normal form method and center manifold theoremSpecially the global existence results of periodic solutionsbifurcating from Hopf bifurcations are also established byusing a global Hopf bifurcation result due to Wu [24]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to thank the anonymous referees fortheir careful reading and constructive suggestions which leadto truly significant improvement of the paper This researchis supported by the National Natural Science Foundation ofChina (no 11061016)
16 Abstract and Applied Analysis
References
[1] R Xu and M A J Chaplain ldquoPersistence and global stabilityin a delayed predator-prey system with Michaelis-Menten typefunctional responserdquo Applied Mathematics and Computationvol 130 no 2-3 pp 441ndash455 2002
[2] Y L Song Y H Peng and J JWei ldquoBifurcations for a predator-prey system with two delaysrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 466ndash479 2008
[3] G-P Hu W-T Li and X-P Yan ldquoHopf bifurcations in apredator-prey system with multiple delaysrdquo Chaos Solitons ampFractals vol 42 no 2 pp 1273ndash1285 2009
[4] M Liao X Tang and C Xu ldquoBifurcation analysis for a three-species predator-prey systemwith two delaysrdquoCommunicationsin Nonlinear Science and Numerical Simulation vol 17 no 1 pp183ndash194 2012
[5] C Xu and P Li ldquoDynamical analysis in a delayed predator-prey model with two delaysrdquo Discrete Dynamics in Nature andSociety vol 2012 Article ID 652947 22 pages 2012
[6] J-F Zhang ldquoStability and bifurcation periodic solutions ina Lotka-Volterra competition system with multiple delaysrdquoNonlinear Dynamics vol 70 no 1 pp 849ndash860 2012
[7] J Xia Z Yu and R Yuan ldquoStability and Hopf bifurcation ina symmetric Lotka-Volterra predator-prey system with delaysrdquoElectronic Journal of Differential Equations vol 2013 no 9 pp1ndash16 2013
[8] H Wan and J Cui ldquoA malaria model with two delaysrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 601265 8pages 2013
[9] M Zhao ldquoHopf bifurcation analysis for a semiratio-dependentpredator-prey system with two delaysrdquo Abstract and AppliedAnalysis vol 2013 Article ID 495072 13 pages 2013
[10] S Feyissa and S Banerjee ldquoDelay-induced oscillatory dynamicsin humoral mediated immune response with two time delaysrdquoNonlinear Analysis Real World Applications vol 14 no 1 pp35ndash52 2013
[11] G Zhang Y Shen andB Chen ldquoHopf bifurcation of a predator-prey systemwith predator harvesting and twodelaysrdquoNonlinearDynamics vol 73 no 4 pp 2119ndash2131 2013
[12] X-Y Meng H-F Huo and H Xiang ldquoHopf bifurcation in athree-species systemwith delaysrdquo Journal of AppliedMathemat-ics and Computing vol 35 no 1-2 pp 635ndash661 2011
[13] J-F Zhang ldquoGlobal existence of bifurcated periodic solutions ina commensalism model with delaysrdquo Applied Mathematics andComputation vol 218 no 23 pp 11688ndash11699 2012
[14] Y Wang W Jiang and H Wang ldquoStability and global Hopfbifurcation in toxic phytoplankton-zooplankton model withdelay and selective harvestingrdquoNonlinear Dynamics vol 73 no1-2 pp 881ndash896 2013
[15] Y Ma ldquoGlobal Hopf bifurcation in the Leslie-Gower predator-prey model with two delaysrdquo Nonlinear Analysis Real WorldApplications vol 13 no 1 pp 370ndash375 2012
[16] X-Y Meng H-F Huo and X-B Zhang ldquoStability and globalHopf bifurcation in a delayed food web consisting of a preyand two predatorsrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4335ndash4348 2011
[17] Y Song J Wei andM Han ldquoLocal and global Hopf bifurcationin a delayed hematopoiesis modelrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol14 no 11 pp 3909ndash3919 2004
[18] Y Song and J Wei ldquoLocal Hopf bifurcation and global periodicsolutions in a delayed predator-prey systemrdquo Journal of Mathe-matical Analysis and Applications vol 301 no 1 pp 1ndash21 2005
[19] J Wei and M Y Li ldquoGlobal existence of periodic solutions in atri-neuron network model with delaysrdquo Physica D vol 198 no1-2 pp 106ndash119 2004
[20] T Zhao Y Kuang and H L Smith ldquoGlobal existence of peri-odic solutions in a class of delayed Gause-type predator-preysystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 28 no 8 pp 1373ndash1394 1997
[21] S J Gao L S Chen and Z D Teng ldquoHopf bifurcation andglobal stability for a delayed predator-prey system with stagestructure for predatorrdquo Applied Mathematics and Computationvol 202 no 2 pp 721ndash729 2008
[22] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977
[23] B Hassard D Kazarinoff and Y WanTheory and Applicationsof Hopf Bifurcation Cambridge University Press CambridgeUK 1981
[24] J H Wu ldquoSymmetric functional-differential equations andneural networks with memoryrdquo Transactions of the AmericanMathematical Society vol 350 no 12 pp 4799ndash4838 1998
[25] V Hutson ldquoThe existence of an equilibrium for permanentsystemsrdquo The Rocky Mountain Journal of Mathematics vol 20no 4 pp 1033ndash1040 1990
[26] S G Ruan and J J Wei ldquoOn the zeros of transcendentalfunctions with applications to stability of delay differentialequations with two delaysrdquo Dynamics of Continuous Discreteamp Impulsive Systems A vol 10 no 6 pp 863ndash874 2003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
14 Abstract and Applied Analysis
other Thus there are no solutions crossing the coordinateaxes On the other hand note the fact that if system (83) hasa periodic solution then there must be the equilibrium in itsinterior and that 119864
1 1198642 1198643are located on the coordinate
axis Thus we conclude that the periodic orbit of system (83)must lie in the first quadrant If (119867
7) holds it is well known
that the positive equilibrium 119864lowastis globally asymptotically
stable in the first quadrant (see [1])Thus there is no periodicorbit in the first quadrant too The above discussion meansthat (83) does not have any nontrivial periodic solution It isa contradiction Therefore the lemma is confirmed
Theorem 12 Suppose the conditions of Theorem 8 and (1198677)
hold let120596119896and 120591(119895)1119896
be defined in Section 2 then when 1205911gt 120591(119895)
1119896
system (3) has at least 119895 minus 1 periodic solutions
Proof It is sufficient to prove that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is [120591
1 +infin) for each 119895 ge 1 where
1205911le 120591(119895)
1119896
In following we prove that the hypotheses (A1)ndash(A4) in[24] hold
(1) From system (3) we know easily that the followingconditions hold
(A1) 119865 isin 1198622(1198773
+times119877+times119877+) where 119865 = 119865|
1198773
+times119877+times119877+
rarr
1198773
+
(A3) 119865(120601 1205911 119901) is differential with respect to 120601
(2) It follows from system (3) that
119863119911119865 (119911 120591
1 119901) =
[[[[[[[[[[[
[
1198861minus 2119886111199091minus
1198861211989811199092
(1198981+ 1199091)2
minus
119886121199091
1198981+ 1199091
0
1198862111989811199092
(1198981+ 1199091)2
minus1198862+
119886211199091
1198981+ 1199091
minus 2119886221199092minus
1198862311989821199093
(1198982+ 1199092)2
minus
119886231199092
1198982+ 1199092
0
1198863211989821199093
(1198982+ 1199092)2
minus1198863+
119886321199092
1198982+ 1199092
minus 2119886331199093
]]]]]]]]]]]
]
(85)
Then under the assumption (1198671)ndash(1198674) we have
det119863119911119865 (119911lowast 1205911 119901)
= det
[[[[[[[[
[
minus11988611119909lowast
1+
11988612119909lowast
1119909lowast
2
(1198981+ 119909lowast
1)2
11988612119909lowast
1
1198981+ 119909lowast
1
0
119886211198981119909lowast
2
(1198981+ 119909lowast
1)2
minus11988622119909lowast
2+
11988623119909lowast
2119909lowast
3
(1198982+ 119909lowast
2)2
minus
11988623119909lowast
2
1198982+ 119909lowast
2
0
119886321198982119909lowast
3
(1198982+ 119909lowast
2)2
minus11988633119909lowast
3
]]]]]]]]
]
= minus
1198862
21199101lowast1199102lowast
(1 + 11988721199101lowast)3[minus119909lowast+
11988611198871119909lowast1199101lowast
(1 + 1198871119909lowast)2] = 0
(86)
From (86) we know that the hypothesis (A2) in [24]is satisfied
(3) The characteristic matrix of (71) at a stationarysolution (119911 120591
0 1199010) where 119911 = (119911
(1) 119911(2) 119911(3)) isin 119877
3
takes the following form
Δ (119911 1205911 119901) (120582) = 120582119868119889 minus 119863
120601119865 (119911 120591
1 119901) (119890
120582119868) (87)
that is
Δ (119911 1205911 119901) (120582)
=
[[[[[[[[[[[[[[[[[
[
120582 minus 1198861+ 211988611119911(1)
+
119886121198981119911(2)
(1198981+ 119911(1))
2
11988612119911(1)
1198981+ 119911(1)
0
minus
119886211198981119911(2)
(1198981+ 119911(1))
2119890minus1205821205911
120582 + 1198862minus
11988621119911(1)
1198981+ 119911(1)
+ 211988622119911(2)
+
119886231198982119911(3)
(1198982+ 119911(2))
2
11988623119911(2)
1198982+ 119911(2)
0 minus
119886321198982119911(3)
(1198982+ 119911(2))
2119890minus120582120591lowast
2120582 + 1198863minus
11988632119911(2)
1198982+ 119911(2)
+ 211988633119911(3)
]]]]]]]]]]]]]]]]]
]
(88)
Abstract and Applied Analysis 15
From (88) we have
det (Δ (119864lowast 1205911 119901) (120582))
= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902
0+ 1199030)] 119890minus1205821205911
(89)
Note that (89) is the same as (20) from the discussion inSection 2 about the local Hopf bifurcation it is easy to verifythat (119864
lowast 120591(119895)
1119896 2120587120596
119896) is an isolated center and there exist 120598 gt
0 120575 gt 0 and a smooth curve 120582 (120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575) rarr C
such that det(Δ(120582(1205911))) = 0 |120582(120591
1) minus 120596119896| lt 120598 for all 120591
1isin
[120591(119895)
1119896minus 120575 120591(119895)
1119896+ 120575] and
120582 (120591(119895)
1119896) = 120596119896119894
119889Re 120582 (1205911)
1198891205911
1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)
1119896
gt 0 (90)
Let
Ω1205982120587120596
119896
= (120578 119901) 0 lt 120578 lt 120598
10038161003816100381610038161003816100381610038161003816
119901 minus
2120587
120596119896
10038161003816100381610038161003816100381610038161003816
lt 120598 (91)
It is easy to see that on [120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575] times 120597Ω
1205982120587120596119896
det(Δ(119864
lowast 1205911 119901)(120578 + (2120587119901)119894)) = 0 if and only if 120578 = 0
1205911= 120591(119895)
1119896 119901 = 2120587120596
119896 119896 = 1 2 3 119895 = 0 1 2
Therefore the hypothesis (A4) in [24] is satisfiedIf we define
119867plusmn(119864lowast 120591(119895)
1119896
2120587
120596119896
) (120578 119901)
= det(Δ (119864lowast 120591(119895)
1119896plusmn 120575 119901) (120578 +
2120587
119901
119894))
(92)
then we have the crossing number of isolated center(119864lowast 120591(119895)
1119896 2120587120596
119896) as follows
120574(119864lowast 120591(119895)
1119896
2120587
120596119896
) = deg119861(119867minus(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
minus deg119861(119867+(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
= minus1
(93)
Thus we have
sum
(1199111205911119901)isinC
(119864lowast120591(119895)
11198962120587120596119896)
120574 (119911 1205911 119901) lt 0
(94)
where (119911 1205911 119901) has all or parts of the form
(119864lowast 120591(119896)
1119895 2120587120596
119896) (119895 = 0 1 ) It follows from
Lemma 9 that the connected component ℓ(119864lowast120591(119895)
11198962120587120596
119896)
through (119864lowast 120591(119895)
1119896 2120587120596
119896) is unbounded for each center
(119911lowast 1205911 119901) (119895 = 0 1 ) From the discussion in Section 2
we have
120591(119895)
1119896=
1
120596119896
times arccos(((11990121205962
119896minus1199010) (1199030+1199020)
+ 1205962(1205962minus1199011) (1199031+1199021))
times ((1199030+1199020)2
+ (1199031+1199021)2
1205962
119896)
minus1
) +2119895120587
(95)
where 119896 = 1 2 3 119895 = 0 1 Thus one can get 2120587120596119896le 120591(119895)
1119896
for 119895 ge 1Now we prove that the projection of ℓ
(119864lowast120591(119895)
11198962120587120596
119896)onto
1205911-space is [120591
1 +infin) where 120591
1le 120591(119895)
1119896 Clearly it follows
from the proof of Lemma 11 that system (3) with 1205911
= 0
has no nontrivial periodic solution Hence the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is away from zero
For a contradiction we suppose that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is bounded this means that the
projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is included in a
interval (0 120591lowast) Noticing 2120587120596119896lt 120591119895
1119896and applying Lemma 11
we have 119901 lt 120591lowast for (119911(119905) 120591
1 119901) belonging to ℓ
(119864lowast120591(119895)
11198962120587120596
119896)
This implies that the projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 119901-
space is bounded Then applying Lemma 10 we get thatthe connected component ℓ
(119864lowast120591(119895)
11198962120587120596
119896)is bounded This
contradiction completes the proof
6 Conclusion
In this paper we take our attention to the stability and Hopfbifurcation analysis of a predator-prey systemwithMichaelis-Menten type functional response and two unequal delaysWe obtained some conditions for local stability and Hopfbifurcation occurring When 120591
1= 1205912 we derived the explicit
formulas to determine the properties of periodic solutionsby the normal form method and center manifold theoremSpecially the global existence results of periodic solutionsbifurcating from Hopf bifurcations are also established byusing a global Hopf bifurcation result due to Wu [24]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to thank the anonymous referees fortheir careful reading and constructive suggestions which leadto truly significant improvement of the paper This researchis supported by the National Natural Science Foundation ofChina (no 11061016)
16 Abstract and Applied Analysis
References
[1] R Xu and M A J Chaplain ldquoPersistence and global stabilityin a delayed predator-prey system with Michaelis-Menten typefunctional responserdquo Applied Mathematics and Computationvol 130 no 2-3 pp 441ndash455 2002
[2] Y L Song Y H Peng and J JWei ldquoBifurcations for a predator-prey system with two delaysrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 466ndash479 2008
[3] G-P Hu W-T Li and X-P Yan ldquoHopf bifurcations in apredator-prey system with multiple delaysrdquo Chaos Solitons ampFractals vol 42 no 2 pp 1273ndash1285 2009
[4] M Liao X Tang and C Xu ldquoBifurcation analysis for a three-species predator-prey systemwith two delaysrdquoCommunicationsin Nonlinear Science and Numerical Simulation vol 17 no 1 pp183ndash194 2012
[5] C Xu and P Li ldquoDynamical analysis in a delayed predator-prey model with two delaysrdquo Discrete Dynamics in Nature andSociety vol 2012 Article ID 652947 22 pages 2012
[6] J-F Zhang ldquoStability and bifurcation periodic solutions ina Lotka-Volterra competition system with multiple delaysrdquoNonlinear Dynamics vol 70 no 1 pp 849ndash860 2012
[7] J Xia Z Yu and R Yuan ldquoStability and Hopf bifurcation ina symmetric Lotka-Volterra predator-prey system with delaysrdquoElectronic Journal of Differential Equations vol 2013 no 9 pp1ndash16 2013
[8] H Wan and J Cui ldquoA malaria model with two delaysrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 601265 8pages 2013
[9] M Zhao ldquoHopf bifurcation analysis for a semiratio-dependentpredator-prey system with two delaysrdquo Abstract and AppliedAnalysis vol 2013 Article ID 495072 13 pages 2013
[10] S Feyissa and S Banerjee ldquoDelay-induced oscillatory dynamicsin humoral mediated immune response with two time delaysrdquoNonlinear Analysis Real World Applications vol 14 no 1 pp35ndash52 2013
[11] G Zhang Y Shen andB Chen ldquoHopf bifurcation of a predator-prey systemwith predator harvesting and twodelaysrdquoNonlinearDynamics vol 73 no 4 pp 2119ndash2131 2013
[12] X-Y Meng H-F Huo and H Xiang ldquoHopf bifurcation in athree-species systemwith delaysrdquo Journal of AppliedMathemat-ics and Computing vol 35 no 1-2 pp 635ndash661 2011
[13] J-F Zhang ldquoGlobal existence of bifurcated periodic solutions ina commensalism model with delaysrdquo Applied Mathematics andComputation vol 218 no 23 pp 11688ndash11699 2012
[14] Y Wang W Jiang and H Wang ldquoStability and global Hopfbifurcation in toxic phytoplankton-zooplankton model withdelay and selective harvestingrdquoNonlinear Dynamics vol 73 no1-2 pp 881ndash896 2013
[15] Y Ma ldquoGlobal Hopf bifurcation in the Leslie-Gower predator-prey model with two delaysrdquo Nonlinear Analysis Real WorldApplications vol 13 no 1 pp 370ndash375 2012
[16] X-Y Meng H-F Huo and X-B Zhang ldquoStability and globalHopf bifurcation in a delayed food web consisting of a preyand two predatorsrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4335ndash4348 2011
[17] Y Song J Wei andM Han ldquoLocal and global Hopf bifurcationin a delayed hematopoiesis modelrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol14 no 11 pp 3909ndash3919 2004
[18] Y Song and J Wei ldquoLocal Hopf bifurcation and global periodicsolutions in a delayed predator-prey systemrdquo Journal of Mathe-matical Analysis and Applications vol 301 no 1 pp 1ndash21 2005
[19] J Wei and M Y Li ldquoGlobal existence of periodic solutions in atri-neuron network model with delaysrdquo Physica D vol 198 no1-2 pp 106ndash119 2004
[20] T Zhao Y Kuang and H L Smith ldquoGlobal existence of peri-odic solutions in a class of delayed Gause-type predator-preysystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 28 no 8 pp 1373ndash1394 1997
[21] S J Gao L S Chen and Z D Teng ldquoHopf bifurcation andglobal stability for a delayed predator-prey system with stagestructure for predatorrdquo Applied Mathematics and Computationvol 202 no 2 pp 721ndash729 2008
[22] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977
[23] B Hassard D Kazarinoff and Y WanTheory and Applicationsof Hopf Bifurcation Cambridge University Press CambridgeUK 1981
[24] J H Wu ldquoSymmetric functional-differential equations andneural networks with memoryrdquo Transactions of the AmericanMathematical Society vol 350 no 12 pp 4799ndash4838 1998
[25] V Hutson ldquoThe existence of an equilibrium for permanentsystemsrdquo The Rocky Mountain Journal of Mathematics vol 20no 4 pp 1033ndash1040 1990
[26] S G Ruan and J J Wei ldquoOn the zeros of transcendentalfunctions with applications to stability of delay differentialequations with two delaysrdquo Dynamics of Continuous Discreteamp Impulsive Systems A vol 10 no 6 pp 863ndash874 2003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Abstract and Applied Analysis 15
From (88) we have
det (Δ (119864lowast 1205911 119901) (120582))
= 1205823+ 11990121205822+ 1199011120582 + 1199010+ [(1199031+ 1199021) 120582 + (119902
0+ 1199030)] 119890minus1205821205911
(89)
Note that (89) is the same as (20) from the discussion inSection 2 about the local Hopf bifurcation it is easy to verifythat (119864
lowast 120591(119895)
1119896 2120587120596
119896) is an isolated center and there exist 120598 gt
0 120575 gt 0 and a smooth curve 120582 (120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575) rarr C
such that det(Δ(120582(1205911))) = 0 |120582(120591
1) minus 120596119896| lt 120598 for all 120591
1isin
[120591(119895)
1119896minus 120575 120591(119895)
1119896+ 120575] and
120582 (120591(119895)
1119896) = 120596119896119894
119889Re 120582 (1205911)
1198891205911
1003816100381610038161003816100381610038161003816100381610038161205911=120591(119895)
1119896
gt 0 (90)
Let
Ω1205982120587120596
119896
= (120578 119901) 0 lt 120578 lt 120598
10038161003816100381610038161003816100381610038161003816
119901 minus
2120587
120596119896
10038161003816100381610038161003816100381610038161003816
lt 120598 (91)
It is easy to see that on [120591(119895)
1119896minus 120575 120591
(119895)
1119896+ 120575] times 120597Ω
1205982120587120596119896
det(Δ(119864
lowast 1205911 119901)(120578 + (2120587119901)119894)) = 0 if and only if 120578 = 0
1205911= 120591(119895)
1119896 119901 = 2120587120596
119896 119896 = 1 2 3 119895 = 0 1 2
Therefore the hypothesis (A4) in [24] is satisfiedIf we define
119867plusmn(119864lowast 120591(119895)
1119896
2120587
120596119896
) (120578 119901)
= det(Δ (119864lowast 120591(119895)
1119896plusmn 120575 119901) (120578 +
2120587
119901
119894))
(92)
then we have the crossing number of isolated center(119864lowast 120591(119895)
1119896 2120587120596
119896) as follows
120574(119864lowast 120591(119895)
1119896
2120587
120596119896
) = deg119861(119867minus(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
minus deg119861(119867+(119864lowast 120591(119895)
1119896
2120587
120596119896
) Ω1205982120587120596
119896
)
= minus1
(93)
Thus we have
sum
(1199111205911119901)isinC
(119864lowast120591(119895)
11198962120587120596119896)
120574 (119911 1205911 119901) lt 0
(94)
where (119911 1205911 119901) has all or parts of the form
(119864lowast 120591(119896)
1119895 2120587120596
119896) (119895 = 0 1 ) It follows from
Lemma 9 that the connected component ℓ(119864lowast120591(119895)
11198962120587120596
119896)
through (119864lowast 120591(119895)
1119896 2120587120596
119896) is unbounded for each center
(119911lowast 1205911 119901) (119895 = 0 1 ) From the discussion in Section 2
we have
120591(119895)
1119896=
1
120596119896
times arccos(((11990121205962
119896minus1199010) (1199030+1199020)
+ 1205962(1205962minus1199011) (1199031+1199021))
times ((1199030+1199020)2
+ (1199031+1199021)2
1205962
119896)
minus1
) +2119895120587
(95)
where 119896 = 1 2 3 119895 = 0 1 Thus one can get 2120587120596119896le 120591(119895)
1119896
for 119895 ge 1Now we prove that the projection of ℓ
(119864lowast120591(119895)
11198962120587120596
119896)onto
1205911-space is [120591
1 +infin) where 120591
1le 120591(119895)
1119896 Clearly it follows
from the proof of Lemma 11 that system (3) with 1205911
= 0
has no nontrivial periodic solution Hence the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is away from zero
For a contradiction we suppose that the projection ofℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is bounded this means that the
projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 120591
1-space is included in a
interval (0 120591lowast) Noticing 2120587120596119896lt 120591119895
1119896and applying Lemma 11
we have 119901 lt 120591lowast for (119911(119905) 120591
1 119901) belonging to ℓ
(119864lowast120591(119895)
11198962120587120596
119896)
This implies that the projection of ℓ(119864lowast120591(119895)
11198962120587120596
119896)onto 119901-
space is bounded Then applying Lemma 10 we get thatthe connected component ℓ
(119864lowast120591(119895)
11198962120587120596
119896)is bounded This
contradiction completes the proof
6 Conclusion
In this paper we take our attention to the stability and Hopfbifurcation analysis of a predator-prey systemwithMichaelis-Menten type functional response and two unequal delaysWe obtained some conditions for local stability and Hopfbifurcation occurring When 120591
1= 1205912 we derived the explicit
formulas to determine the properties of periodic solutionsby the normal form method and center manifold theoremSpecially the global existence results of periodic solutionsbifurcating from Hopf bifurcations are also established byusing a global Hopf bifurcation result due to Wu [24]
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to thank the anonymous referees fortheir careful reading and constructive suggestions which leadto truly significant improvement of the paper This researchis supported by the National Natural Science Foundation ofChina (no 11061016)
16 Abstract and Applied Analysis
References
[1] R Xu and M A J Chaplain ldquoPersistence and global stabilityin a delayed predator-prey system with Michaelis-Menten typefunctional responserdquo Applied Mathematics and Computationvol 130 no 2-3 pp 441ndash455 2002
[2] Y L Song Y H Peng and J JWei ldquoBifurcations for a predator-prey system with two delaysrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 466ndash479 2008
[3] G-P Hu W-T Li and X-P Yan ldquoHopf bifurcations in apredator-prey system with multiple delaysrdquo Chaos Solitons ampFractals vol 42 no 2 pp 1273ndash1285 2009
[4] M Liao X Tang and C Xu ldquoBifurcation analysis for a three-species predator-prey systemwith two delaysrdquoCommunicationsin Nonlinear Science and Numerical Simulation vol 17 no 1 pp183ndash194 2012
[5] C Xu and P Li ldquoDynamical analysis in a delayed predator-prey model with two delaysrdquo Discrete Dynamics in Nature andSociety vol 2012 Article ID 652947 22 pages 2012
[6] J-F Zhang ldquoStability and bifurcation periodic solutions ina Lotka-Volterra competition system with multiple delaysrdquoNonlinear Dynamics vol 70 no 1 pp 849ndash860 2012
[7] J Xia Z Yu and R Yuan ldquoStability and Hopf bifurcation ina symmetric Lotka-Volterra predator-prey system with delaysrdquoElectronic Journal of Differential Equations vol 2013 no 9 pp1ndash16 2013
[8] H Wan and J Cui ldquoA malaria model with two delaysrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 601265 8pages 2013
[9] M Zhao ldquoHopf bifurcation analysis for a semiratio-dependentpredator-prey system with two delaysrdquo Abstract and AppliedAnalysis vol 2013 Article ID 495072 13 pages 2013
[10] S Feyissa and S Banerjee ldquoDelay-induced oscillatory dynamicsin humoral mediated immune response with two time delaysrdquoNonlinear Analysis Real World Applications vol 14 no 1 pp35ndash52 2013
[11] G Zhang Y Shen andB Chen ldquoHopf bifurcation of a predator-prey systemwith predator harvesting and twodelaysrdquoNonlinearDynamics vol 73 no 4 pp 2119ndash2131 2013
[12] X-Y Meng H-F Huo and H Xiang ldquoHopf bifurcation in athree-species systemwith delaysrdquo Journal of AppliedMathemat-ics and Computing vol 35 no 1-2 pp 635ndash661 2011
[13] J-F Zhang ldquoGlobal existence of bifurcated periodic solutions ina commensalism model with delaysrdquo Applied Mathematics andComputation vol 218 no 23 pp 11688ndash11699 2012
[14] Y Wang W Jiang and H Wang ldquoStability and global Hopfbifurcation in toxic phytoplankton-zooplankton model withdelay and selective harvestingrdquoNonlinear Dynamics vol 73 no1-2 pp 881ndash896 2013
[15] Y Ma ldquoGlobal Hopf bifurcation in the Leslie-Gower predator-prey model with two delaysrdquo Nonlinear Analysis Real WorldApplications vol 13 no 1 pp 370ndash375 2012
[16] X-Y Meng H-F Huo and X-B Zhang ldquoStability and globalHopf bifurcation in a delayed food web consisting of a preyand two predatorsrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4335ndash4348 2011
[17] Y Song J Wei andM Han ldquoLocal and global Hopf bifurcationin a delayed hematopoiesis modelrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol14 no 11 pp 3909ndash3919 2004
[18] Y Song and J Wei ldquoLocal Hopf bifurcation and global periodicsolutions in a delayed predator-prey systemrdquo Journal of Mathe-matical Analysis and Applications vol 301 no 1 pp 1ndash21 2005
[19] J Wei and M Y Li ldquoGlobal existence of periodic solutions in atri-neuron network model with delaysrdquo Physica D vol 198 no1-2 pp 106ndash119 2004
[20] T Zhao Y Kuang and H L Smith ldquoGlobal existence of peri-odic solutions in a class of delayed Gause-type predator-preysystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 28 no 8 pp 1373ndash1394 1997
[21] S J Gao L S Chen and Z D Teng ldquoHopf bifurcation andglobal stability for a delayed predator-prey system with stagestructure for predatorrdquo Applied Mathematics and Computationvol 202 no 2 pp 721ndash729 2008
[22] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977
[23] B Hassard D Kazarinoff and Y WanTheory and Applicationsof Hopf Bifurcation Cambridge University Press CambridgeUK 1981
[24] J H Wu ldquoSymmetric functional-differential equations andneural networks with memoryrdquo Transactions of the AmericanMathematical Society vol 350 no 12 pp 4799ndash4838 1998
[25] V Hutson ldquoThe existence of an equilibrium for permanentsystemsrdquo The Rocky Mountain Journal of Mathematics vol 20no 4 pp 1033ndash1040 1990
[26] S G Ruan and J J Wei ldquoOn the zeros of transcendentalfunctions with applications to stability of delay differentialequations with two delaysrdquo Dynamics of Continuous Discreteamp Impulsive Systems A vol 10 no 6 pp 863ndash874 2003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
16 Abstract and Applied Analysis
References
[1] R Xu and M A J Chaplain ldquoPersistence and global stabilityin a delayed predator-prey system with Michaelis-Menten typefunctional responserdquo Applied Mathematics and Computationvol 130 no 2-3 pp 441ndash455 2002
[2] Y L Song Y H Peng and J JWei ldquoBifurcations for a predator-prey system with two delaysrdquo Journal of Mathematical Analysisand Applications vol 337 no 1 pp 466ndash479 2008
[3] G-P Hu W-T Li and X-P Yan ldquoHopf bifurcations in apredator-prey system with multiple delaysrdquo Chaos Solitons ampFractals vol 42 no 2 pp 1273ndash1285 2009
[4] M Liao X Tang and C Xu ldquoBifurcation analysis for a three-species predator-prey systemwith two delaysrdquoCommunicationsin Nonlinear Science and Numerical Simulation vol 17 no 1 pp183ndash194 2012
[5] C Xu and P Li ldquoDynamical analysis in a delayed predator-prey model with two delaysrdquo Discrete Dynamics in Nature andSociety vol 2012 Article ID 652947 22 pages 2012
[6] J-F Zhang ldquoStability and bifurcation periodic solutions ina Lotka-Volterra competition system with multiple delaysrdquoNonlinear Dynamics vol 70 no 1 pp 849ndash860 2012
[7] J Xia Z Yu and R Yuan ldquoStability and Hopf bifurcation ina symmetric Lotka-Volterra predator-prey system with delaysrdquoElectronic Journal of Differential Equations vol 2013 no 9 pp1ndash16 2013
[8] H Wan and J Cui ldquoA malaria model with two delaysrdquo DiscreteDynamics in Nature and Society vol 2013 Article ID 601265 8pages 2013
[9] M Zhao ldquoHopf bifurcation analysis for a semiratio-dependentpredator-prey system with two delaysrdquo Abstract and AppliedAnalysis vol 2013 Article ID 495072 13 pages 2013
[10] S Feyissa and S Banerjee ldquoDelay-induced oscillatory dynamicsin humoral mediated immune response with two time delaysrdquoNonlinear Analysis Real World Applications vol 14 no 1 pp35ndash52 2013
[11] G Zhang Y Shen andB Chen ldquoHopf bifurcation of a predator-prey systemwith predator harvesting and twodelaysrdquoNonlinearDynamics vol 73 no 4 pp 2119ndash2131 2013
[12] X-Y Meng H-F Huo and H Xiang ldquoHopf bifurcation in athree-species systemwith delaysrdquo Journal of AppliedMathemat-ics and Computing vol 35 no 1-2 pp 635ndash661 2011
[13] J-F Zhang ldquoGlobal existence of bifurcated periodic solutions ina commensalism model with delaysrdquo Applied Mathematics andComputation vol 218 no 23 pp 11688ndash11699 2012
[14] Y Wang W Jiang and H Wang ldquoStability and global Hopfbifurcation in toxic phytoplankton-zooplankton model withdelay and selective harvestingrdquoNonlinear Dynamics vol 73 no1-2 pp 881ndash896 2013
[15] Y Ma ldquoGlobal Hopf bifurcation in the Leslie-Gower predator-prey model with two delaysrdquo Nonlinear Analysis Real WorldApplications vol 13 no 1 pp 370ndash375 2012
[16] X-Y Meng H-F Huo and X-B Zhang ldquoStability and globalHopf bifurcation in a delayed food web consisting of a preyand two predatorsrdquo Communications in Nonlinear Science andNumerical Simulation vol 16 no 11 pp 4335ndash4348 2011
[17] Y Song J Wei andM Han ldquoLocal and global Hopf bifurcationin a delayed hematopoiesis modelrdquo International Journal ofBifurcation and Chaos in Applied Sciences and Engineering vol14 no 11 pp 3909ndash3919 2004
[18] Y Song and J Wei ldquoLocal Hopf bifurcation and global periodicsolutions in a delayed predator-prey systemrdquo Journal of Mathe-matical Analysis and Applications vol 301 no 1 pp 1ndash21 2005
[19] J Wei and M Y Li ldquoGlobal existence of periodic solutions in atri-neuron network model with delaysrdquo Physica D vol 198 no1-2 pp 106ndash119 2004
[20] T Zhao Y Kuang and H L Smith ldquoGlobal existence of peri-odic solutions in a class of delayed Gause-type predator-preysystemsrdquo Nonlinear Analysis Theory Methods amp Applicationsvol 28 no 8 pp 1373ndash1394 1997
[21] S J Gao L S Chen and Z D Teng ldquoHopf bifurcation andglobal stability for a delayed predator-prey system with stagestructure for predatorrdquo Applied Mathematics and Computationvol 202 no 2 pp 721ndash729 2008
[22] J Hale Theory of Functional Differential Equations SpringerNew York NY USA 1977
[23] B Hassard D Kazarinoff and Y WanTheory and Applicationsof Hopf Bifurcation Cambridge University Press CambridgeUK 1981
[24] J H Wu ldquoSymmetric functional-differential equations andneural networks with memoryrdquo Transactions of the AmericanMathematical Society vol 350 no 12 pp 4799ndash4838 1998
[25] V Hutson ldquoThe existence of an equilibrium for permanentsystemsrdquo The Rocky Mountain Journal of Mathematics vol 20no 4 pp 1033ndash1040 1990
[26] S G Ruan and J J Wei ldquoOn the zeros of transcendentalfunctions with applications to stability of delay differentialequations with two delaysrdquo Dynamics of Continuous Discreteamp Impulsive Systems A vol 10 no 6 pp 863ndash874 2003
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of