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Research Article Stability in a Simple Food Chain System ...Stability in a Simple Food Chain System...
15
Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2013, Article ID 936952, 14 pages http://dx.doi.org/10.1155/2013/936952 Research Article Stability in a Simple Food Chain System with Michaelis-Menten Functional Response and Nonlocal Delays Wenzhen Gan, 1 Canrong Tian, 2 Qunying Zhang, 3 and Zhigui Lin 3 1 School of Mathematics and Physics, Jiangsu Teachers University of Technology, Changzhou 213001, China 2 Basic Department, Yancheng Institute of Technology, Yangcheng 224003, China 3 School of Mathematical Science, Yangzhou University, Yangzhou 225002, China Correspondence should be addressed to Zhigui Lin; [email protected] Received 26 April 2013; Accepted 8 July 2013 Academic Editor: Rodrigo Lopez Pouso Copyright © 2013 Wenzhen Gan et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper is concerned with the asymptotical behavior of solutions to the reaction-diffusion system under homogeneous Neumann boundary condition. By taking food ingestion and species’ moving into account, the model is further coupled with Michaelis- Menten type functional response and nonlocal delay. Sufficient conditions are derived for the global stability of the positive steady state and the semitrivial steady state of the proposed problem by using the Lyapunov functional. Our results show that intraspecific competition benefits the coexistence of prey and predator. Furthermore, the introduction of Michaelis-Menten type functional response positively affects the coexistence of prey and predator, and the nonlocal delay is harmless for stabilities of all nonnegative steady states of the system. Numerical simulations are carried out to illustrate the main results. 1. Introduction e overall behavior of ecological systems continues to be of great interest to both applied mathematicians and ecol- ogists. Two species predator-prey models have been exten- sively investigated in the literature. But recently more and more attention has been focused on systems with three or more trophic levels. For example, the predator-prey system for three species with Michaelis-Menten type functional response was studied by many authors [1–4]. However, the systems in [1–4] are either with discrete delay or without delay or without diffusion. In view of individuals taking time to move, spatial dispersal was dealt with by introducing diffusion term to corresponding delayed ODE model in previous literatures, namely, adding a Laplacian term to the ODE model. In recent years, it has been recognized that there are modelling difficulties with this approach. e difficulty is that diffusion and time delay are independent of each other, since individuals have not been at the same point at previous times. Britton [5] made a first comprehensive attempt to address this difficulty by introducing a nonlocal delay; that is, the delay term involves a weighted-temporal average over the whole of the infinite domain and the whole of the previous times. ere are many results for reaction-diffusion equations with nonlocal delays [5–18]. e existence and stability of traveling wave fronts were studied in reaction-diffusion equa- tions with nonlocal delay [5–9]. e stability of impulsive cellular neural networks with time varying was discussed in [10] by means of new Poincare integral inequality. e asymptotic behavior of solutions of the reaction-diffusion equations with nonlocal delay was investigated in [11, 12] by using an iterative technique and in [13–15] by the Lyapunov functional. e stability and Hopf bifurcation were discussed in [16] for a diffusive logistic population model with nonlocal delay effect. Motivated by the work above, we are concerned with the following food chain model with Michaelis-Menten type functional response: 1 − 1 Δ 1 = 1 ( 1 − 11 1 − 12 2 1 + 1 ),
Transcript of Research Article Stability in a Simple Food Chain System ...Stability in a Simple Food Chain System...
Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2013 Article ID 936952 14 pageshttpdxdoiorg1011552013936952
Research ArticleStability in a Simple Food Chain System with Michaelis-MentenFunctional Response and Nonlocal Delays
Wenzhen Gan1 Canrong Tian2 Qunying Zhang3 and Zhigui Lin3
1 School of Mathematics and Physics Jiangsu Teachers University of Technology Changzhou 213001 China2 Basic Department Yancheng Institute of Technology Yangcheng 224003 China3 School of Mathematical Science Yangzhou University Yangzhou 225002 China
Correspondence should be addressed to Zhigui Lin zglin68hotmailcom
Received 26 April 2013 Accepted 8 July 2013
Academic Editor Rodrigo Lopez Pouso
Copyright copy 2013 Wenzhen Gan 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
This paper is concernedwith the asymptotical behavior of solutions to the reaction-diffusion systemunder homogeneousNeumannboundary condition By taking food ingestion and speciesrsquo moving into account the model is further coupled with Michaelis-Menten type functional response and nonlocal delay Sufficient conditions are derived for the global stability of the positive steadystate and the semitrivial steady state of the proposed problem by using the Lyapunov functional Our results show that intraspecificcompetition benefits the coexistence of prey and predator Furthermore the introduction of Michaelis-Menten type functionalresponse positively affects the coexistence of prey and predator and the nonlocal delay is harmless for stabilities of all nonnegativesteady states of the system Numerical simulations are carried out to illustrate the main results
1 Introduction
The overall behavior of ecological systems continues to beof great interest to both applied mathematicians and ecol-ogists Two species predator-prey models have been exten-sively investigated in the literature But recently more andmore attention has been focused on systems with three ormore trophic levels For example the predator-prey systemfor three species with Michaelis-Menten type functionalresponse was studied by many authors [1ndash4] However thesystems in [1ndash4] are either with discrete delay or withoutdelay or without diffusion In view of individuals takingtime to move spatial dispersal was dealt with by introducingdiffusion term to corresponding delayed ODE model inprevious literatures namely adding a Laplacian term to theODEmodel In recent years it has been recognized that thereare modelling difficulties with this approach The difficulty isthat diffusion and time delay are independent of each othersince individuals have not been at the same point at previoustimes Britton [5] made a first comprehensive attempt toaddress this difficulty by introducing a nonlocal delay that
is the delay term involves a weighted-temporal average overthewhole of the infinite domain and thewhole of the previoustimes
There are many results for reaction-diffusion equationswith nonlocal delays [5ndash18] The existence and stability oftravelingwave fronts were studied in reaction-diffusion equa-tions with nonlocal delay [5ndash9] The stability of impulsivecellular neural networks with time varying was discussedin [10] by means of new Poincare integral inequality Theasymptotic behavior of solutions of the reaction-diffusionequations with nonlocal delay was investigated in [11 12] byusing an iterative technique and in [13ndash15] by the LyapunovfunctionalThe stability and Hopf bifurcation were discussedin [16] for a diffusive logistic populationmodel with nonlocaldelay effect
Motivated by the work above we are concerned withthe following food chain model with Michaelis-Menten typefunctional response
1205971199061
120597119905minus 1198891Δ1199061= 1199061(1198861minus 119886111199061minus119886121199062
1198981+ 1199061
)
2 Abstract and Applied Analysis
1205971199062
120597119905minus 1198892Δ1199062
= 1199062(minus1198862+ 11988621intΩ
int
119905
minusinfin
1199061(119904 119910)
1198981+ 1199061(119904 119910)
times 1198701(119909 119910 119905 minus 119904) 119889119904 119889119910
minus119886221199062minus119886231199063
1198982+ 1199062
)
1205971199063
120597119905minus 1198893Δ1199063
= 1199063(minus1198863+ 11988632intΩ
int
119905
minusinfin
1199062(119904 119910)
1198982+ 1199062(119904 119910)
times 1198702(119909 119910 119905 minus 119904) 119889119904 119889119910
minus119886331199063)
(1)
for 119905 gt 0 119909 isin Ω with homogeneous Neumann boundaryconditions
1205971199061
120597]=1205971199062
120597]=1205971199063
120597]= 0 119905 gt 0 119909 isin 120597Ω (2)
and initial conditions
119906119894(119905 119909) = 120601
119894(119905 119909) ge 0 (119894 = 1 2) (119905 119909) isin (minusinfin 0] times Ω
1199063(0 119909) = 120601
3(119909) ge 0 119909 isin Ω
(3)
where 120601119894is bounded Holder continuous function and satis-
fies 120597120601119894120597] = 0 (119894 = 1 2 3) on (minusinfin 0] times 120597Ω Here Ω is a
bounded domain in R119899 with smooth boundary 120597Ω and 120597120597]is the outward normal derivative on 120597Ω 119906
119894(119905 119909) represents
the density of the 119894th species (prey predator and top predatorresp) at time 119905 and location 119909 and thus only nonnegative119906119894(119905 119909) is of interest The parameter 119886
1is the intrinsic growth
rate of the prey and 1198862and 119886
3are the death rates of the
predator and top-predator 119886119894119894is the intraspecific competitive
rate of the 119894th species 11988612is the maximum predation rate 119886
23
and 11988632are the efficiencies of food utilization of the predator
and top predator respectively We assume the predator andtop predator show the Michaelis-Menten (or Holling typeII) functional response with 119906
1(1198981+ 1199061) and 119906
2(1198982+ 1199062)
respectively where 1198981and 119898
2are half-saturation constants
For a through biological background of similar models see[18 19] As our most knowledge the tritrophic food chainmodel has been found to have many interesting biologicalproperties such as the coexistence and the Hopf bifurcationHowever the effect of nonlocal time delays on the coexistencehas not been reported Our paper mainly concerns thisperspective
Additionally intΩint119905
minusinfin(119906119894(119904 119910)(119898
119894+ 119906119894(119904 119910)))119870
119894(119909 119910 119905 minus
119904)119889119904 119889119910 (119894 = 1 2) represents the nonlocal delay due tothe ingestion of predator that is mature adult predator can
only contribute to the production of predator biomass Theboundary condition in (2) implies that there is no migrationacross the boundary ofΩ
The main purpose of this paper is to study the globalasymptotic behavior of the solution of system (1)ndash(3) Thepreliminary results are presented in Section 2 Section 3 con-tains sufficient conditions for the global asymptotic behaviorsof the equilibria of system (1)ndash(3) by means of the Lyapunovfunctional Numerical simulations are carried out to show thefeasibility of the conditions in Theorems 8ndash10 in Section 4Finally a brief discussion is given to conclude this work
2 Preliminary Results
In this section we present several preliminary results that willbe employed in the sequel
Lemma 1 (see [3]) Let 119886 and 119887 be positive constants Assumethat 120601 120593 isin 1198621(119886 +infin) 120593(119905) ge 0 and 120601 is bounded frombelow If 1206011015840(119905) le minus119887120593(119905) and 1205931015840(119905) le 119870 in [119886 +infin) for somepositive constant 119870 then lim
119905rarr+infin120593(119905) = 0
The following lemma is the Positivity Lemma in [20]
Lemma 2 Let 119906119894isin 119862([0 119879] times Ω)⋂119862
12((0 119879] times Ω) (119894 =
1 2 3) and satisfy
119906119894119905minus 119889119894Δ119906119894ge
3
sum
119895=1
119887119894119895(119905 119909) 119906
119895(119905 119909)
+
3
sum
119895=1
119888119894119895119906119895(119905 minus 120591119895 119909) in (0 119879] times Ω
120597119906119894(119905 119909)
120597]ge 0 on (0 119879] times 120597Ω
119906119894(119905 119909) ge 0 in [minus120591
119894 0] times Ω
(4)
where 119887119894119895 119888119894119895isin 119862([0 119879] times Ω) 119906
119894119905= (119906119894)119905 If 119887119894119895ge 0 for 119895 = 119894
and 119888119894119895ge 0 for all 119894 119895 = 1 2 3 Then 119906
119894(119905 119909) ge 0 on [0 119879] times Ω
Moreover if the initial function is nontrivial then 119906119894gt 0 in
(0 119879] times Ω
Lemma 3 (see [20]) Let c and c be a pair of constant vectorsatisfying c ge c and let the reaction functions satisfy localLipschitz condition withΛ = ⟨c c⟩Then system (1)ndash(3) admitsa unique global solution u(119905 119909) such that
c le u (119905 119909) le c forall119905 gt 0 119909 isin Ω (5)
whenever c le 120601(119905 119909) le c (119905 119909) isin (minusinfin 0] times Ω
The following result was obtained by themethod of upperand lower solutions and the associated iterations in [21]
Abstract and Applied Analysis 3
Lemma 4 (see [21]) Let 119906(119905 119909) isin 119862([0infin) times Ω) cap
11986221((0infin)timesΩ) be the nontrivial positive solution of the system
119906119905minus 119889Δ119906 = 119861119906 (119905 minus 120591 119909) plusmn 119860
1119906 (119905 119909)
minus 11986021199062(119905 119909) (119905 119909) isin (0infin) times Ω
120597119906
120597]= 0 (119905 119909) isin (0infin) times 120597Ω
119906 (119905 119909) = 120601 (119905 119909) ge 0 (119905 119909) isin (minus120591 0) times Ω
(6)
where 1198601ge 0 119861 119860
2 120591 gt 0 The following results hold
(1) if 119861plusmn1198601gt 0 then 119906 rarr (119861 plusmn119860
1)1198602uniformly onΩ
as 119905 rarr infin
(2) if 119861plusmn1198601le 0 then 119906 rarr 0 uniformly onΩ as 119905 rarr infin
Throughout this paper we assume that
119870119894(119909 119910 119905) = 119866
119894(119909 119910 119905) 119896
119894(119905) 119909 119910 isin Ω 119896
119894(119905) ge 0
intΩ
119866119894(119909 119910 119905) 119889119909 = int
Ω
119866119894(119909 119910 119905) 119889119910 = 1 119905 ge 0
int
+infin
0
119896119894(119905) 119889119905 = 1 119905119896
119894(119905) isin 119871
1((0 +infin) 119877)
(7)
where119866119894(119909 119910 119905) is a nonnegative function which is continu-
ous in (119909 119910) isin ΩtimesΩ for each 119905 isin [0 +infin) and measurable in119905 isin [0 +infin) for each pair (119909 119910) isin Ω times Ω
Now we prove the following propositions which will beused in the sequel
Proposition 5 For any nonnegative initial function thecorresponding solution of system (1)ndash(3) is nonnegative
Proof Suppose that (1199061 1199062 1199063) is a solution and satisfies 119906
119894isin
119862([0 119879) times Ω) cap 11986212((0 119879) times Ω) with 119879 le +infin Choose 0 lt
120591 lt 119879 Then
1199061119905minus 1198891Δ1199061= 11986011199061 0 lt 119905 lt 120591 119909 isin Ω
1199062119905minus 1198892Δ1199062= 11986021199062 0 lt 119905 lt 120591 119909 isin Ω
1199063119905minus 1198893Δ1199063= 11986031199063 0 lt 119905 lt 120591 119909 isin Ω
120597119906119894
120597]= 0 0 lt 119905 lt 120591 119909 isin 120597Ω
119906119894(119905 119909) ge 0 (119894 = 1 2 3) 119905 le 0 119909 isin Ω
(8)
where
1198601= 1198861minus 119886111199061minus119886121199062
1198981+ 1199061
1198602= minus1198862+ 11988621intΩ
int
119905
minusinfin
1199061(119904 119910)
1198981+ 1199061(119904 119910)
times 1198701(119909 119910 119905 minus 119904) 119889119904 119889119910
minus 119886221199062minus119886231199063
1198982+ 1199063
1198603= minus1198863+ 11988632intΩ
int
119905
minusinfin
1199062(119904 119910)
1198982+ 1199062(119904 119910)
times 1198702(119909 119910 119905 minus 119904) 119889119904 119889119910 minus 119886
331199063
(9)
It follows from Lemma 2 that 119906119894ge 0 (119905 119909) isin [0 120591] times Ω Due
to the arbitrariness of 120591 we have 119906119894ge 0 for (119905 119909) isin [0 119879) times
Ω
Proposition 6 System (1)ndash(3) with initial functions 120601119894admits
a unique global solution (1199061 1199062 1199063) satisfying 0 le 119906
119894(119905 119909) le
119872119894for 119894 = 1 2 3 where119872
119894is defined as
1198721= max 1198861
11988611
10038171003817100381710038171206011 (119905 119909)
1003817100381710038171003817119871infin((minusinfin0]timesΩ)
1198722= max
(11988621minus 1198862)1198721minus 11989811198862
11988622(1198981+1198721)
10038171003817100381710038171206012 (119905 119909)
1003817100381710038171003817119871infin((minusinfin0]timesΩ)
1198723= max
(11988632minus 1198863)1198722minus 11989821198863
11988622(1198982+1198722)
10038171003817100381710038171206013 (119909)
1003817100381710038171003817119871infin(Ω)
(10)
Proof It follows from standard PDE theory that there existsa 119879 gt 0 such that problem (1)ndash(3) admits a unique solutionin [0 119879) times Ω From Lemma 2 we know that 119906
119894(119905 119909) ge 0 for
(119905 119909) isin [0 119879) times Ω Considering the first equation in (1) wehave
1205971199061
120597119905minus 1198891Δ1199061le 1199061(1198861minus 119886111199061) (11)
Using themaximumprinciple gives that 1199061le 1198721 In a similar
way we have 119906119894le 119872119894for 119894 = 2 3 It is easy to see that (0 0 0)
and (119872111987221198723) are a pair of coupled upper and lower
solutions to system (1)ndash(3) from the direct computation Invirtue of Lemma 3 system (1)ndash(3) admits a unique globalsolution (119906
1 1199062 1199063) satisfying 0 le 119906
119894(119905 119909) le 119872
119894for 119894 = 1 2 3
In addition if 1206011(119905 119909) 120601
2(119905 119909) 120601
3(119909) (119905 119909) isin (minusinfin 0] times Ω is
nontrivial it follows from the strongmaximumprinciple that119906119894(119905 119909) gt 0 (119894 = 1 2 3) for all 119905 gt 0 119909 isin Ω
3 Global Stability
In this section we study the asymptotic behavior of theequilibrium of system (1)ndash(3) In the beginning we show theexistence and uniqueness of positive steady state
4 Abstract and Applied Analysis
L1
L2
L3
025
02
015
01
005
0
0 02 04 06 081
2
Figure 1 Graphs of the equations in (12) 1198713is the boundary
condition (1198863= 11988632V2(1198982+ V2)) and 119871
2corresponds to minus119886
2+
11988621(V1(1198981+ V1)) minus 119886
22V2= 0 while 119871
1corresponds to 119886
1minus 11988611V1minus
11988612V2(1198981+ V1) = 0 Intersection point between 119871
1and 119871
2gives the
positive solution (Vlowast1 Vlowast2) to (12) Parameter values are listed in the
example in Section 4
Let us consider the following equations
1198861minus 11988611V1minus11988612V2
1198981+ V1
= 0
minus 1198862+11988621V1
1198981+ V1
minus 11988622V2= 0
(12)
A direct computation shows that the above equations haveonly one positive solution (Vlowast
1 Vlowast2) if and only if
1198671 1198861(11988621minus 1198862) gt 119898
1119886211988611 (13)
is satisfied Taking1198671into account we consider the equation
1198863= 11988632Vlowast2(1198982+ Vlowast2) which corresponds to 119871
3in Figure 1
It is clear that if 1198863lt 11988632Vlowast2(1198982+ Vlowast2) the intersection
point (Vlowast1 Vlowast2) always lies in 119871
3 Let
1198672 11988631198982lt (11988632minus 1198863) Vlowast
2 (14)
Suppose that1198671and119867
2hold We take 119906
3as a parameter and
consider the following system
1198861minus 119886111199061minus119886121199062
1198981+ 1199061
= 0
minus 1198862+119886211199061
1198981+ 1199061
minus 119886221199062minus119886231199063
1198982+ 1199062
= 0
minus 1198863+119886321199062
1198982+ 1199062
minus 119886331199063= 0
(15)
When the parameter 1199063is sufficiently small the first two
equations in (15) can be approximated by (12) Moreover
by continuously increasing the value of 1199063 1198713goes up and
meanwhile the intersection point between 1198711and 119871
2also
goes up However 1198713goes up faster than 119871
2 while 119871
1keeps
still In other words there exists a critical value 1199063such
that the intersection point lies in 1198713 which implies that
there is a unique positive solution 119864lowast(119906lowast1 119906lowast
2 119906lowast
3) to (15) or
equivalently (1)ndash(3)
Lemma 7 Assume that1198672and 119866
2hold and then the positive
steady state119864lowast of system (1)ndash(3) is locally asymptotically stable
Proof We get the local stability of 119864lowast by performing alinearization and analyzing the corresponding characteristicequation Similarly as in [22] let 0 lt 120583
1lt 1205832lt 1205833lt
be the eigenvalues of minusΔ on Ω with the homogeneousNeumann boundary condition Let 119864(120583
119894) be the eigenspace
corresponding to 120583119894in 1198621(Ω) for 119894 = 1 2 3 Let
X = u = (1199061 1199062 1199063) isin [119862
1(Ω)]3
120597120578u = 0 119909 isin 120597Ω (16)
120593119894119895 119895 = 1 dim119864(120583
119894) be an orthonormal basis of 119864(120583
119894)
and X119894119895= 119888 sdot 120593
119894119895| 119888 isin 119877
3 Then
X119894=
dim119864(120583119894)
⨁
119895=1
X119894119895 X =
infin
⨁
119894=1
X119894 (17)
Let 119863 = diag(1198631 1198632 1198633) and k = (V
1 V2 V3) V119894=
119906119894minus 119906lowast
119894 (119894 = 1 2 3) Then the linearization of (1)
is k119905= 119871k = 119863Δk + Fk(Elowast)k Fk(Elowast)k = 119888
119894119895
where 119888119894119894
= 119887119894119894V119894 119894 = 1 2 3 119888
12= 11988712V2 11988823
=
11988723V3 11988821= 11988721intΩint119905
minusinfinV1(119904 119910)119870
1(119909 119910 119905 minus 119904)119889119904 119889119910 and 119888
32=
11988732intΩint119905
minusinfinV2(119904 119910)119870
2(119909 119910 119905 minus 119904)119889119904 119889119910 The coefficients 119887
119894119895are
defined as follows
11988711= minus11988611119906lowast
1+11988612119906lowast
1119906lowast
2
(1198981+ 119906lowast1)2
11988712= minus
11988612119906lowast
1
1198981+ 119906lowast1
11988713= 0
11988721=119886211198981119906lowast
2
(1198981+ 119906lowast1)2
11988722= minus11988622119906lowast
2+11988623119906lowast
2119906lowast
3
(1198982+ 119906lowast2)2 119887
23= minus
11988623119906lowast
2
1198982+ 119906lowast2
11988731= 0 119887
32=119886321198982119906lowast
3
(1198982+ 119906lowast2)2 119887
33= minus11988633119906lowast
3
(18)
SinceX119894is invariant under the operator 119871 for each 119894 ge 1 then
the operator 119871 on X119894is k119905= 119871k = 119863120583
119894k + Fk(Elowast)k Let V119894 =
119888119894119890120582119905(119894 = 1 2 3) and we can get the characteristic equation
Abstract and Applied Analysis 5
120593119894(120582) = 120582
3+ 1198601198941205822+ 119861119894120582 + 119862
119894= 0 where 119860
119894 119861119894 and 119862
119894are
defined as follows
119860119894= (1198891+ 1198892+ 1198893) 120583119894minus 11988711minus 11988722minus 11988733
119861119894= (11988911198892+ 11988921198893+ 11988931198891) 1205832
119894
+ [1198893(minus11988711minus 11988722) + 1198891(minus11988733minus 11988722)
+1198892(minus11988711minus 11988733)] 120583119894
+ 1198871111988722+ 1198871111988733+ 1198872211988733
minus 1198871211988721int
infin
0
1198961(119904) 119890minus120582119904119889119904 minus 119887
2311988732int
infin
0
1198962(119904) 119890minus120582119904119889119904
119862119894= 1198891119889211988931205833
119894+ [minus1198873311988911198892minus 1198871111988921198893minus 1198872211988911198893] 1205832
119894
+ [11988711119887221198893+ 11988711119887331198892+ 11988722119887331198891+ 11988722119887331198891
minus 11988931198871211988721int
infin
0
1198961(119904) 119890minus120582119904119889119904
minus11988911198872311988732int
infin
0
1198962(119904) 119890minus120582119904119889119904] 120583119894
+ [minus119887111198872211988733+ 119887121198872111988733int
infin
0
1198961(119904) 119890minus120582119904119889119904
+119887111198872311988732int
infin
0
1198962(119904) 119890minus120582119904119889119904]
(19)
It is easy to see that 11988712 11988723 11988733lt 0 and 119887
21 11988732gt 0 It
follows from assumption 1198661and 119866
2that 119886
11lt 0 119886
22lt 0
So 119860119894gt 0 119861
119894gt 0 119862
119894gt 0 and 119860
119894119861119894minus 119862119894gt 0 for 119894 ge 1
from the direct calculation According to the Routh-Hurwitzcriterion the three roots 120582
1198941 1205821198942 1205821198943
of 120593119894(120582) = 0 all have
negative real partsBy continuity of the roots with respect to 120583
119894and Routh-
Hurwitz criterion we can conclude that there exists a positiveconstant 120576 such that
Re 1205821198941 Re 120582
1198942 Re 120582
1198943 le minus120576 119894 ge 1 (20)
Consequently the spectrum of 119871 consisting only of eigen-values lies in Re 120582 le minus120576 It is easy to see that Elowast islocally asymptotically stable and follows fromTheorem 511of [23]
Theorem 8 Assume that
11986731198861(11988621minus 1198862)
11989811198862
gt 11988611gt11988612119906lowast
2
11989821
(21)
1198672and 119866
2hold and the positive steady state 119864lowast of system (1)ndash
(3) with nontrivial initial function is globally asymptoticallystable
Proof It is easy to see that the equations in (1) can be rewrittenas
1205971199061
120597119905minus 1198891Δ1199061= 1199061[ minus 11988611(1199061minus 119906lowast
1)
minus11988612(1199062minus 119906lowast
2)
1198981+ 1199061
+11988612119906lowast
2(1199061minus 119906lowast
1)
(1198981+ 1199061) (1198981+ 119906lowast1)]
1205971199062
120597119905minus 1198892Δ1199062= 1199062[intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus 119906
lowast
1)
(1198981+ 1199061(119904 119910)) (119898
1+ 119906lowast1)119889119904 119889119910
minus 11988622(1199062minus 119906lowast
2) minus11988623(1199063minus 119906lowast
3)
1198982+ 1199062
+11988623119906lowast
3(1199062minus 119906lowast
2)
(1198982+ 1199062) (1198982+ 119906lowast2)]
1205971199063
120597119905minus 1198893Δ1199063= 1199063[intΩ
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321198982(1199062(119904 119910) minus 119906
lowast
2)
(1198982+ 1199062(119904 119910)) (119898
2+ 119906lowast2)119889119904 119889119910
minus11988633(1199063minus 119906lowast
3) ]
(22)
Define
119881 (119905) = 120572intΩ
(1199061minus 119906lowast
1minus 119906lowast
1ln 1199061119906lowast1
)119889119909
+ intΩ
(1199062minus 119906lowast
2minus 119906lowast
2ln 1199062119906lowast2
)119889119909
+ 120573intΩ
(1199062minus 119906lowast
3minus 119906lowast
3ln1199063
119906lowast3
)119889119909
(23)
where 120572 and 120573 are positive constants to be determinedCalculating the derivatives 119881(119905) along the positive solutionto the system (1)ndash(3) yields
1198811015840(119905) = Φ
1(119905) + Φ
2(119905) (24)
where
Φ1(119905) = minusint
Ω
(1205721198891119906lowast
1
11990621
1003816100381610038161003816nabla119906110038161003816100381610038162
+1198892119906lowast
2
11990622
1003816100381610038161003816nabla119906210038161003816100381610038162
+1205731198893119906lowast
3
11990623
1003816100381610038161003816nabla119906310038161003816100381610038162
)119889119909
6 Abstract and Applied Analysis
Φ2(119905) = minusint
Ω
120572[11988611minus
11988612119906lowast
2
(1198981+ 1199061) (1198981+ 119906lowast1)] (1199061minus 119906lowast
1)2
119889119909
minus intΩ
[11988622minus
11988623119906lowast
3
(1198982+ 1199062) (1198982+ 119906lowast2)] (1199062minus 119906lowast
2)2
119889119909
minus intΩ
11988633120573(1199063minus 119906lowast
3)2
119889119909
minus intΩ
11988623(1199063minus 119906lowast
3) (1199062minus 119906lowast
2)
1198982+ 1199062
119889119909
minus intΩ
12057211988612(1199062minus 119906lowast
2) (1199061minus 119906lowast
1)
1198981+ 1199061
119889119909
+∬Ω
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus 119906
lowast
1) (1199062(119905 119909) minus 119906
lowast
2)
(1198981+ 1199061(119904 119910)) (119898
1+ 119906lowast1)
119889119904 119889119910 119889119909
+ 120573∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321198982(1199062(119904 119910) minus 119906
lowast
2) (1199063(119905 119909) minus 119906
lowast
3)
(1198982+ 1199062(119904 119910)) (119898
2+ 119906lowast2)
119889119904 119889119910 119889119909
(25)
Applying the inequality 119886119887 le 1205981198862 + (14120598)1198872 we derive from(25) that
Φ2(119905) le minusint
Ω
120572[11988611minus
11988612119906lowast
2
(1198981+ 1199061) (1198981+ 119906lowast1)] (1199061minus 119906lowast
1)2
119889119909
minus intΩ
11988633120573(1199063minus 119906lowast
3)2
119889119909
minus intΩ
[11988622minus
11988623119906lowast
3
(1198982+ 1199062) (1198982+ 119906lowast2)] (1199062minus 119906lowast
2)2
119889119909
+ intΩ
12057211988612
1198981+ 1199061
[1205981(1199061minus 119906lowast
1)2
+1
41205981
(1199062minus 119906lowast
2)2
] 119889119909
+ intΩ
11988623
1198982+ 1199062
[1
41205982
(1199062minus 119906lowast
2)2
+ 1205982(1199063minus 119906lowast
3)2
] 119889119909
+∬Ω
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981
(1198981+ 1199061(119904 119910)) (119898
1+ 119906lowast1)
times [1205983(1199061(119904 119910) minus 119906
lowast
1)2
+1
41205983
(1199062(119905 119909) minus 119906
lowast
2)2
] 119889119904 119889119910 119889119909
+∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321205731198982
(1198982+ 1199062(119904 119910)) (119898
2+ 119906lowast2)
times [1
41205984
(1199062(119904 119910) minus 119906
lowast
2)2
+1205984(1199063(119905 119909) minus 119906
lowast
3)2
] 119889119904 119889119910 119889119909
(26)
According to the property of the Kernel functions119870119894(119909 119910 119905)
(119894 = 1 2) we know that
Φ2(119905) le minusint
Ω
[12057211988611minus12057211988612119906lowast
2
11989821
minus120572119886121205981
1198981
] (1199061minus 119906lowast
1)2
119889119909
minus intΩ
[11988622minus11988623119906lowast
3
11989822
minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
] (1199062minus 119906lowast
2)2
119889119909
minus intΩ
[11988633120573 minus
119886231205982
1198982
minus119886321205731205984
1198982
] (1199063minus 119906lowast
3)2
119889119909
+119886211205983
1198981
∬Ω
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times (1199061(119904 119910) minus 119906
lowast
1)2
119889119904 119889119910 119889119909
+11988632120573
411989821205984
∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times (1199062(119904 119910) minus 119906
lowast
2)2
119889119904 119889119910 119889119909
(27)
Define a new Lyapunov functional as follows
119864 (119905) = 119881 (119905)
+119886211205983
1198981
∬Ω
int
+infin
0
int
119905
119905minus119903
1198701(119909 119910 119903)
times (1199061(119904 119910) minus 119906
lowast
1)2
119889119904 119889119903 119889119910 119889119909
+11988632120573
411989821205984
∬Ω
int
+infin
0
int
119905
119905minus119903
1198702(119909 119910 119903)
times (1199062(119904 119910) minus 119906
lowast
2)2
119889119904 119889119903 119889119910 119889119909
(28)
Abstract and Applied Analysis 7
It is derived from (27) and (28) that
1198641015840(119905) le minusint
Ω
(1205721198891119906lowast
1
11990621
1003816100381610038161003816nabla119906110038161003816100381610038162
+1198892119906lowast
2
11990622
1003816100381610038161003816nabla119906210038161003816100381610038162
+1205731198893119906lowast
3
11990623
1003816100381610038161003816nabla119906310038161003816100381610038162
)119889119909
minus [12057211988611minus12057211988612119906lowast
2
11989821
minus120572119886121205981
1198981
minus119886211205983
1198981
]intΩ
(1199061minus 119906lowast
1)2
119889119909
minus [11988622minus11988623119906lowast
3
11989822
minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632120573
411989821205984
]intΩ
(1199062minus 119906lowast
2)2
119889119909
minus [11988633120573 minus
119886231205982
1198982
minus120573119886321205984
1198982
]intΩ
(1199063minus 119906lowast
3)2
119889119909
(29)
Since (1199061 1199062 1199063) is the unique positive solution of system (1)
Using Proposition 5 there exists a constant119862which does notdepend on119909 isin Ω or 119905 ge 0 such that 119906
119894(sdot 119905)infinle 119862 (119894 = 1 2 3)
for 119905 ge 0 By Theorem 1198602in [24] we have
1003817100381710038171003817119906119894 (sdot 119905)10038171003817100381710038171198622120572(Ω) le 119862 forall119905 ge 1 (30)
Assume that
1198661 119886111198982
1gt 11988612119906lowast
2
1198662 119886221198982
2gt 11988623119906lowast
3+21198862311988632
11988633
+211988612119886211198982
2
1198861111989821minus 11988612119906lowast2
(31)
and denote
1198971= 12057211988611minus12057211988612119906lowast
2
11989821
minus120572119886121205981
1198981
minus119886211205983
1198981
1198972= 11988622minus11988623119906lowast
3
11989822
minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632120573
411989821205984
1198973= 11988633120573 minus
119886231205982
1198982
minus120573119886321205984
1198982
(32)
Then (29) is transformed into
1198641015840(119905) le minusint
Ω
(1205721198891119906lowast
1
11990621
1003816100381610038161003816nabla119906110038161003816100381610038162
+1198892119906lowast
2
11990622
1003816100381610038161003816nabla119906210038161003816100381610038162
+1205731198893119906lowast
3
11990623
1003816100381610038161003816nabla119906310038161003816100381610038162
)119889119909
minus
3
sum
119894=1
119897119894intΩ
(119906119894minus 119906lowast
119894)2
119889119909
(33)
If we choose
120572 =11988621
11988612
120573 =11988623
11988632
1205981= 1205983=119886111198982
1minus 11988612119906lowast
2
4119898111988612
1205982= 1205984=119898211988633
411988632
(34)
then 119897119894gt 0 (119894 = 1 2 3) Therefore we have
1198641015840(119905) le minus
3
sum
119894=1
119897119894intΩ
(119906119894minus 119906lowast
119894)2
119889119909 (35)
FromProposition 6 we can see that the solution of system(1) and (3) is bounded and so are the derivatives of (119906
119894minus 119906lowast
119894)
(119894 = 1 2 3) by the equations in (1) Applying Lemma 1 weobtain that
lim119905rarrinfin
intΩ
(119906119894minus 119906lowast
119894) 119889119909 = 0 (119894 = 1 2 3) (36)
Recomputing 1198641015840(119905) gives
1198641015840(119905) le minusint
Ω
(1205721198891119906lowast
1
11990621
1003816100381610038161003816nabla119906110038161003816100381610038162
+1198892119906lowast
2
11990622
1003816100381610038161003816nabla119906210038161003816100381610038162
+1205731198893119906lowast
3
11990623
1003816100381610038161003816nabla119906310038161003816100381610038162
)119889119909
le minus119888intΩ
(1003816100381610038161003816nabla1199061
10038161003816100381610038162
+1003816100381610038161003816nabla1199062
10038161003816100381610038162
+1003816100381610038161003816nabla1199063
10038161003816100381610038162
) 119889119909 = minus119892 (119905)
(37)
where 119888 = min1205721198891119906lowast
11198722
1 1198892119906lowast
21198722
2 1205731198893119906lowast
311987223 Using (30)
and (1) we obtain that the derivative of 119892(119905) is bounded in[1 +infin) From Lemma 1 we conclude that 119892(119905) rarr 0 as 119905 rarrinfin Therefore
lim119905rarrinfin
intΩ
(1003816100381610038161003816nabla1199061
10038161003816100381610038162
+1003816100381610038161003816nabla1199062
10038161003816100381610038162
+1003816100381610038161003816nabla1199063
10038161003816100381610038162
) 119889119909 = 0 (38)
Applying the Poincare inequality
intΩ
120582|119906 minus 119906|2119889119909 le int
Ω
|nabla119906|2119889119909 (39)
leads to
lim119905rarrinfin
intΩ
(119906119894minus 119906119894)2
119889119909 = 0 (119894 = 1 2 3) (40)
where 119906119894= (1|Ω|) int
Ω119906119894119889119909 and 120582 is the smallest positive
eigenvalue ofminusΔwith the homogeneousNeumann conditionTherefore
|Ω| (1199061 (119905) minus 119906lowast
1)2
= intΩ
(1199061(119905) minus 119906
lowast
1)2
119889119909
= intΩ
(1199061(119905) minus 119906
1(119905 119909) + 119906
1(119905 119909) minus 119906
lowast
1)2
119889119909
le 2intΩ
(1199061(119905) minus 119906
1(119905 119909))
2
119889119909
+ 2intΩ
(1199061(119905 119909) minus 119906
lowast
1)2
119889119909
(41)
8 Abstract and Applied Analysis
So we have 1199061(119905) rarr 119906
lowast
1as 119905 rarr infin Similarly 119906
2(119905) rarr 119906
lowast
2
and 1199063(119905) rarr 119906
lowast
3as 119905 rarr infin According to (30) there exists
a subsequence 119905119898 and non-negative functions 119908
119894isin 1198622(Ω)
such thatlim119898rarrinfin
1003817100381710038171003817119906119894 (sdot 119905119898) minus 119908119894 (sdot)10038171003817100381710038171198622(Ω) = 0 (119894 = 1 2 3) (42)
Applying (40) and noting that 119906119894(119905) rarr 119906
lowast
119894 we then have119908
119894=
119906lowast
119894 (119894 = 1 2 3) That is
lim119898rarrinfin
1003817100381710038171003817119906119894 (sdot 119905119898) minus 119906lowast
119894
10038171003817100381710038171198622(Ω) = 0 (119894 = 1 2 3) (43)
Furthermore the local stability ofElowast combiningwith (43)gives the following global stability
Theorem 9 Assume that
11986751198861(11988621minus 1198862)
11989811198862
gt 11988611gt11988612Vlowast2
11989821
(44)
1198674and 119866
4hold and then the semi-trivial steady state 119864
2
of system (1)ndash(3) with non-trivial initial functions is globallyasymptotically stable
Proof It is obvious that system (1)ndash(3) always has two non-negative equilibria (119906
1 1199062 1199063) as follows 119864
0= (0 0 0) and
1198641= (119886111988611 0 0) If 119867
1and 119867
4are satisfied system (1) has
the other semitrivial solution denoted by 1198642(Vlowast1 Vlowast2 0) where
1198674 11988631198982gt (11988632minus 1198863) Vlowast
2 (45)
We consider the stability of 1198642under condition 119867
1and
1198674 Equation (1) can be rewritten as
1205971199061
120597119905minus 1198891Δ1199061= 1199061[minus11988611(1199061minus Vlowast
1) minus11988612(1199062minus Vlowast2)
1198981+ 1199061
+11988612Vlowast2(1199061minus Vlowast1)
(1198981+ 1199061) (1198981+ Vlowast1)]
1205971199062
120597119905minus 1198892Δ1199062
= 1199062[intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus Vlowast
1)
(1198981+ 1199061(119904 119910)) (119898
1+ Vlowast1)119889119904 119889119910
minus11988622(1199062minus Vlowast
2) minus
119886231199063
1198982+ 1199062
]
1205971199063
120597119905minus 1198893Δ1199063
= 1199063[intΩ
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321198982(1199062(119904 119910) minus Vlowast
2)
(1198982+ 1199062(119904 119910)) (119898
2+ Vlowast2)119889119904 119889119910
minus119886331199063]
(46)
Similar to the argument ofTheorem 8 we have 1199061(119909 119905) rarr Vlowast
1
and 1199062(119909 119905) rarr Vlowast
2as 119905 rarr infin uniformly on Ω provided that
the following additional condition holds
1198663 119886111198982
1gt 11988612Vlowast
2
1198664 119886221198982
2gt1198862311988632
11988633
+211988612119886211198982
2
1198861111989821minus 11988612Vlowast2
(47)
Next we consider the asymptotic behavior of 1199063(119905 119909) Let
120579 =1198863minus (11988632minus 1198863) Vlowast2
2120575 120575 =
100381610038161003816100381611988632 minus 11988631003816100381610038161003816
(48)
Then there exists 1199051gt 0 such that
Vlowast
2minus 120579 lt 119906
2(119905 119909) lt V
lowast
2+ 120579 forall119905 gt 119905
1 119909 isin Ω (49)
Consider the following two systems
1199083119905minus 1198893Δ1199083= 1199083[minus1198863+11988632(Vlowast2+ 120579)
1198982+ Vlowast2+ 120579
minus 119886331199083]
(119905 119909) isin [1199051 +infin) times Ω
1205971199083
120597]= 0 (119905 119909) isin [119905
1 +infin) times 120597Ω
1199083= 1199063 (119905 119909) isin (minusinfin 119905
1] times Ω
1198823119905minus 1198893Δ1198823= 1198823[minus1198863+11988632(Vlowast2minus 120579)
1198982+ Vlowast2minus 120579
minus 11988633V3]
(119905 119909) isin [1199051 +infin) times Ω
1205971198823
120597]= 0 (119905 119909) isin [119905
1 +infin) times 120597Ω
1198823= 1199063 (119905 119909) isin (minusinfin 119905
1] times Ω
(50)
Combining comparison principle with (50) we obtain that
1198823(119905 119909) le 119906
3(119905 119909) le 119908
3(119905 119909) forall119905 gt 119905
1 119909 isin Ω (51)
By Lemma 4 we obtain
0 = lim119905rarrinfin
1198823(119905 119909) le inf 119906
3(119905 119909) le sup 119906
3(119905 119909)
le lim119905rarrinfin
1199083(119905 119909) = 0 forall119909 isin Ω
(52)
which implies that lim119905rarrinfin
1199063(119905 119909) = 0 uniformly onΩ
Theorem 10 Suppose that1198665and119866
6hold and then the semi-
trivial steady state 1198641of system (1)ndash(3) with non-trivial initial
functions is globally asymptotically stable
Abstract and Applied Analysis 9
Proof We study the stability of the semi-trivial solution 1198641=
(1 0 0) Similarly the equations in (1) can be written as
1205971199061
120597119905minus 1198891Δ1199061= 1199061[minus11988611(1199061minus 1) minus
119886121199062
1198981+ 1199061
]
1205971199062
120597119905minus 1198892Δ1199062
= 1199062[minus1198862+ intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus
1)
(1198981+ 1199061(119904 119910)) (119898
1+ 1)119889119904 119889119910
+119886211
1198981+ 1
minus 119886221199062minus119886231199063
1198982+ 1199062
]
1205971199063
120597119905minus 1198893Δ1199063
= 1199063[minus1198863+ intΩ
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321199062(119904 119910)
1198982+ 1199062(119904 119910)
119889119904 119889119910 minus 119886331199063]
(53)
Define
119881 (119905) = 120572intΩ
[1199061minus 1minus 1log 11990611
] 119889119909 + intΩ
1199062119889119909
+ 120573intΩ
1199063119889119909
(54)
Calculating the derivative of 119881(119905) along 1198641 we get from (54)
that
1198811015840(119905) le minusint
Ω
[120572(11988611minus119886121205981
1198981
)] (1199061minus 1)2
119889119909
+ intΩ
[11988622minus12057211988612
411989811205981
minus11988623
412059821198982
minus11988621
411989811205983
] 1199062
2119889119909
+ intΩ
[12057311988633minus119886231205982
1198982
] 1199062
3119889119909
minus intΩ
[(1198862minus119886211
1198981+ 1
)1199062+ 11988631199063] 119889119909
+119886211205983
1198981
intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times (1199061(119904 119910) minus
1)2
119889119904 119889119910 119889119909
+11988632120573
412059841198982
∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times 1199062
2(119904 119910) 119889119904 119889119910 119889119909
(55)
Define119864 (119905) = 119881 (119905)
+119886211205983
1198981
∬Ω
int
+infin
0
int
119905
119905minus119903
1198701(119909 119910 119903)
times (1199061(119904 119910) minus 119906
lowast
1)2
119889119904 119889119903 119889119910 119889119909
+11988632120573
412059841198982
∬Ω
int
+infin
0
int
119905
119905minus119903
1198702(119909 119910 119903)
times (1199062(119904 119910) minus 119906
lowast
2)2
119889119904 119889119903 119889119910 119889119909
(56)
It is easy to see that
1198641015840(119905) le minus120572119889
11intΩ
1003816100381610038161003816nabla119906110038161003816100381610038162
11990621
119889119909
minus intΩ
[120572(11988611minus119886121205981
1198981
) minus119886211205983
1198981
] (1199061minus 1)2
119889119909
+ intΩ
[11988622minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632
411989821205984
] 1199062
2119889119909
+ intΩ
[12057311988633minus119886231205982
1198982
minus119886321205731205984
1198982
] 1199062
3119889119909
minus intΩ
[(1198862minus119886211
1198981+ 1
)1199062minus 11988631205731199063] 119889119909
(57)
Assume that1198665 1198861(11988621minus 1198862) lt 119898
1119886211988611
1198666 119886221198982
2gt1198862311988632
11988633
+211988612119886211198982
2
1198861111989821
(58)
Let
1198971= 12057211988611minus120572119886121205981
1198981
minus119886211205983
1198981
1198972= 11988622minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632120573
411989821205984
1198973= 11988633120573 minus
119886231205982
1198982
minus120573119886321205984
1198982
(59)
Choose
120572 =11988621
11988612
120573 =11988623
11988632
1205981= 1205983=119886111198981
411988612
1205982= 1205984=119898211988633
411988632
(60)
Then we get 119897119894gt 0 (119894 = 1 2 3) Therefore we have
lim119905rarrinfin
1199061(119905 119909) =
1 (61)
uniformly onΩ
10 Abstract and Applied Analysis
tx
u1
02
46
8
01
23
40
1
2
3
4
(a)
02
46
8
01
23
40
1
2
3
4
t
x
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 2 Global stability of the positive equilibrium 119864lowast with an initial condition (1206011 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos 119909 27 + 05 sin119909)
Next we consider the asymptotic behavior of 1199062(119905 119909) and
1199063(119905 119909) For any 119879 gt 0 integrating (57) over [0 119879] yields
119864 (119879) + 1205721198891
10038171003817100381710038171003817100381710038171003817
10038161003816100381610038161003816100381610038161003816
nabla1199061
1199061
10038161003816100381610038161003816100381610038161003816
10038171003817100381710038171003817100381710038171003817
2
2
+ 1198971
100381710038171003817100381710038161003816100381610038161199061 minus 1
100381610038161003816100381610038171003817100381710038172
2
+ 1198972
100381710038171003817100381710038161003816100381610038161199062100381610038161003816100381610038171003817100381710038172
2+ 1198973
100381710038171003817100381710038161003816100381610038161199063100381610038161003816100381610038171003817100381710038172
2le 119864 (0)
(62)
where |119906119894|2
2= int119879
0intΩ1199062
119894119889119909 119889119905 It implies that |119906
119894|2le 119862119894(119894 =
2 3) for the constant 119862119894which is independent of 119879 Now we
consider the boundedness of |nabla1199062|2and |nabla119906
3|2 From the
Greenrsquos identity we obtain
1198892int
119879
0
intΩ
1003816100381610038161003816nabla1199062 (119905 119909)10038161003816100381610038162
119889119909 119889119905
= minus1198892int
119879
0
intΩ
1199062(119905 119909) Δ119906
2(119905 119909) 119889119909 119889119905
= minusint
119879
0
intΩ
1199062
1205971199062
120597119905119889119909 119889119905 minus 119886
2int
119879
0
intΩ
1199062
2119889119909 119889119905
minus 11988622int
119879
0
intΩ
1199063
2119889119909 119889119905 minus 119886
23int
119879
0
intΩ
1199062
21199063
1198982+ 1199063
119889119909 119889119905
+ 11988621int
119879
0
∬Ω
int
119905
minusinfin
1199061(119904 119910) 119906
2
2(119905 119909)
1198981+ 1199061(119904 119910)
times 1198701(119909 119910 119905 minus 119904) 119889119904 119889119910 119889119909 119889119905
(63)
Note that
int
119879
0
intΩ
1199062(119905 119909)
1205971199062(119905 119909)
120597119905119889119909 119889119905
= int
119879
0
intΩ
1
2
1205971199062
2
120597119905119889119905 119889119909
=1
2intΩ
1199062
2(119879 119909) 119889119909 minus
1
2intΩ
1199062
2(0 119909) 119889119909 le 119872
2
2|Ω|
int
119879
0
intΩ
1199062
21199063119889119909 119889119905 le 119872
3int
119879
0
intΩ
1199062
2119889119909 119889119905
Abstract and Applied Analysis 11
02
46
8
01
23
40
1
2
3
4
tx
u1
(a)
02
46
8
01
23
40
1
2
3
4
tx
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 3 Global stability of the positive equilibrium 1198642with an initial condition (120601
1 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos 119909 27 + 05 sin119909)
int
119879
0
intΩ
1199063
2119889119909 119889119905 le 119872
2int
119879
0
intΩ
1199062
2119889119909 119889119905
int
119879
0
∬Ω
int
119905
minusinfin
1199061(119904 119910) 119906
2
2(119905 119909)
1198981+ 1199061(119904 119910)
1198701(119909 119910 119905 minus 119904) 119889119904 119889119910 119889119909 119889119905
le int
119879
0
intΩ
1199062
2(119905 119909) 119889119909 119889119905
(64)
Thus we get |nabla1199062|2le 1198624 In a similar way we have
|nabla1199063|2le 1198625 Here 119862
4and 119862
5are independent of 119879
It is easy to see that 1199062(119905 119909) 119906
3(119905 119909) isin 119871
2((0infin)
11988212(Ω)) These imply that
lim119905rarrinfin
1003817100381710038171003817119906119894 (sdot 119905)100381710038171003817100381711988212(Ω) = 0 119894 = 2 3 (65)
From the Sobolev compact embedding theorem we know
lim119905rarrinfin
1003817100381710038171003817119906119894 (sdot 119905)1003817100381710038171003817119862(Ω) = 0 119894 = 2 3 (66)
In the end we show that the trivial solution 1198640is an
unstable equilibrium Similarly to the local stability to 119864lowast wecan get the characteristic equation of 119864
0as
(120582 + 1205831198941198631minus 1198861) (120582 + 120583
1198941198632+ 1198862) (120582 + 120583
1198941198633+ 1198863) = 0 (67)
If 119894 = 1 then 1205831= 0 It is easy to see that this equation admits
a positive solution 120582 = 1198861 According to Theorem 51 in [23]
we have the following result
Theorem 11 The trivial equilibrium 1198640is an unstable equilib-
rium of system (1)ndash(3)
4 Numerical Illustrations
In this section we performnumerical simulations to illustratethe theoretical results given in Section 3
12 Abstract and Applied Analysis
u1
02
46
8
01
23
40
1
2
3
4
tx
(a)
02
46
8
01
23
40
1
2
3
4
tx
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 4 Global stability of the positive equilibrium 1198641with an initial condition (120601
1 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos119909 27 + 05 sin119909)
In the following we always take Ω = [0 120587] 119870119894(119909 119910 119905) =
119866119894(119909 119910 119905)119896
119894(119905) where 119896
119894(119905) = (1120591
lowast)119890minus119905120591lowast
(119894 = 1 2) and
1198661(119909 119910 119905) =
1
120587+2
120587
infin
sum
119899=1
119890minus11988921198992
119905 cos 119899119909 cos 119899119910
1198662(119909 119910 119905) =
1
120587+2
120587
infin
sum
119899=1
119890minus11988931198992
119905 cos 119899119909 cos 119899119910
(68)
However it is difficult for us to simulate our results directlybecause of the nonlocal term Similar to [25] the equationsin (1) can be rewritten as follows
1205971199061
120597119905minus 1198891Δ1199061= 1199061(1198861minus 119886111199061minus119886121199062
1198981+ 1199061
)
1205971199062
120597119905minus 1198892Δ1199062= 1199062(minus1198862+ 11988621V1minus 119886221199062minus119886231199063
1198982+ 1199063
)
1205971199063
120597119905minus 1198893Δ1199063= 1199063(minus1198863+ 11988632V2minus 119886331199063)
120597V1
120597119905minus 1198892ΔV1=1
120591lowast(
1199061
1198981+ 1199061
minus V1)
120597V2
120597119905minus 1198893ΔV2=1
120591lowast(
1199062
1198982+ 1199062
minus V2)
(69)
where
V119894= int
120587
0
int
119905
minusinfin
119866119894(119909 119910 119905 minus 119904)
1
120591lowast119890minus(119905minus119904)120591
lowast 119906119894(119904 119910)
119898119894+ 119906119894(119904 119910)
119889119910 119889119904
119894 = 1 2
(70)
Each component is considered with homogeneous Neumannboundary conditions and the initial condition of V
119894is
V119894(0 119909) = int
120587
0
int
0
minusinfin
119866119894(119909 119910 minus119904)
1
120591lowast119890119904120591lowast 119906119894(119904 119910)
119898119894+ 119906119894(119904 119910)
119889119910 119889119904
119894 = 1 2
(71)
Abstract and Applied Analysis 13
In the following examples we fix some coefficients andassume that 119889
1= 1198892= 1198893= 1 119886
1= 3 119886
12= 1 119898
1= 1198982=
1 1198862= 1 119886
22= 12 119886
23= 1 119886
33= 12 and 120591lowast = 1 The
asymptotic behaviors of system (1)ndash(3) are shownby choosingdifferent coefficients 119886
11 11988621 1198863 and 119886
32
Example 12 Let 11988611= 539 119886
21= 8113 119886
3= 113 and
11988632= 14Then it is easy to see that the system admits a unique
positive equilibrium 119864lowast(12 112 112) By Theorem 8 we
see that the positive solution (1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of
system (1)ndash(3) converges to 119864lowast as 119905 rarr infin See Figure 2
Example 13 Let 11988611
= 6 11988621
= 12 1198863= 14 and
11988632= 1 Clearly 119867
2does not hold Hence the positive
steady state is not feasible System (1) admits two semi-trivial steady state 119864
2(04732 02372 0) and 119864
1(12 0 0)
According to Theorem 9 we know that the positive solution(1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of system (1)ndash(3) converges to 119864
2
as 119905 rarr infin See Figure 3
Example 14 Let 11988611= 6 119886
21= 2 119886
3= 14 and 119886
32= 1
Clearly1198671does not hold Hence 119864lowast and 119864
2are not feasible
System (1) has a unique semi-trivial steady state 1198641(12 0 0)
According to Theorem 10 we know that the positive solution(1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of system (1)ndash(3) converges to 119864
1
as 119905 rarr infin See Figure 4
5 Discussion
In this paper we incorporate nonlocal delay into a three-species food chain model with Michaelis-Menten functionalresponse to represent a delay due to the gestation of the preda-tor The conditions under which the spatial homogeneousequilibria are asymptotically stable are given by using theLyapunov functional
We now summarize the ecological meanings of ourtheoretical results Firstly the positive equilibrium 119864
lowast ofsystem (1)ndash(3) exists under the high birth rate of the prey(1198861) and low death rates (119886
2and 119886
3) of predator and top
predator 119864lowast is globally stable if the intraspecific competition11988611is neither too big nor too small and the maximum harvest
rates 11988612 11988623
are small enough Secondly the semi-trivialequilibrium 119864
2of system (1)ndash(3) exists if the birth rate of the
prey (1198861) is high death rate of predator 119886
2is low and the death
rate (1198863) exceeds the conversion rate from predator to top
predator (11988632) 1198642is globally stable if the maximum harvest
rates 11988612 11988623are small and the intra-specific competition 119886
11
is neither too big nor too small Thirdly system (1)ndash(3) hasonly one semi-trivial equilibrium 119864
1when the death rate (119886
2)
exceeds the conversion rate from prey to predator (11988621) 1198641
is globally stable if intra-specific competitions (11988611 11988622 and
11988633) are strong Finally 119864
0is unstable and the non-stability of
trivial equilibrium tells us that not all of the populations goto extinction Furthermore our main results imply that thenonlocal delay is harmless for stabilities of all non-negativesteady states of system (1)ndash(3)
There are still many interesting and challenging problemswith respect to system (1)ndash(3) for example the permanence
and stability of periodic solution or almost periodic solutionThese problems are clearly worthy for further investigations
Acknowledgments
This work is partially supported by PRC Grants NSFC11102076 and 11071209 andNSF of theHigher Education Insti-tutions of Jiangsu Province (12KJD110002 and 12KJD110008)
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] H F Huo andW T Li ldquoPeriodic solution of a delayed predator-prey system with Michaelis-Menten type functional responserdquoJournal of Computational and AppliedMathematics vol 166 no2 pp 453ndash463 2004
[3] Z Lin and M Pedersen ldquoStability in a diffusive food-chainmodel with Michaelis-Menten functional responserdquo NonlinearAnalysis Theory Methods amp Applications vol 57 no 3 pp 421ndash433 2004
[4] B X Dai N Zhang and J Z Zou ldquoPermanence for theMichaelis-Menten type discrete three-species ratio-dependentfood chain model with delayrdquo Journal of Mathematical Analysisand Applications vol 324 no 1 pp 728ndash738 2006
[5] N F Britton ldquoSpatial structures and periodic travelling wavesin an integro-differential reaction-diffusion population modelrdquoSIAM Journal on Applied Mathematics vol 50 no 6 pp 1663ndash1688 1990
[6] S A Gourley and N F Britton ldquoInstability of travelling wavesolutions of a population model with nonlocal effectsrdquo IMAJournal of Applied Mathematics vol 51 no 3 pp 299ndash310 1993
[7] X Zhang and R Xu ldquoTraveling waves of a diffusive predator-prey model with nonlocal delay and stage structurerdquo Journal ofMathematical Analysis and Applications vol 373 no 2 pp 475ndash484 2011
[8] Z C Wang W T Li and S G Ruan ldquoExistence and stability oftraveling wave fronts in reaction advection diffusion equationswith nonlocal delayrdquo Journal of Differential Equations vol 238no 1 pp 153ndash200 2007
[9] S A Gourley and S G Ruan ldquoConvergence and travellingfronts in functional differential equations with nonlocal termsa competition modelrdquo SIAM Journal on Mathematical Analysisvol 35 no 3 pp 806ndash822 2003
[10] X H Lai and T X Yao ldquoExponential stability of impul-sive delayed reaction-diffusion cellular neural networks viaPoincare integral inequalityrdquoAbstract and Applied Analysis vol2013 Article ID 131836 10 pages 2013
[11] R Xu and Z E Ma ldquoGlobal stability of a reaction-diffusionpredator-prey model with a nonlocal delayrdquo Mathematical andComputer Modelling vol 50 no 1-2 pp 194ndash206 2009
[12] Z Li F D Chen and M X He ldquoAsymptotic behavior of thereaction-diffusion model of plankton allelopathy with nonlocaldelaysrdquo Nonlinear Analysis Real World Applications vol 12 no3 pp 1748ndash1758 2011
[13] Y Kuang and H L Smith ldquoGlobal stability in diffusive delayLotka-Volterra systemsrdquoDifferential and Integral Equations vol4 no 1 pp 117ndash128 1991
14 Abstract and Applied Analysis
[14] Z G Lin M Pedersen and L Zhang ldquoA predator-prey systemwith stage-structure for predator and nonlocal delayrdquoNonlinearAnalysis Theory Methods amp Applications vol 72 no 3-4 pp2019ndash2030 2010
[15] R Xu ldquoA reaction-diffusion predator-prey model with stagestructure and nonlocal delayrdquo Applied Mathematics and Com-putation vol 175 no 2 pp 984ndash1006 2006
[16] S S Chen and J P Shi ldquoStability and Hopf bifurcation in adiffusive logistic population model with nonlocal delay effectrdquoJournal of Differential Equations vol 253 no 12 pp 3440ndash34702012
[17] H I Freedman Deterministic Mathematical Models in Popula-tion Ecology vol 57 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1980
[18] P A Abrams and L R Ginzburg ldquoThe nature of predation preydepedent ratio dependent or neitherrdquo Trends in Ecology andEvolution vol 15 no 8 pp 337ndash341 2000
[19] D L DeAngelis R A Goldstein and R V OrsquoNeill ldquoA modelfor trophic iterationrdquo Ecology vol 56 no 4 pp 881ndash892 1975
[20] C V Pao ldquoDynamics of nonlinear parabolic systems with timedelaysrdquo Journal of Mathematical Analysis and Applications vol198 no 3 pp 751ndash779 1996
[21] F Xu Y Hou and Z G Lin ldquoTime delay parabolic system ina predator-prey model with stage structurerdquo Acta MathematicaSinica vol 48 no 6 pp 1121ndash1130 2005 (Chinese)
[22] P Y H Pang and M X Wang ldquoStrategy and stationary patternin a three-species predator-prey modelrdquo Journal of DifferentialEquations vol 200 no 2 pp 245ndash273 2004
[23] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981
[24] K J Brown P C Dunne and R A Gardner ldquoA semilinearparabolic system arising in the theory of superconductivityrdquoJournal of Differential Equations vol 40 no 2 pp 232ndash252 1981
[25] S A Gourley and J W H So ldquoDynamics of a food-limitedpopulation model incorporating nonlocal delays on a finitedomainrdquo Journal of Mathematical Biology vol 44 no 1 pp 49ndash78 2002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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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
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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
1205971199062
120597119905minus 1198892Δ1199062
= 1199062(minus1198862+ 11988621intΩ
int
119905
minusinfin
1199061(119904 119910)
1198981+ 1199061(119904 119910)
times 1198701(119909 119910 119905 minus 119904) 119889119904 119889119910
minus119886221199062minus119886231199063
1198982+ 1199062
)
1205971199063
120597119905minus 1198893Δ1199063
= 1199063(minus1198863+ 11988632intΩ
int
119905
minusinfin
1199062(119904 119910)
1198982+ 1199062(119904 119910)
times 1198702(119909 119910 119905 minus 119904) 119889119904 119889119910
minus119886331199063)
(1)
for 119905 gt 0 119909 isin Ω with homogeneous Neumann boundaryconditions
1205971199061
120597]=1205971199062
120597]=1205971199063
120597]= 0 119905 gt 0 119909 isin 120597Ω (2)
and initial conditions
119906119894(119905 119909) = 120601
119894(119905 119909) ge 0 (119894 = 1 2) (119905 119909) isin (minusinfin 0] times Ω
1199063(0 119909) = 120601
3(119909) ge 0 119909 isin Ω
(3)
where 120601119894is bounded Holder continuous function and satis-
fies 120597120601119894120597] = 0 (119894 = 1 2 3) on (minusinfin 0] times 120597Ω Here Ω is a
bounded domain in R119899 with smooth boundary 120597Ω and 120597120597]is the outward normal derivative on 120597Ω 119906
119894(119905 119909) represents
the density of the 119894th species (prey predator and top predatorresp) at time 119905 and location 119909 and thus only nonnegative119906119894(119905 119909) is of interest The parameter 119886
1is the intrinsic growth
rate of the prey and 1198862and 119886
3are the death rates of the
predator and top-predator 119886119894119894is the intraspecific competitive
rate of the 119894th species 11988612is the maximum predation rate 119886
23
and 11988632are the efficiencies of food utilization of the predator
and top predator respectively We assume the predator andtop predator show the Michaelis-Menten (or Holling typeII) functional response with 119906
1(1198981+ 1199061) and 119906
2(1198982+ 1199062)
respectively where 1198981and 119898
2are half-saturation constants
For a through biological background of similar models see[18 19] As our most knowledge the tritrophic food chainmodel has been found to have many interesting biologicalproperties such as the coexistence and the Hopf bifurcationHowever the effect of nonlocal time delays on the coexistencehas not been reported Our paper mainly concerns thisperspective
Additionally intΩint119905
minusinfin(119906119894(119904 119910)(119898
119894+ 119906119894(119904 119910)))119870
119894(119909 119910 119905 minus
119904)119889119904 119889119910 (119894 = 1 2) represents the nonlocal delay due tothe ingestion of predator that is mature adult predator can
only contribute to the production of predator biomass Theboundary condition in (2) implies that there is no migrationacross the boundary ofΩ
The main purpose of this paper is to study the globalasymptotic behavior of the solution of system (1)ndash(3) Thepreliminary results are presented in Section 2 Section 3 con-tains sufficient conditions for the global asymptotic behaviorsof the equilibria of system (1)ndash(3) by means of the Lyapunovfunctional Numerical simulations are carried out to show thefeasibility of the conditions in Theorems 8ndash10 in Section 4Finally a brief discussion is given to conclude this work
2 Preliminary Results
In this section we present several preliminary results that willbe employed in the sequel
Lemma 1 (see [3]) Let 119886 and 119887 be positive constants Assumethat 120601 120593 isin 1198621(119886 +infin) 120593(119905) ge 0 and 120601 is bounded frombelow If 1206011015840(119905) le minus119887120593(119905) and 1205931015840(119905) le 119870 in [119886 +infin) for somepositive constant 119870 then lim
119905rarr+infin120593(119905) = 0
The following lemma is the Positivity Lemma in [20]
Lemma 2 Let 119906119894isin 119862([0 119879] times Ω)⋂119862
12((0 119879] times Ω) (119894 =
1 2 3) and satisfy
119906119894119905minus 119889119894Δ119906119894ge
3
sum
119895=1
119887119894119895(119905 119909) 119906
119895(119905 119909)
+
3
sum
119895=1
119888119894119895119906119895(119905 minus 120591119895 119909) in (0 119879] times Ω
120597119906119894(119905 119909)
120597]ge 0 on (0 119879] times 120597Ω
119906119894(119905 119909) ge 0 in [minus120591
119894 0] times Ω
(4)
where 119887119894119895 119888119894119895isin 119862([0 119879] times Ω) 119906
119894119905= (119906119894)119905 If 119887119894119895ge 0 for 119895 = 119894
and 119888119894119895ge 0 for all 119894 119895 = 1 2 3 Then 119906
119894(119905 119909) ge 0 on [0 119879] times Ω
Moreover if the initial function is nontrivial then 119906119894gt 0 in
(0 119879] times Ω
Lemma 3 (see [20]) Let c and c be a pair of constant vectorsatisfying c ge c and let the reaction functions satisfy localLipschitz condition withΛ = ⟨c c⟩Then system (1)ndash(3) admitsa unique global solution u(119905 119909) such that
c le u (119905 119909) le c forall119905 gt 0 119909 isin Ω (5)
whenever c le 120601(119905 119909) le c (119905 119909) isin (minusinfin 0] times Ω
The following result was obtained by themethod of upperand lower solutions and the associated iterations in [21]
Abstract and Applied Analysis 3
Lemma 4 (see [21]) Let 119906(119905 119909) isin 119862([0infin) times Ω) cap
11986221((0infin)timesΩ) be the nontrivial positive solution of the system
119906119905minus 119889Δ119906 = 119861119906 (119905 minus 120591 119909) plusmn 119860
1119906 (119905 119909)
minus 11986021199062(119905 119909) (119905 119909) isin (0infin) times Ω
120597119906
120597]= 0 (119905 119909) isin (0infin) times 120597Ω
119906 (119905 119909) = 120601 (119905 119909) ge 0 (119905 119909) isin (minus120591 0) times Ω
(6)
where 1198601ge 0 119861 119860
2 120591 gt 0 The following results hold
(1) if 119861plusmn1198601gt 0 then 119906 rarr (119861 plusmn119860
1)1198602uniformly onΩ
as 119905 rarr infin
(2) if 119861plusmn1198601le 0 then 119906 rarr 0 uniformly onΩ as 119905 rarr infin
Throughout this paper we assume that
119870119894(119909 119910 119905) = 119866
119894(119909 119910 119905) 119896
119894(119905) 119909 119910 isin Ω 119896
119894(119905) ge 0
intΩ
119866119894(119909 119910 119905) 119889119909 = int
Ω
119866119894(119909 119910 119905) 119889119910 = 1 119905 ge 0
int
+infin
0
119896119894(119905) 119889119905 = 1 119905119896
119894(119905) isin 119871
1((0 +infin) 119877)
(7)
where119866119894(119909 119910 119905) is a nonnegative function which is continu-
ous in (119909 119910) isin ΩtimesΩ for each 119905 isin [0 +infin) and measurable in119905 isin [0 +infin) for each pair (119909 119910) isin Ω times Ω
Now we prove the following propositions which will beused in the sequel
Proposition 5 For any nonnegative initial function thecorresponding solution of system (1)ndash(3) is nonnegative
Proof Suppose that (1199061 1199062 1199063) is a solution and satisfies 119906
119894isin
119862([0 119879) times Ω) cap 11986212((0 119879) times Ω) with 119879 le +infin Choose 0 lt
120591 lt 119879 Then
1199061119905minus 1198891Δ1199061= 11986011199061 0 lt 119905 lt 120591 119909 isin Ω
1199062119905minus 1198892Δ1199062= 11986021199062 0 lt 119905 lt 120591 119909 isin Ω
1199063119905minus 1198893Δ1199063= 11986031199063 0 lt 119905 lt 120591 119909 isin Ω
120597119906119894
120597]= 0 0 lt 119905 lt 120591 119909 isin 120597Ω
119906119894(119905 119909) ge 0 (119894 = 1 2 3) 119905 le 0 119909 isin Ω
(8)
where
1198601= 1198861minus 119886111199061minus119886121199062
1198981+ 1199061
1198602= minus1198862+ 11988621intΩ
int
119905
minusinfin
1199061(119904 119910)
1198981+ 1199061(119904 119910)
times 1198701(119909 119910 119905 minus 119904) 119889119904 119889119910
minus 119886221199062minus119886231199063
1198982+ 1199063
1198603= minus1198863+ 11988632intΩ
int
119905
minusinfin
1199062(119904 119910)
1198982+ 1199062(119904 119910)
times 1198702(119909 119910 119905 minus 119904) 119889119904 119889119910 minus 119886
331199063
(9)
It follows from Lemma 2 that 119906119894ge 0 (119905 119909) isin [0 120591] times Ω Due
to the arbitrariness of 120591 we have 119906119894ge 0 for (119905 119909) isin [0 119879) times
Ω
Proposition 6 System (1)ndash(3) with initial functions 120601119894admits
a unique global solution (1199061 1199062 1199063) satisfying 0 le 119906
119894(119905 119909) le
119872119894for 119894 = 1 2 3 where119872
119894is defined as
1198721= max 1198861
11988611
10038171003817100381710038171206011 (119905 119909)
1003817100381710038171003817119871infin((minusinfin0]timesΩ)
1198722= max
(11988621minus 1198862)1198721minus 11989811198862
11988622(1198981+1198721)
10038171003817100381710038171206012 (119905 119909)
1003817100381710038171003817119871infin((minusinfin0]timesΩ)
1198723= max
(11988632minus 1198863)1198722minus 11989821198863
11988622(1198982+1198722)
10038171003817100381710038171206013 (119909)
1003817100381710038171003817119871infin(Ω)
(10)
Proof It follows from standard PDE theory that there existsa 119879 gt 0 such that problem (1)ndash(3) admits a unique solutionin [0 119879) times Ω From Lemma 2 we know that 119906
119894(119905 119909) ge 0 for
(119905 119909) isin [0 119879) times Ω Considering the first equation in (1) wehave
1205971199061
120597119905minus 1198891Δ1199061le 1199061(1198861minus 119886111199061) (11)
Using themaximumprinciple gives that 1199061le 1198721 In a similar
way we have 119906119894le 119872119894for 119894 = 2 3 It is easy to see that (0 0 0)
and (119872111987221198723) are a pair of coupled upper and lower
solutions to system (1)ndash(3) from the direct computation Invirtue of Lemma 3 system (1)ndash(3) admits a unique globalsolution (119906
1 1199062 1199063) satisfying 0 le 119906
119894(119905 119909) le 119872
119894for 119894 = 1 2 3
In addition if 1206011(119905 119909) 120601
2(119905 119909) 120601
3(119909) (119905 119909) isin (minusinfin 0] times Ω is
nontrivial it follows from the strongmaximumprinciple that119906119894(119905 119909) gt 0 (119894 = 1 2 3) for all 119905 gt 0 119909 isin Ω
3 Global Stability
In this section we study the asymptotic behavior of theequilibrium of system (1)ndash(3) In the beginning we show theexistence and uniqueness of positive steady state
4 Abstract and Applied Analysis
L1
L2
L3
025
02
015
01
005
0
0 02 04 06 081
2
Figure 1 Graphs of the equations in (12) 1198713is the boundary
condition (1198863= 11988632V2(1198982+ V2)) and 119871
2corresponds to minus119886
2+
11988621(V1(1198981+ V1)) minus 119886
22V2= 0 while 119871
1corresponds to 119886
1minus 11988611V1minus
11988612V2(1198981+ V1) = 0 Intersection point between 119871
1and 119871
2gives the
positive solution (Vlowast1 Vlowast2) to (12) Parameter values are listed in the
example in Section 4
Let us consider the following equations
1198861minus 11988611V1minus11988612V2
1198981+ V1
= 0
minus 1198862+11988621V1
1198981+ V1
minus 11988622V2= 0
(12)
A direct computation shows that the above equations haveonly one positive solution (Vlowast
1 Vlowast2) if and only if
1198671 1198861(11988621minus 1198862) gt 119898
1119886211988611 (13)
is satisfied Taking1198671into account we consider the equation
1198863= 11988632Vlowast2(1198982+ Vlowast2) which corresponds to 119871
3in Figure 1
It is clear that if 1198863lt 11988632Vlowast2(1198982+ Vlowast2) the intersection
point (Vlowast1 Vlowast2) always lies in 119871
3 Let
1198672 11988631198982lt (11988632minus 1198863) Vlowast
2 (14)
Suppose that1198671and119867
2hold We take 119906
3as a parameter and
consider the following system
1198861minus 119886111199061minus119886121199062
1198981+ 1199061
= 0
minus 1198862+119886211199061
1198981+ 1199061
minus 119886221199062minus119886231199063
1198982+ 1199062
= 0
minus 1198863+119886321199062
1198982+ 1199062
minus 119886331199063= 0
(15)
When the parameter 1199063is sufficiently small the first two
equations in (15) can be approximated by (12) Moreover
by continuously increasing the value of 1199063 1198713goes up and
meanwhile the intersection point between 1198711and 119871
2also
goes up However 1198713goes up faster than 119871
2 while 119871
1keeps
still In other words there exists a critical value 1199063such
that the intersection point lies in 1198713 which implies that
there is a unique positive solution 119864lowast(119906lowast1 119906lowast
2 119906lowast
3) to (15) or
equivalently (1)ndash(3)
Lemma 7 Assume that1198672and 119866
2hold and then the positive
steady state119864lowast of system (1)ndash(3) is locally asymptotically stable
Proof We get the local stability of 119864lowast by performing alinearization and analyzing the corresponding characteristicequation Similarly as in [22] let 0 lt 120583
1lt 1205832lt 1205833lt
be the eigenvalues of minusΔ on Ω with the homogeneousNeumann boundary condition Let 119864(120583
119894) be the eigenspace
corresponding to 120583119894in 1198621(Ω) for 119894 = 1 2 3 Let
X = u = (1199061 1199062 1199063) isin [119862
1(Ω)]3
120597120578u = 0 119909 isin 120597Ω (16)
120593119894119895 119895 = 1 dim119864(120583
119894) be an orthonormal basis of 119864(120583
119894)
and X119894119895= 119888 sdot 120593
119894119895| 119888 isin 119877
3 Then
X119894=
dim119864(120583119894)
⨁
119895=1
X119894119895 X =
infin
⨁
119894=1
X119894 (17)
Let 119863 = diag(1198631 1198632 1198633) and k = (V
1 V2 V3) V119894=
119906119894minus 119906lowast
119894 (119894 = 1 2 3) Then the linearization of (1)
is k119905= 119871k = 119863Δk + Fk(Elowast)k Fk(Elowast)k = 119888
119894119895
where 119888119894119894
= 119887119894119894V119894 119894 = 1 2 3 119888
12= 11988712V2 11988823
=
11988723V3 11988821= 11988721intΩint119905
minusinfinV1(119904 119910)119870
1(119909 119910 119905 minus 119904)119889119904 119889119910 and 119888
32=
11988732intΩint119905
minusinfinV2(119904 119910)119870
2(119909 119910 119905 minus 119904)119889119904 119889119910 The coefficients 119887
119894119895are
defined as follows
11988711= minus11988611119906lowast
1+11988612119906lowast
1119906lowast
2
(1198981+ 119906lowast1)2
11988712= minus
11988612119906lowast
1
1198981+ 119906lowast1
11988713= 0
11988721=119886211198981119906lowast
2
(1198981+ 119906lowast1)2
11988722= minus11988622119906lowast
2+11988623119906lowast
2119906lowast
3
(1198982+ 119906lowast2)2 119887
23= minus
11988623119906lowast
2
1198982+ 119906lowast2
11988731= 0 119887
32=119886321198982119906lowast
3
(1198982+ 119906lowast2)2 119887
33= minus11988633119906lowast
3
(18)
SinceX119894is invariant under the operator 119871 for each 119894 ge 1 then
the operator 119871 on X119894is k119905= 119871k = 119863120583
119894k + Fk(Elowast)k Let V119894 =
119888119894119890120582119905(119894 = 1 2 3) and we can get the characteristic equation
Abstract and Applied Analysis 5
120593119894(120582) = 120582
3+ 1198601198941205822+ 119861119894120582 + 119862
119894= 0 where 119860
119894 119861119894 and 119862
119894are
defined as follows
119860119894= (1198891+ 1198892+ 1198893) 120583119894minus 11988711minus 11988722minus 11988733
119861119894= (11988911198892+ 11988921198893+ 11988931198891) 1205832
119894
+ [1198893(minus11988711minus 11988722) + 1198891(minus11988733minus 11988722)
+1198892(minus11988711minus 11988733)] 120583119894
+ 1198871111988722+ 1198871111988733+ 1198872211988733
minus 1198871211988721int
infin
0
1198961(119904) 119890minus120582119904119889119904 minus 119887
2311988732int
infin
0
1198962(119904) 119890minus120582119904119889119904
119862119894= 1198891119889211988931205833
119894+ [minus1198873311988911198892minus 1198871111988921198893minus 1198872211988911198893] 1205832
119894
+ [11988711119887221198893+ 11988711119887331198892+ 11988722119887331198891+ 11988722119887331198891
minus 11988931198871211988721int
infin
0
1198961(119904) 119890minus120582119904119889119904
minus11988911198872311988732int
infin
0
1198962(119904) 119890minus120582119904119889119904] 120583119894
+ [minus119887111198872211988733+ 119887121198872111988733int
infin
0
1198961(119904) 119890minus120582119904119889119904
+119887111198872311988732int
infin
0
1198962(119904) 119890minus120582119904119889119904]
(19)
It is easy to see that 11988712 11988723 11988733lt 0 and 119887
21 11988732gt 0 It
follows from assumption 1198661and 119866
2that 119886
11lt 0 119886
22lt 0
So 119860119894gt 0 119861
119894gt 0 119862
119894gt 0 and 119860
119894119861119894minus 119862119894gt 0 for 119894 ge 1
from the direct calculation According to the Routh-Hurwitzcriterion the three roots 120582
1198941 1205821198942 1205821198943
of 120593119894(120582) = 0 all have
negative real partsBy continuity of the roots with respect to 120583
119894and Routh-
Hurwitz criterion we can conclude that there exists a positiveconstant 120576 such that
Re 1205821198941 Re 120582
1198942 Re 120582
1198943 le minus120576 119894 ge 1 (20)
Consequently the spectrum of 119871 consisting only of eigen-values lies in Re 120582 le minus120576 It is easy to see that Elowast islocally asymptotically stable and follows fromTheorem 511of [23]
Theorem 8 Assume that
11986731198861(11988621minus 1198862)
11989811198862
gt 11988611gt11988612119906lowast
2
11989821
(21)
1198672and 119866
2hold and the positive steady state 119864lowast of system (1)ndash
(3) with nontrivial initial function is globally asymptoticallystable
Proof It is easy to see that the equations in (1) can be rewrittenas
1205971199061
120597119905minus 1198891Δ1199061= 1199061[ minus 11988611(1199061minus 119906lowast
1)
minus11988612(1199062minus 119906lowast
2)
1198981+ 1199061
+11988612119906lowast
2(1199061minus 119906lowast
1)
(1198981+ 1199061) (1198981+ 119906lowast1)]
1205971199062
120597119905minus 1198892Δ1199062= 1199062[intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus 119906
lowast
1)
(1198981+ 1199061(119904 119910)) (119898
1+ 119906lowast1)119889119904 119889119910
minus 11988622(1199062minus 119906lowast
2) minus11988623(1199063minus 119906lowast
3)
1198982+ 1199062
+11988623119906lowast
3(1199062minus 119906lowast
2)
(1198982+ 1199062) (1198982+ 119906lowast2)]
1205971199063
120597119905minus 1198893Δ1199063= 1199063[intΩ
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321198982(1199062(119904 119910) minus 119906
lowast
2)
(1198982+ 1199062(119904 119910)) (119898
2+ 119906lowast2)119889119904 119889119910
minus11988633(1199063minus 119906lowast
3) ]
(22)
Define
119881 (119905) = 120572intΩ
(1199061minus 119906lowast
1minus 119906lowast
1ln 1199061119906lowast1
)119889119909
+ intΩ
(1199062minus 119906lowast
2minus 119906lowast
2ln 1199062119906lowast2
)119889119909
+ 120573intΩ
(1199062minus 119906lowast
3minus 119906lowast
3ln1199063
119906lowast3
)119889119909
(23)
where 120572 and 120573 are positive constants to be determinedCalculating the derivatives 119881(119905) along the positive solutionto the system (1)ndash(3) yields
1198811015840(119905) = Φ
1(119905) + Φ
2(119905) (24)
where
Φ1(119905) = minusint
Ω
(1205721198891119906lowast
1
11990621
1003816100381610038161003816nabla119906110038161003816100381610038162
+1198892119906lowast
2
11990622
1003816100381610038161003816nabla119906210038161003816100381610038162
+1205731198893119906lowast
3
11990623
1003816100381610038161003816nabla119906310038161003816100381610038162
)119889119909
6 Abstract and Applied Analysis
Φ2(119905) = minusint
Ω
120572[11988611minus
11988612119906lowast
2
(1198981+ 1199061) (1198981+ 119906lowast1)] (1199061minus 119906lowast
1)2
119889119909
minus intΩ
[11988622minus
11988623119906lowast
3
(1198982+ 1199062) (1198982+ 119906lowast2)] (1199062minus 119906lowast
2)2
119889119909
minus intΩ
11988633120573(1199063minus 119906lowast
3)2
119889119909
minus intΩ
11988623(1199063minus 119906lowast
3) (1199062minus 119906lowast
2)
1198982+ 1199062
119889119909
minus intΩ
12057211988612(1199062minus 119906lowast
2) (1199061minus 119906lowast
1)
1198981+ 1199061
119889119909
+∬Ω
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus 119906
lowast
1) (1199062(119905 119909) minus 119906
lowast
2)
(1198981+ 1199061(119904 119910)) (119898
1+ 119906lowast1)
119889119904 119889119910 119889119909
+ 120573∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321198982(1199062(119904 119910) minus 119906
lowast
2) (1199063(119905 119909) minus 119906
lowast
3)
(1198982+ 1199062(119904 119910)) (119898
2+ 119906lowast2)
119889119904 119889119910 119889119909
(25)
Applying the inequality 119886119887 le 1205981198862 + (14120598)1198872 we derive from(25) that
Φ2(119905) le minusint
Ω
120572[11988611minus
11988612119906lowast
2
(1198981+ 1199061) (1198981+ 119906lowast1)] (1199061minus 119906lowast
1)2
119889119909
minus intΩ
11988633120573(1199063minus 119906lowast
3)2
119889119909
minus intΩ
[11988622minus
11988623119906lowast
3
(1198982+ 1199062) (1198982+ 119906lowast2)] (1199062minus 119906lowast
2)2
119889119909
+ intΩ
12057211988612
1198981+ 1199061
[1205981(1199061minus 119906lowast
1)2
+1
41205981
(1199062minus 119906lowast
2)2
] 119889119909
+ intΩ
11988623
1198982+ 1199062
[1
41205982
(1199062minus 119906lowast
2)2
+ 1205982(1199063minus 119906lowast
3)2
] 119889119909
+∬Ω
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981
(1198981+ 1199061(119904 119910)) (119898
1+ 119906lowast1)
times [1205983(1199061(119904 119910) minus 119906
lowast
1)2
+1
41205983
(1199062(119905 119909) minus 119906
lowast
2)2
] 119889119904 119889119910 119889119909
+∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321205731198982
(1198982+ 1199062(119904 119910)) (119898
2+ 119906lowast2)
times [1
41205984
(1199062(119904 119910) minus 119906
lowast
2)2
+1205984(1199063(119905 119909) minus 119906
lowast
3)2
] 119889119904 119889119910 119889119909
(26)
According to the property of the Kernel functions119870119894(119909 119910 119905)
(119894 = 1 2) we know that
Φ2(119905) le minusint
Ω
[12057211988611minus12057211988612119906lowast
2
11989821
minus120572119886121205981
1198981
] (1199061minus 119906lowast
1)2
119889119909
minus intΩ
[11988622minus11988623119906lowast
3
11989822
minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
] (1199062minus 119906lowast
2)2
119889119909
minus intΩ
[11988633120573 minus
119886231205982
1198982
minus119886321205731205984
1198982
] (1199063minus 119906lowast
3)2
119889119909
+119886211205983
1198981
∬Ω
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times (1199061(119904 119910) minus 119906
lowast
1)2
119889119904 119889119910 119889119909
+11988632120573
411989821205984
∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times (1199062(119904 119910) minus 119906
lowast
2)2
119889119904 119889119910 119889119909
(27)
Define a new Lyapunov functional as follows
119864 (119905) = 119881 (119905)
+119886211205983
1198981
∬Ω
int
+infin
0
int
119905
119905minus119903
1198701(119909 119910 119903)
times (1199061(119904 119910) minus 119906
lowast
1)2
119889119904 119889119903 119889119910 119889119909
+11988632120573
411989821205984
∬Ω
int
+infin
0
int
119905
119905minus119903
1198702(119909 119910 119903)
times (1199062(119904 119910) minus 119906
lowast
2)2
119889119904 119889119903 119889119910 119889119909
(28)
Abstract and Applied Analysis 7
It is derived from (27) and (28) that
1198641015840(119905) le minusint
Ω
(1205721198891119906lowast
1
11990621
1003816100381610038161003816nabla119906110038161003816100381610038162
+1198892119906lowast
2
11990622
1003816100381610038161003816nabla119906210038161003816100381610038162
+1205731198893119906lowast
3
11990623
1003816100381610038161003816nabla119906310038161003816100381610038162
)119889119909
minus [12057211988611minus12057211988612119906lowast
2
11989821
minus120572119886121205981
1198981
minus119886211205983
1198981
]intΩ
(1199061minus 119906lowast
1)2
119889119909
minus [11988622minus11988623119906lowast
3
11989822
minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632120573
411989821205984
]intΩ
(1199062minus 119906lowast
2)2
119889119909
minus [11988633120573 minus
119886231205982
1198982
minus120573119886321205984
1198982
]intΩ
(1199063minus 119906lowast
3)2
119889119909
(29)
Since (1199061 1199062 1199063) is the unique positive solution of system (1)
Using Proposition 5 there exists a constant119862which does notdepend on119909 isin Ω or 119905 ge 0 such that 119906
119894(sdot 119905)infinle 119862 (119894 = 1 2 3)
for 119905 ge 0 By Theorem 1198602in [24] we have
1003817100381710038171003817119906119894 (sdot 119905)10038171003817100381710038171198622120572(Ω) le 119862 forall119905 ge 1 (30)
Assume that
1198661 119886111198982
1gt 11988612119906lowast
2
1198662 119886221198982
2gt 11988623119906lowast
3+21198862311988632
11988633
+211988612119886211198982
2
1198861111989821minus 11988612119906lowast2
(31)
and denote
1198971= 12057211988611minus12057211988612119906lowast
2
11989821
minus120572119886121205981
1198981
minus119886211205983
1198981
1198972= 11988622minus11988623119906lowast
3
11989822
minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632120573
411989821205984
1198973= 11988633120573 minus
119886231205982
1198982
minus120573119886321205984
1198982
(32)
Then (29) is transformed into
1198641015840(119905) le minusint
Ω
(1205721198891119906lowast
1
11990621
1003816100381610038161003816nabla119906110038161003816100381610038162
+1198892119906lowast
2
11990622
1003816100381610038161003816nabla119906210038161003816100381610038162
+1205731198893119906lowast
3
11990623
1003816100381610038161003816nabla119906310038161003816100381610038162
)119889119909
minus
3
sum
119894=1
119897119894intΩ
(119906119894minus 119906lowast
119894)2
119889119909
(33)
If we choose
120572 =11988621
11988612
120573 =11988623
11988632
1205981= 1205983=119886111198982
1minus 11988612119906lowast
2
4119898111988612
1205982= 1205984=119898211988633
411988632
(34)
then 119897119894gt 0 (119894 = 1 2 3) Therefore we have
1198641015840(119905) le minus
3
sum
119894=1
119897119894intΩ
(119906119894minus 119906lowast
119894)2
119889119909 (35)
FromProposition 6 we can see that the solution of system(1) and (3) is bounded and so are the derivatives of (119906
119894minus 119906lowast
119894)
(119894 = 1 2 3) by the equations in (1) Applying Lemma 1 weobtain that
lim119905rarrinfin
intΩ
(119906119894minus 119906lowast
119894) 119889119909 = 0 (119894 = 1 2 3) (36)
Recomputing 1198641015840(119905) gives
1198641015840(119905) le minusint
Ω
(1205721198891119906lowast
1
11990621
1003816100381610038161003816nabla119906110038161003816100381610038162
+1198892119906lowast
2
11990622
1003816100381610038161003816nabla119906210038161003816100381610038162
+1205731198893119906lowast
3
11990623
1003816100381610038161003816nabla119906310038161003816100381610038162
)119889119909
le minus119888intΩ
(1003816100381610038161003816nabla1199061
10038161003816100381610038162
+1003816100381610038161003816nabla1199062
10038161003816100381610038162
+1003816100381610038161003816nabla1199063
10038161003816100381610038162
) 119889119909 = minus119892 (119905)
(37)
where 119888 = min1205721198891119906lowast
11198722
1 1198892119906lowast
21198722
2 1205731198893119906lowast
311987223 Using (30)
and (1) we obtain that the derivative of 119892(119905) is bounded in[1 +infin) From Lemma 1 we conclude that 119892(119905) rarr 0 as 119905 rarrinfin Therefore
lim119905rarrinfin
intΩ
(1003816100381610038161003816nabla1199061
10038161003816100381610038162
+1003816100381610038161003816nabla1199062
10038161003816100381610038162
+1003816100381610038161003816nabla1199063
10038161003816100381610038162
) 119889119909 = 0 (38)
Applying the Poincare inequality
intΩ
120582|119906 minus 119906|2119889119909 le int
Ω
|nabla119906|2119889119909 (39)
leads to
lim119905rarrinfin
intΩ
(119906119894minus 119906119894)2
119889119909 = 0 (119894 = 1 2 3) (40)
where 119906119894= (1|Ω|) int
Ω119906119894119889119909 and 120582 is the smallest positive
eigenvalue ofminusΔwith the homogeneousNeumann conditionTherefore
|Ω| (1199061 (119905) minus 119906lowast
1)2
= intΩ
(1199061(119905) minus 119906
lowast
1)2
119889119909
= intΩ
(1199061(119905) minus 119906
1(119905 119909) + 119906
1(119905 119909) minus 119906
lowast
1)2
119889119909
le 2intΩ
(1199061(119905) minus 119906
1(119905 119909))
2
119889119909
+ 2intΩ
(1199061(119905 119909) minus 119906
lowast
1)2
119889119909
(41)
8 Abstract and Applied Analysis
So we have 1199061(119905) rarr 119906
lowast
1as 119905 rarr infin Similarly 119906
2(119905) rarr 119906
lowast
2
and 1199063(119905) rarr 119906
lowast
3as 119905 rarr infin According to (30) there exists
a subsequence 119905119898 and non-negative functions 119908
119894isin 1198622(Ω)
such thatlim119898rarrinfin
1003817100381710038171003817119906119894 (sdot 119905119898) minus 119908119894 (sdot)10038171003817100381710038171198622(Ω) = 0 (119894 = 1 2 3) (42)
Applying (40) and noting that 119906119894(119905) rarr 119906
lowast
119894 we then have119908
119894=
119906lowast
119894 (119894 = 1 2 3) That is
lim119898rarrinfin
1003817100381710038171003817119906119894 (sdot 119905119898) minus 119906lowast
119894
10038171003817100381710038171198622(Ω) = 0 (119894 = 1 2 3) (43)
Furthermore the local stability ofElowast combiningwith (43)gives the following global stability
Theorem 9 Assume that
11986751198861(11988621minus 1198862)
11989811198862
gt 11988611gt11988612Vlowast2
11989821
(44)
1198674and 119866
4hold and then the semi-trivial steady state 119864
2
of system (1)ndash(3) with non-trivial initial functions is globallyasymptotically stable
Proof It is obvious that system (1)ndash(3) always has two non-negative equilibria (119906
1 1199062 1199063) as follows 119864
0= (0 0 0) and
1198641= (119886111988611 0 0) If 119867
1and 119867
4are satisfied system (1) has
the other semitrivial solution denoted by 1198642(Vlowast1 Vlowast2 0) where
1198674 11988631198982gt (11988632minus 1198863) Vlowast
2 (45)
We consider the stability of 1198642under condition 119867
1and
1198674 Equation (1) can be rewritten as
1205971199061
120597119905minus 1198891Δ1199061= 1199061[minus11988611(1199061minus Vlowast
1) minus11988612(1199062minus Vlowast2)
1198981+ 1199061
+11988612Vlowast2(1199061minus Vlowast1)
(1198981+ 1199061) (1198981+ Vlowast1)]
1205971199062
120597119905minus 1198892Δ1199062
= 1199062[intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus Vlowast
1)
(1198981+ 1199061(119904 119910)) (119898
1+ Vlowast1)119889119904 119889119910
minus11988622(1199062minus Vlowast
2) minus
119886231199063
1198982+ 1199062
]
1205971199063
120597119905minus 1198893Δ1199063
= 1199063[intΩ
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321198982(1199062(119904 119910) minus Vlowast
2)
(1198982+ 1199062(119904 119910)) (119898
2+ Vlowast2)119889119904 119889119910
minus119886331199063]
(46)
Similar to the argument ofTheorem 8 we have 1199061(119909 119905) rarr Vlowast
1
and 1199062(119909 119905) rarr Vlowast
2as 119905 rarr infin uniformly on Ω provided that
the following additional condition holds
1198663 119886111198982
1gt 11988612Vlowast
2
1198664 119886221198982
2gt1198862311988632
11988633
+211988612119886211198982
2
1198861111989821minus 11988612Vlowast2
(47)
Next we consider the asymptotic behavior of 1199063(119905 119909) Let
120579 =1198863minus (11988632minus 1198863) Vlowast2
2120575 120575 =
100381610038161003816100381611988632 minus 11988631003816100381610038161003816
(48)
Then there exists 1199051gt 0 such that
Vlowast
2minus 120579 lt 119906
2(119905 119909) lt V
lowast
2+ 120579 forall119905 gt 119905
1 119909 isin Ω (49)
Consider the following two systems
1199083119905minus 1198893Δ1199083= 1199083[minus1198863+11988632(Vlowast2+ 120579)
1198982+ Vlowast2+ 120579
minus 119886331199083]
(119905 119909) isin [1199051 +infin) times Ω
1205971199083
120597]= 0 (119905 119909) isin [119905
1 +infin) times 120597Ω
1199083= 1199063 (119905 119909) isin (minusinfin 119905
1] times Ω
1198823119905minus 1198893Δ1198823= 1198823[minus1198863+11988632(Vlowast2minus 120579)
1198982+ Vlowast2minus 120579
minus 11988633V3]
(119905 119909) isin [1199051 +infin) times Ω
1205971198823
120597]= 0 (119905 119909) isin [119905
1 +infin) times 120597Ω
1198823= 1199063 (119905 119909) isin (minusinfin 119905
1] times Ω
(50)
Combining comparison principle with (50) we obtain that
1198823(119905 119909) le 119906
3(119905 119909) le 119908
3(119905 119909) forall119905 gt 119905
1 119909 isin Ω (51)
By Lemma 4 we obtain
0 = lim119905rarrinfin
1198823(119905 119909) le inf 119906
3(119905 119909) le sup 119906
3(119905 119909)
le lim119905rarrinfin
1199083(119905 119909) = 0 forall119909 isin Ω
(52)
which implies that lim119905rarrinfin
1199063(119905 119909) = 0 uniformly onΩ
Theorem 10 Suppose that1198665and119866
6hold and then the semi-
trivial steady state 1198641of system (1)ndash(3) with non-trivial initial
functions is globally asymptotically stable
Abstract and Applied Analysis 9
Proof We study the stability of the semi-trivial solution 1198641=
(1 0 0) Similarly the equations in (1) can be written as
1205971199061
120597119905minus 1198891Δ1199061= 1199061[minus11988611(1199061minus 1) minus
119886121199062
1198981+ 1199061
]
1205971199062
120597119905minus 1198892Δ1199062
= 1199062[minus1198862+ intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus
1)
(1198981+ 1199061(119904 119910)) (119898
1+ 1)119889119904 119889119910
+119886211
1198981+ 1
minus 119886221199062minus119886231199063
1198982+ 1199062
]
1205971199063
120597119905minus 1198893Δ1199063
= 1199063[minus1198863+ intΩ
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321199062(119904 119910)
1198982+ 1199062(119904 119910)
119889119904 119889119910 minus 119886331199063]
(53)
Define
119881 (119905) = 120572intΩ
[1199061minus 1minus 1log 11990611
] 119889119909 + intΩ
1199062119889119909
+ 120573intΩ
1199063119889119909
(54)
Calculating the derivative of 119881(119905) along 1198641 we get from (54)
that
1198811015840(119905) le minusint
Ω
[120572(11988611minus119886121205981
1198981
)] (1199061minus 1)2
119889119909
+ intΩ
[11988622minus12057211988612
411989811205981
minus11988623
412059821198982
minus11988621
411989811205983
] 1199062
2119889119909
+ intΩ
[12057311988633minus119886231205982
1198982
] 1199062
3119889119909
minus intΩ
[(1198862minus119886211
1198981+ 1
)1199062+ 11988631199063] 119889119909
+119886211205983
1198981
intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times (1199061(119904 119910) minus
1)2
119889119904 119889119910 119889119909
+11988632120573
412059841198982
∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times 1199062
2(119904 119910) 119889119904 119889119910 119889119909
(55)
Define119864 (119905) = 119881 (119905)
+119886211205983
1198981
∬Ω
int
+infin
0
int
119905
119905minus119903
1198701(119909 119910 119903)
times (1199061(119904 119910) minus 119906
lowast
1)2
119889119904 119889119903 119889119910 119889119909
+11988632120573
412059841198982
∬Ω
int
+infin
0
int
119905
119905minus119903
1198702(119909 119910 119903)
times (1199062(119904 119910) minus 119906
lowast
2)2
119889119904 119889119903 119889119910 119889119909
(56)
It is easy to see that
1198641015840(119905) le minus120572119889
11intΩ
1003816100381610038161003816nabla119906110038161003816100381610038162
11990621
119889119909
minus intΩ
[120572(11988611minus119886121205981
1198981
) minus119886211205983
1198981
] (1199061minus 1)2
119889119909
+ intΩ
[11988622minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632
411989821205984
] 1199062
2119889119909
+ intΩ
[12057311988633minus119886231205982
1198982
minus119886321205731205984
1198982
] 1199062
3119889119909
minus intΩ
[(1198862minus119886211
1198981+ 1
)1199062minus 11988631205731199063] 119889119909
(57)
Assume that1198665 1198861(11988621minus 1198862) lt 119898
1119886211988611
1198666 119886221198982
2gt1198862311988632
11988633
+211988612119886211198982
2
1198861111989821
(58)
Let
1198971= 12057211988611minus120572119886121205981
1198981
minus119886211205983
1198981
1198972= 11988622minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632120573
411989821205984
1198973= 11988633120573 minus
119886231205982
1198982
minus120573119886321205984
1198982
(59)
Choose
120572 =11988621
11988612
120573 =11988623
11988632
1205981= 1205983=119886111198981
411988612
1205982= 1205984=119898211988633
411988632
(60)
Then we get 119897119894gt 0 (119894 = 1 2 3) Therefore we have
lim119905rarrinfin
1199061(119905 119909) =
1 (61)
uniformly onΩ
10 Abstract and Applied Analysis
tx
u1
02
46
8
01
23
40
1
2
3
4
(a)
02
46
8
01
23
40
1
2
3
4
t
x
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 2 Global stability of the positive equilibrium 119864lowast with an initial condition (1206011 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos 119909 27 + 05 sin119909)
Next we consider the asymptotic behavior of 1199062(119905 119909) and
1199063(119905 119909) For any 119879 gt 0 integrating (57) over [0 119879] yields
119864 (119879) + 1205721198891
10038171003817100381710038171003817100381710038171003817
10038161003816100381610038161003816100381610038161003816
nabla1199061
1199061
10038161003816100381610038161003816100381610038161003816
10038171003817100381710038171003817100381710038171003817
2
2
+ 1198971
100381710038171003817100381710038161003816100381610038161199061 minus 1
100381610038161003816100381610038171003817100381710038172
2
+ 1198972
100381710038171003817100381710038161003816100381610038161199062100381610038161003816100381610038171003817100381710038172
2+ 1198973
100381710038171003817100381710038161003816100381610038161199063100381610038161003816100381610038171003817100381710038172
2le 119864 (0)
(62)
where |119906119894|2
2= int119879
0intΩ1199062
119894119889119909 119889119905 It implies that |119906
119894|2le 119862119894(119894 =
2 3) for the constant 119862119894which is independent of 119879 Now we
consider the boundedness of |nabla1199062|2and |nabla119906
3|2 From the
Greenrsquos identity we obtain
1198892int
119879
0
intΩ
1003816100381610038161003816nabla1199062 (119905 119909)10038161003816100381610038162
119889119909 119889119905
= minus1198892int
119879
0
intΩ
1199062(119905 119909) Δ119906
2(119905 119909) 119889119909 119889119905
= minusint
119879
0
intΩ
1199062
1205971199062
120597119905119889119909 119889119905 minus 119886
2int
119879
0
intΩ
1199062
2119889119909 119889119905
minus 11988622int
119879
0
intΩ
1199063
2119889119909 119889119905 minus 119886
23int
119879
0
intΩ
1199062
21199063
1198982+ 1199063
119889119909 119889119905
+ 11988621int
119879
0
∬Ω
int
119905
minusinfin
1199061(119904 119910) 119906
2
2(119905 119909)
1198981+ 1199061(119904 119910)
times 1198701(119909 119910 119905 minus 119904) 119889119904 119889119910 119889119909 119889119905
(63)
Note that
int
119879
0
intΩ
1199062(119905 119909)
1205971199062(119905 119909)
120597119905119889119909 119889119905
= int
119879
0
intΩ
1
2
1205971199062
2
120597119905119889119905 119889119909
=1
2intΩ
1199062
2(119879 119909) 119889119909 minus
1
2intΩ
1199062
2(0 119909) 119889119909 le 119872
2
2|Ω|
int
119879
0
intΩ
1199062
21199063119889119909 119889119905 le 119872
3int
119879
0
intΩ
1199062
2119889119909 119889119905
Abstract and Applied Analysis 11
02
46
8
01
23
40
1
2
3
4
tx
u1
(a)
02
46
8
01
23
40
1
2
3
4
tx
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 3 Global stability of the positive equilibrium 1198642with an initial condition (120601
1 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos 119909 27 + 05 sin119909)
int
119879
0
intΩ
1199063
2119889119909 119889119905 le 119872
2int
119879
0
intΩ
1199062
2119889119909 119889119905
int
119879
0
∬Ω
int
119905
minusinfin
1199061(119904 119910) 119906
2
2(119905 119909)
1198981+ 1199061(119904 119910)
1198701(119909 119910 119905 minus 119904) 119889119904 119889119910 119889119909 119889119905
le int
119879
0
intΩ
1199062
2(119905 119909) 119889119909 119889119905
(64)
Thus we get |nabla1199062|2le 1198624 In a similar way we have
|nabla1199063|2le 1198625 Here 119862
4and 119862
5are independent of 119879
It is easy to see that 1199062(119905 119909) 119906
3(119905 119909) isin 119871
2((0infin)
11988212(Ω)) These imply that
lim119905rarrinfin
1003817100381710038171003817119906119894 (sdot 119905)100381710038171003817100381711988212(Ω) = 0 119894 = 2 3 (65)
From the Sobolev compact embedding theorem we know
lim119905rarrinfin
1003817100381710038171003817119906119894 (sdot 119905)1003817100381710038171003817119862(Ω) = 0 119894 = 2 3 (66)
In the end we show that the trivial solution 1198640is an
unstable equilibrium Similarly to the local stability to 119864lowast wecan get the characteristic equation of 119864
0as
(120582 + 1205831198941198631minus 1198861) (120582 + 120583
1198941198632+ 1198862) (120582 + 120583
1198941198633+ 1198863) = 0 (67)
If 119894 = 1 then 1205831= 0 It is easy to see that this equation admits
a positive solution 120582 = 1198861 According to Theorem 51 in [23]
we have the following result
Theorem 11 The trivial equilibrium 1198640is an unstable equilib-
rium of system (1)ndash(3)
4 Numerical Illustrations
In this section we performnumerical simulations to illustratethe theoretical results given in Section 3
12 Abstract and Applied Analysis
u1
02
46
8
01
23
40
1
2
3
4
tx
(a)
02
46
8
01
23
40
1
2
3
4
tx
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 4 Global stability of the positive equilibrium 1198641with an initial condition (120601
1 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos119909 27 + 05 sin119909)
In the following we always take Ω = [0 120587] 119870119894(119909 119910 119905) =
119866119894(119909 119910 119905)119896
119894(119905) where 119896
119894(119905) = (1120591
lowast)119890minus119905120591lowast
(119894 = 1 2) and
1198661(119909 119910 119905) =
1
120587+2
120587
infin
sum
119899=1
119890minus11988921198992
119905 cos 119899119909 cos 119899119910
1198662(119909 119910 119905) =
1
120587+2
120587
infin
sum
119899=1
119890minus11988931198992
119905 cos 119899119909 cos 119899119910
(68)
However it is difficult for us to simulate our results directlybecause of the nonlocal term Similar to [25] the equationsin (1) can be rewritten as follows
1205971199061
120597119905minus 1198891Δ1199061= 1199061(1198861minus 119886111199061minus119886121199062
1198981+ 1199061
)
1205971199062
120597119905minus 1198892Δ1199062= 1199062(minus1198862+ 11988621V1minus 119886221199062minus119886231199063
1198982+ 1199063
)
1205971199063
120597119905minus 1198893Δ1199063= 1199063(minus1198863+ 11988632V2minus 119886331199063)
120597V1
120597119905minus 1198892ΔV1=1
120591lowast(
1199061
1198981+ 1199061
minus V1)
120597V2
120597119905minus 1198893ΔV2=1
120591lowast(
1199062
1198982+ 1199062
minus V2)
(69)
where
V119894= int
120587
0
int
119905
minusinfin
119866119894(119909 119910 119905 minus 119904)
1
120591lowast119890minus(119905minus119904)120591
lowast 119906119894(119904 119910)
119898119894+ 119906119894(119904 119910)
119889119910 119889119904
119894 = 1 2
(70)
Each component is considered with homogeneous Neumannboundary conditions and the initial condition of V
119894is
V119894(0 119909) = int
120587
0
int
0
minusinfin
119866119894(119909 119910 minus119904)
1
120591lowast119890119904120591lowast 119906119894(119904 119910)
119898119894+ 119906119894(119904 119910)
119889119910 119889119904
119894 = 1 2
(71)
Abstract and Applied Analysis 13
In the following examples we fix some coefficients andassume that 119889
1= 1198892= 1198893= 1 119886
1= 3 119886
12= 1 119898
1= 1198982=
1 1198862= 1 119886
22= 12 119886
23= 1 119886
33= 12 and 120591lowast = 1 The
asymptotic behaviors of system (1)ndash(3) are shownby choosingdifferent coefficients 119886
11 11988621 1198863 and 119886
32
Example 12 Let 11988611= 539 119886
21= 8113 119886
3= 113 and
11988632= 14Then it is easy to see that the system admits a unique
positive equilibrium 119864lowast(12 112 112) By Theorem 8 we
see that the positive solution (1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of
system (1)ndash(3) converges to 119864lowast as 119905 rarr infin See Figure 2
Example 13 Let 11988611
= 6 11988621
= 12 1198863= 14 and
11988632= 1 Clearly 119867
2does not hold Hence the positive
steady state is not feasible System (1) admits two semi-trivial steady state 119864
2(04732 02372 0) and 119864
1(12 0 0)
According to Theorem 9 we know that the positive solution(1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of system (1)ndash(3) converges to 119864
2
as 119905 rarr infin See Figure 3
Example 14 Let 11988611= 6 119886
21= 2 119886
3= 14 and 119886
32= 1
Clearly1198671does not hold Hence 119864lowast and 119864
2are not feasible
System (1) has a unique semi-trivial steady state 1198641(12 0 0)
According to Theorem 10 we know that the positive solution(1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of system (1)ndash(3) converges to 119864
1
as 119905 rarr infin See Figure 4
5 Discussion
In this paper we incorporate nonlocal delay into a three-species food chain model with Michaelis-Menten functionalresponse to represent a delay due to the gestation of the preda-tor The conditions under which the spatial homogeneousequilibria are asymptotically stable are given by using theLyapunov functional
We now summarize the ecological meanings of ourtheoretical results Firstly the positive equilibrium 119864
lowast ofsystem (1)ndash(3) exists under the high birth rate of the prey(1198861) and low death rates (119886
2and 119886
3) of predator and top
predator 119864lowast is globally stable if the intraspecific competition11988611is neither too big nor too small and the maximum harvest
rates 11988612 11988623
are small enough Secondly the semi-trivialequilibrium 119864
2of system (1)ndash(3) exists if the birth rate of the
prey (1198861) is high death rate of predator 119886
2is low and the death
rate (1198863) exceeds the conversion rate from predator to top
predator (11988632) 1198642is globally stable if the maximum harvest
rates 11988612 11988623are small and the intra-specific competition 119886
11
is neither too big nor too small Thirdly system (1)ndash(3) hasonly one semi-trivial equilibrium 119864
1when the death rate (119886
2)
exceeds the conversion rate from prey to predator (11988621) 1198641
is globally stable if intra-specific competitions (11988611 11988622 and
11988633) are strong Finally 119864
0is unstable and the non-stability of
trivial equilibrium tells us that not all of the populations goto extinction Furthermore our main results imply that thenonlocal delay is harmless for stabilities of all non-negativesteady states of system (1)ndash(3)
There are still many interesting and challenging problemswith respect to system (1)ndash(3) for example the permanence
and stability of periodic solution or almost periodic solutionThese problems are clearly worthy for further investigations
Acknowledgments
This work is partially supported by PRC Grants NSFC11102076 and 11071209 andNSF of theHigher Education Insti-tutions of Jiangsu Province (12KJD110002 and 12KJD110008)
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] H F Huo andW T Li ldquoPeriodic solution of a delayed predator-prey system with Michaelis-Menten type functional responserdquoJournal of Computational and AppliedMathematics vol 166 no2 pp 453ndash463 2004
[3] Z Lin and M Pedersen ldquoStability in a diffusive food-chainmodel with Michaelis-Menten functional responserdquo NonlinearAnalysis Theory Methods amp Applications vol 57 no 3 pp 421ndash433 2004
[4] B X Dai N Zhang and J Z Zou ldquoPermanence for theMichaelis-Menten type discrete three-species ratio-dependentfood chain model with delayrdquo Journal of Mathematical Analysisand Applications vol 324 no 1 pp 728ndash738 2006
[5] N F Britton ldquoSpatial structures and periodic travelling wavesin an integro-differential reaction-diffusion population modelrdquoSIAM Journal on Applied Mathematics vol 50 no 6 pp 1663ndash1688 1990
[6] S A Gourley and N F Britton ldquoInstability of travelling wavesolutions of a population model with nonlocal effectsrdquo IMAJournal of Applied Mathematics vol 51 no 3 pp 299ndash310 1993
[7] X Zhang and R Xu ldquoTraveling waves of a diffusive predator-prey model with nonlocal delay and stage structurerdquo Journal ofMathematical Analysis and Applications vol 373 no 2 pp 475ndash484 2011
[8] Z C Wang W T Li and S G Ruan ldquoExistence and stability oftraveling wave fronts in reaction advection diffusion equationswith nonlocal delayrdquo Journal of Differential Equations vol 238no 1 pp 153ndash200 2007
[9] S A Gourley and S G Ruan ldquoConvergence and travellingfronts in functional differential equations with nonlocal termsa competition modelrdquo SIAM Journal on Mathematical Analysisvol 35 no 3 pp 806ndash822 2003
[10] X H Lai and T X Yao ldquoExponential stability of impul-sive delayed reaction-diffusion cellular neural networks viaPoincare integral inequalityrdquoAbstract and Applied Analysis vol2013 Article ID 131836 10 pages 2013
[11] R Xu and Z E Ma ldquoGlobal stability of a reaction-diffusionpredator-prey model with a nonlocal delayrdquo Mathematical andComputer Modelling vol 50 no 1-2 pp 194ndash206 2009
[12] Z Li F D Chen and M X He ldquoAsymptotic behavior of thereaction-diffusion model of plankton allelopathy with nonlocaldelaysrdquo Nonlinear Analysis Real World Applications vol 12 no3 pp 1748ndash1758 2011
[13] Y Kuang and H L Smith ldquoGlobal stability in diffusive delayLotka-Volterra systemsrdquoDifferential and Integral Equations vol4 no 1 pp 117ndash128 1991
14 Abstract and Applied Analysis
[14] Z G Lin M Pedersen and L Zhang ldquoA predator-prey systemwith stage-structure for predator and nonlocal delayrdquoNonlinearAnalysis Theory Methods amp Applications vol 72 no 3-4 pp2019ndash2030 2010
[15] R Xu ldquoA reaction-diffusion predator-prey model with stagestructure and nonlocal delayrdquo Applied Mathematics and Com-putation vol 175 no 2 pp 984ndash1006 2006
[16] S S Chen and J P Shi ldquoStability and Hopf bifurcation in adiffusive logistic population model with nonlocal delay effectrdquoJournal of Differential Equations vol 253 no 12 pp 3440ndash34702012
[17] H I Freedman Deterministic Mathematical Models in Popula-tion Ecology vol 57 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1980
[18] P A Abrams and L R Ginzburg ldquoThe nature of predation preydepedent ratio dependent or neitherrdquo Trends in Ecology andEvolution vol 15 no 8 pp 337ndash341 2000
[19] D L DeAngelis R A Goldstein and R V OrsquoNeill ldquoA modelfor trophic iterationrdquo Ecology vol 56 no 4 pp 881ndash892 1975
[20] C V Pao ldquoDynamics of nonlinear parabolic systems with timedelaysrdquo Journal of Mathematical Analysis and Applications vol198 no 3 pp 751ndash779 1996
[21] F Xu Y Hou and Z G Lin ldquoTime delay parabolic system ina predator-prey model with stage structurerdquo Acta MathematicaSinica vol 48 no 6 pp 1121ndash1130 2005 (Chinese)
[22] P Y H Pang and M X Wang ldquoStrategy and stationary patternin a three-species predator-prey modelrdquo Journal of DifferentialEquations vol 200 no 2 pp 245ndash273 2004
[23] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981
[24] K J Brown P C Dunne and R A Gardner ldquoA semilinearparabolic system arising in the theory of superconductivityrdquoJournal of Differential Equations vol 40 no 2 pp 232ndash252 1981
[25] S A Gourley and J W H So ldquoDynamics of a food-limitedpopulation model incorporating nonlocal delays on a finitedomainrdquo Journal of Mathematical Biology vol 44 no 1 pp 49ndash78 2002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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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
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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
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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
Lemma 4 (see [21]) Let 119906(119905 119909) isin 119862([0infin) times Ω) cap
11986221((0infin)timesΩ) be the nontrivial positive solution of the system
119906119905minus 119889Δ119906 = 119861119906 (119905 minus 120591 119909) plusmn 119860
1119906 (119905 119909)
minus 11986021199062(119905 119909) (119905 119909) isin (0infin) times Ω
120597119906
120597]= 0 (119905 119909) isin (0infin) times 120597Ω
119906 (119905 119909) = 120601 (119905 119909) ge 0 (119905 119909) isin (minus120591 0) times Ω
(6)
where 1198601ge 0 119861 119860
2 120591 gt 0 The following results hold
(1) if 119861plusmn1198601gt 0 then 119906 rarr (119861 plusmn119860
1)1198602uniformly onΩ
as 119905 rarr infin
(2) if 119861plusmn1198601le 0 then 119906 rarr 0 uniformly onΩ as 119905 rarr infin
Throughout this paper we assume that
119870119894(119909 119910 119905) = 119866
119894(119909 119910 119905) 119896
119894(119905) 119909 119910 isin Ω 119896
119894(119905) ge 0
intΩ
119866119894(119909 119910 119905) 119889119909 = int
Ω
119866119894(119909 119910 119905) 119889119910 = 1 119905 ge 0
int
+infin
0
119896119894(119905) 119889119905 = 1 119905119896
119894(119905) isin 119871
1((0 +infin) 119877)
(7)
where119866119894(119909 119910 119905) is a nonnegative function which is continu-
ous in (119909 119910) isin ΩtimesΩ for each 119905 isin [0 +infin) and measurable in119905 isin [0 +infin) for each pair (119909 119910) isin Ω times Ω
Now we prove the following propositions which will beused in the sequel
Proposition 5 For any nonnegative initial function thecorresponding solution of system (1)ndash(3) is nonnegative
Proof Suppose that (1199061 1199062 1199063) is a solution and satisfies 119906
119894isin
119862([0 119879) times Ω) cap 11986212((0 119879) times Ω) with 119879 le +infin Choose 0 lt
120591 lt 119879 Then
1199061119905minus 1198891Δ1199061= 11986011199061 0 lt 119905 lt 120591 119909 isin Ω
1199062119905minus 1198892Δ1199062= 11986021199062 0 lt 119905 lt 120591 119909 isin Ω
1199063119905minus 1198893Δ1199063= 11986031199063 0 lt 119905 lt 120591 119909 isin Ω
120597119906119894
120597]= 0 0 lt 119905 lt 120591 119909 isin 120597Ω
119906119894(119905 119909) ge 0 (119894 = 1 2 3) 119905 le 0 119909 isin Ω
(8)
where
1198601= 1198861minus 119886111199061minus119886121199062
1198981+ 1199061
1198602= minus1198862+ 11988621intΩ
int
119905
minusinfin
1199061(119904 119910)
1198981+ 1199061(119904 119910)
times 1198701(119909 119910 119905 minus 119904) 119889119904 119889119910
minus 119886221199062minus119886231199063
1198982+ 1199063
1198603= minus1198863+ 11988632intΩ
int
119905
minusinfin
1199062(119904 119910)
1198982+ 1199062(119904 119910)
times 1198702(119909 119910 119905 minus 119904) 119889119904 119889119910 minus 119886
331199063
(9)
It follows from Lemma 2 that 119906119894ge 0 (119905 119909) isin [0 120591] times Ω Due
to the arbitrariness of 120591 we have 119906119894ge 0 for (119905 119909) isin [0 119879) times
Ω
Proposition 6 System (1)ndash(3) with initial functions 120601119894admits
a unique global solution (1199061 1199062 1199063) satisfying 0 le 119906
119894(119905 119909) le
119872119894for 119894 = 1 2 3 where119872
119894is defined as
1198721= max 1198861
11988611
10038171003817100381710038171206011 (119905 119909)
1003817100381710038171003817119871infin((minusinfin0]timesΩ)
1198722= max
(11988621minus 1198862)1198721minus 11989811198862
11988622(1198981+1198721)
10038171003817100381710038171206012 (119905 119909)
1003817100381710038171003817119871infin((minusinfin0]timesΩ)
1198723= max
(11988632minus 1198863)1198722minus 11989821198863
11988622(1198982+1198722)
10038171003817100381710038171206013 (119909)
1003817100381710038171003817119871infin(Ω)
(10)
Proof It follows from standard PDE theory that there existsa 119879 gt 0 such that problem (1)ndash(3) admits a unique solutionin [0 119879) times Ω From Lemma 2 we know that 119906
119894(119905 119909) ge 0 for
(119905 119909) isin [0 119879) times Ω Considering the first equation in (1) wehave
1205971199061
120597119905minus 1198891Δ1199061le 1199061(1198861minus 119886111199061) (11)
Using themaximumprinciple gives that 1199061le 1198721 In a similar
way we have 119906119894le 119872119894for 119894 = 2 3 It is easy to see that (0 0 0)
and (119872111987221198723) are a pair of coupled upper and lower
solutions to system (1)ndash(3) from the direct computation Invirtue of Lemma 3 system (1)ndash(3) admits a unique globalsolution (119906
1 1199062 1199063) satisfying 0 le 119906
119894(119905 119909) le 119872
119894for 119894 = 1 2 3
In addition if 1206011(119905 119909) 120601
2(119905 119909) 120601
3(119909) (119905 119909) isin (minusinfin 0] times Ω is
nontrivial it follows from the strongmaximumprinciple that119906119894(119905 119909) gt 0 (119894 = 1 2 3) for all 119905 gt 0 119909 isin Ω
3 Global Stability
In this section we study the asymptotic behavior of theequilibrium of system (1)ndash(3) In the beginning we show theexistence and uniqueness of positive steady state
4 Abstract and Applied Analysis
L1
L2
L3
025
02
015
01
005
0
0 02 04 06 081
2
Figure 1 Graphs of the equations in (12) 1198713is the boundary
condition (1198863= 11988632V2(1198982+ V2)) and 119871
2corresponds to minus119886
2+
11988621(V1(1198981+ V1)) minus 119886
22V2= 0 while 119871
1corresponds to 119886
1minus 11988611V1minus
11988612V2(1198981+ V1) = 0 Intersection point between 119871
1and 119871
2gives the
positive solution (Vlowast1 Vlowast2) to (12) Parameter values are listed in the
example in Section 4
Let us consider the following equations
1198861minus 11988611V1minus11988612V2
1198981+ V1
= 0
minus 1198862+11988621V1
1198981+ V1
minus 11988622V2= 0
(12)
A direct computation shows that the above equations haveonly one positive solution (Vlowast
1 Vlowast2) if and only if
1198671 1198861(11988621minus 1198862) gt 119898
1119886211988611 (13)
is satisfied Taking1198671into account we consider the equation
1198863= 11988632Vlowast2(1198982+ Vlowast2) which corresponds to 119871
3in Figure 1
It is clear that if 1198863lt 11988632Vlowast2(1198982+ Vlowast2) the intersection
point (Vlowast1 Vlowast2) always lies in 119871
3 Let
1198672 11988631198982lt (11988632minus 1198863) Vlowast
2 (14)
Suppose that1198671and119867
2hold We take 119906
3as a parameter and
consider the following system
1198861minus 119886111199061minus119886121199062
1198981+ 1199061
= 0
minus 1198862+119886211199061
1198981+ 1199061
minus 119886221199062minus119886231199063
1198982+ 1199062
= 0
minus 1198863+119886321199062
1198982+ 1199062
minus 119886331199063= 0
(15)
When the parameter 1199063is sufficiently small the first two
equations in (15) can be approximated by (12) Moreover
by continuously increasing the value of 1199063 1198713goes up and
meanwhile the intersection point between 1198711and 119871
2also
goes up However 1198713goes up faster than 119871
2 while 119871
1keeps
still In other words there exists a critical value 1199063such
that the intersection point lies in 1198713 which implies that
there is a unique positive solution 119864lowast(119906lowast1 119906lowast
2 119906lowast
3) to (15) or
equivalently (1)ndash(3)
Lemma 7 Assume that1198672and 119866
2hold and then the positive
steady state119864lowast of system (1)ndash(3) is locally asymptotically stable
Proof We get the local stability of 119864lowast by performing alinearization and analyzing the corresponding characteristicequation Similarly as in [22] let 0 lt 120583
1lt 1205832lt 1205833lt
be the eigenvalues of minusΔ on Ω with the homogeneousNeumann boundary condition Let 119864(120583
119894) be the eigenspace
corresponding to 120583119894in 1198621(Ω) for 119894 = 1 2 3 Let
X = u = (1199061 1199062 1199063) isin [119862
1(Ω)]3
120597120578u = 0 119909 isin 120597Ω (16)
120593119894119895 119895 = 1 dim119864(120583
119894) be an orthonormal basis of 119864(120583
119894)
and X119894119895= 119888 sdot 120593
119894119895| 119888 isin 119877
3 Then
X119894=
dim119864(120583119894)
⨁
119895=1
X119894119895 X =
infin
⨁
119894=1
X119894 (17)
Let 119863 = diag(1198631 1198632 1198633) and k = (V
1 V2 V3) V119894=
119906119894minus 119906lowast
119894 (119894 = 1 2 3) Then the linearization of (1)
is k119905= 119871k = 119863Δk + Fk(Elowast)k Fk(Elowast)k = 119888
119894119895
where 119888119894119894
= 119887119894119894V119894 119894 = 1 2 3 119888
12= 11988712V2 11988823
=
11988723V3 11988821= 11988721intΩint119905
minusinfinV1(119904 119910)119870
1(119909 119910 119905 minus 119904)119889119904 119889119910 and 119888
32=
11988732intΩint119905
minusinfinV2(119904 119910)119870
2(119909 119910 119905 minus 119904)119889119904 119889119910 The coefficients 119887
119894119895are
defined as follows
11988711= minus11988611119906lowast
1+11988612119906lowast
1119906lowast
2
(1198981+ 119906lowast1)2
11988712= minus
11988612119906lowast
1
1198981+ 119906lowast1
11988713= 0
11988721=119886211198981119906lowast
2
(1198981+ 119906lowast1)2
11988722= minus11988622119906lowast
2+11988623119906lowast
2119906lowast
3
(1198982+ 119906lowast2)2 119887
23= minus
11988623119906lowast
2
1198982+ 119906lowast2
11988731= 0 119887
32=119886321198982119906lowast
3
(1198982+ 119906lowast2)2 119887
33= minus11988633119906lowast
3
(18)
SinceX119894is invariant under the operator 119871 for each 119894 ge 1 then
the operator 119871 on X119894is k119905= 119871k = 119863120583
119894k + Fk(Elowast)k Let V119894 =
119888119894119890120582119905(119894 = 1 2 3) and we can get the characteristic equation
Abstract and Applied Analysis 5
120593119894(120582) = 120582
3+ 1198601198941205822+ 119861119894120582 + 119862
119894= 0 where 119860
119894 119861119894 and 119862
119894are
defined as follows
119860119894= (1198891+ 1198892+ 1198893) 120583119894minus 11988711minus 11988722minus 11988733
119861119894= (11988911198892+ 11988921198893+ 11988931198891) 1205832
119894
+ [1198893(minus11988711minus 11988722) + 1198891(minus11988733minus 11988722)
+1198892(minus11988711minus 11988733)] 120583119894
+ 1198871111988722+ 1198871111988733+ 1198872211988733
minus 1198871211988721int
infin
0
1198961(119904) 119890minus120582119904119889119904 minus 119887
2311988732int
infin
0
1198962(119904) 119890minus120582119904119889119904
119862119894= 1198891119889211988931205833
119894+ [minus1198873311988911198892minus 1198871111988921198893minus 1198872211988911198893] 1205832
119894
+ [11988711119887221198893+ 11988711119887331198892+ 11988722119887331198891+ 11988722119887331198891
minus 11988931198871211988721int
infin
0
1198961(119904) 119890minus120582119904119889119904
minus11988911198872311988732int
infin
0
1198962(119904) 119890minus120582119904119889119904] 120583119894
+ [minus119887111198872211988733+ 119887121198872111988733int
infin
0
1198961(119904) 119890minus120582119904119889119904
+119887111198872311988732int
infin
0
1198962(119904) 119890minus120582119904119889119904]
(19)
It is easy to see that 11988712 11988723 11988733lt 0 and 119887
21 11988732gt 0 It
follows from assumption 1198661and 119866
2that 119886
11lt 0 119886
22lt 0
So 119860119894gt 0 119861
119894gt 0 119862
119894gt 0 and 119860
119894119861119894minus 119862119894gt 0 for 119894 ge 1
from the direct calculation According to the Routh-Hurwitzcriterion the three roots 120582
1198941 1205821198942 1205821198943
of 120593119894(120582) = 0 all have
negative real partsBy continuity of the roots with respect to 120583
119894and Routh-
Hurwitz criterion we can conclude that there exists a positiveconstant 120576 such that
Re 1205821198941 Re 120582
1198942 Re 120582
1198943 le minus120576 119894 ge 1 (20)
Consequently the spectrum of 119871 consisting only of eigen-values lies in Re 120582 le minus120576 It is easy to see that Elowast islocally asymptotically stable and follows fromTheorem 511of [23]
Theorem 8 Assume that
11986731198861(11988621minus 1198862)
11989811198862
gt 11988611gt11988612119906lowast
2
11989821
(21)
1198672and 119866
2hold and the positive steady state 119864lowast of system (1)ndash
(3) with nontrivial initial function is globally asymptoticallystable
Proof It is easy to see that the equations in (1) can be rewrittenas
1205971199061
120597119905minus 1198891Δ1199061= 1199061[ minus 11988611(1199061minus 119906lowast
1)
minus11988612(1199062minus 119906lowast
2)
1198981+ 1199061
+11988612119906lowast
2(1199061minus 119906lowast
1)
(1198981+ 1199061) (1198981+ 119906lowast1)]
1205971199062
120597119905minus 1198892Δ1199062= 1199062[intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus 119906
lowast
1)
(1198981+ 1199061(119904 119910)) (119898
1+ 119906lowast1)119889119904 119889119910
minus 11988622(1199062minus 119906lowast
2) minus11988623(1199063minus 119906lowast
3)
1198982+ 1199062
+11988623119906lowast
3(1199062minus 119906lowast
2)
(1198982+ 1199062) (1198982+ 119906lowast2)]
1205971199063
120597119905minus 1198893Δ1199063= 1199063[intΩ
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321198982(1199062(119904 119910) minus 119906
lowast
2)
(1198982+ 1199062(119904 119910)) (119898
2+ 119906lowast2)119889119904 119889119910
minus11988633(1199063minus 119906lowast
3) ]
(22)
Define
119881 (119905) = 120572intΩ
(1199061minus 119906lowast
1minus 119906lowast
1ln 1199061119906lowast1
)119889119909
+ intΩ
(1199062minus 119906lowast
2minus 119906lowast
2ln 1199062119906lowast2
)119889119909
+ 120573intΩ
(1199062minus 119906lowast
3minus 119906lowast
3ln1199063
119906lowast3
)119889119909
(23)
where 120572 and 120573 are positive constants to be determinedCalculating the derivatives 119881(119905) along the positive solutionto the system (1)ndash(3) yields
1198811015840(119905) = Φ
1(119905) + Φ
2(119905) (24)
where
Φ1(119905) = minusint
Ω
(1205721198891119906lowast
1
11990621
1003816100381610038161003816nabla119906110038161003816100381610038162
+1198892119906lowast
2
11990622
1003816100381610038161003816nabla119906210038161003816100381610038162
+1205731198893119906lowast
3
11990623
1003816100381610038161003816nabla119906310038161003816100381610038162
)119889119909
6 Abstract and Applied Analysis
Φ2(119905) = minusint
Ω
120572[11988611minus
11988612119906lowast
2
(1198981+ 1199061) (1198981+ 119906lowast1)] (1199061minus 119906lowast
1)2
119889119909
minus intΩ
[11988622minus
11988623119906lowast
3
(1198982+ 1199062) (1198982+ 119906lowast2)] (1199062minus 119906lowast
2)2
119889119909
minus intΩ
11988633120573(1199063minus 119906lowast
3)2
119889119909
minus intΩ
11988623(1199063minus 119906lowast
3) (1199062minus 119906lowast
2)
1198982+ 1199062
119889119909
minus intΩ
12057211988612(1199062minus 119906lowast
2) (1199061minus 119906lowast
1)
1198981+ 1199061
119889119909
+∬Ω
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus 119906
lowast
1) (1199062(119905 119909) minus 119906
lowast
2)
(1198981+ 1199061(119904 119910)) (119898
1+ 119906lowast1)
119889119904 119889119910 119889119909
+ 120573∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321198982(1199062(119904 119910) minus 119906
lowast
2) (1199063(119905 119909) minus 119906
lowast
3)
(1198982+ 1199062(119904 119910)) (119898
2+ 119906lowast2)
119889119904 119889119910 119889119909
(25)
Applying the inequality 119886119887 le 1205981198862 + (14120598)1198872 we derive from(25) that
Φ2(119905) le minusint
Ω
120572[11988611minus
11988612119906lowast
2
(1198981+ 1199061) (1198981+ 119906lowast1)] (1199061minus 119906lowast
1)2
119889119909
minus intΩ
11988633120573(1199063minus 119906lowast
3)2
119889119909
minus intΩ
[11988622minus
11988623119906lowast
3
(1198982+ 1199062) (1198982+ 119906lowast2)] (1199062minus 119906lowast
2)2
119889119909
+ intΩ
12057211988612
1198981+ 1199061
[1205981(1199061minus 119906lowast
1)2
+1
41205981
(1199062minus 119906lowast
2)2
] 119889119909
+ intΩ
11988623
1198982+ 1199062
[1
41205982
(1199062minus 119906lowast
2)2
+ 1205982(1199063minus 119906lowast
3)2
] 119889119909
+∬Ω
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981
(1198981+ 1199061(119904 119910)) (119898
1+ 119906lowast1)
times [1205983(1199061(119904 119910) minus 119906
lowast
1)2
+1
41205983
(1199062(119905 119909) minus 119906
lowast
2)2
] 119889119904 119889119910 119889119909
+∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321205731198982
(1198982+ 1199062(119904 119910)) (119898
2+ 119906lowast2)
times [1
41205984
(1199062(119904 119910) minus 119906
lowast
2)2
+1205984(1199063(119905 119909) minus 119906
lowast
3)2
] 119889119904 119889119910 119889119909
(26)
According to the property of the Kernel functions119870119894(119909 119910 119905)
(119894 = 1 2) we know that
Φ2(119905) le minusint
Ω
[12057211988611minus12057211988612119906lowast
2
11989821
minus120572119886121205981
1198981
] (1199061minus 119906lowast
1)2
119889119909
minus intΩ
[11988622minus11988623119906lowast
3
11989822
minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
] (1199062minus 119906lowast
2)2
119889119909
minus intΩ
[11988633120573 minus
119886231205982
1198982
minus119886321205731205984
1198982
] (1199063minus 119906lowast
3)2
119889119909
+119886211205983
1198981
∬Ω
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times (1199061(119904 119910) minus 119906
lowast
1)2
119889119904 119889119910 119889119909
+11988632120573
411989821205984
∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times (1199062(119904 119910) minus 119906
lowast
2)2
119889119904 119889119910 119889119909
(27)
Define a new Lyapunov functional as follows
119864 (119905) = 119881 (119905)
+119886211205983
1198981
∬Ω
int
+infin
0
int
119905
119905minus119903
1198701(119909 119910 119903)
times (1199061(119904 119910) minus 119906
lowast
1)2
119889119904 119889119903 119889119910 119889119909
+11988632120573
411989821205984
∬Ω
int
+infin
0
int
119905
119905minus119903
1198702(119909 119910 119903)
times (1199062(119904 119910) minus 119906
lowast
2)2
119889119904 119889119903 119889119910 119889119909
(28)
Abstract and Applied Analysis 7
It is derived from (27) and (28) that
1198641015840(119905) le minusint
Ω
(1205721198891119906lowast
1
11990621
1003816100381610038161003816nabla119906110038161003816100381610038162
+1198892119906lowast
2
11990622
1003816100381610038161003816nabla119906210038161003816100381610038162
+1205731198893119906lowast
3
11990623
1003816100381610038161003816nabla119906310038161003816100381610038162
)119889119909
minus [12057211988611minus12057211988612119906lowast
2
11989821
minus120572119886121205981
1198981
minus119886211205983
1198981
]intΩ
(1199061minus 119906lowast
1)2
119889119909
minus [11988622minus11988623119906lowast
3
11989822
minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632120573
411989821205984
]intΩ
(1199062minus 119906lowast
2)2
119889119909
minus [11988633120573 minus
119886231205982
1198982
minus120573119886321205984
1198982
]intΩ
(1199063minus 119906lowast
3)2
119889119909
(29)
Since (1199061 1199062 1199063) is the unique positive solution of system (1)
Using Proposition 5 there exists a constant119862which does notdepend on119909 isin Ω or 119905 ge 0 such that 119906
119894(sdot 119905)infinle 119862 (119894 = 1 2 3)
for 119905 ge 0 By Theorem 1198602in [24] we have
1003817100381710038171003817119906119894 (sdot 119905)10038171003817100381710038171198622120572(Ω) le 119862 forall119905 ge 1 (30)
Assume that
1198661 119886111198982
1gt 11988612119906lowast
2
1198662 119886221198982
2gt 11988623119906lowast
3+21198862311988632
11988633
+211988612119886211198982
2
1198861111989821minus 11988612119906lowast2
(31)
and denote
1198971= 12057211988611minus12057211988612119906lowast
2
11989821
minus120572119886121205981
1198981
minus119886211205983
1198981
1198972= 11988622minus11988623119906lowast
3
11989822
minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632120573
411989821205984
1198973= 11988633120573 minus
119886231205982
1198982
minus120573119886321205984
1198982
(32)
Then (29) is transformed into
1198641015840(119905) le minusint
Ω
(1205721198891119906lowast
1
11990621
1003816100381610038161003816nabla119906110038161003816100381610038162
+1198892119906lowast
2
11990622
1003816100381610038161003816nabla119906210038161003816100381610038162
+1205731198893119906lowast
3
11990623
1003816100381610038161003816nabla119906310038161003816100381610038162
)119889119909
minus
3
sum
119894=1
119897119894intΩ
(119906119894minus 119906lowast
119894)2
119889119909
(33)
If we choose
120572 =11988621
11988612
120573 =11988623
11988632
1205981= 1205983=119886111198982
1minus 11988612119906lowast
2
4119898111988612
1205982= 1205984=119898211988633
411988632
(34)
then 119897119894gt 0 (119894 = 1 2 3) Therefore we have
1198641015840(119905) le minus
3
sum
119894=1
119897119894intΩ
(119906119894minus 119906lowast
119894)2
119889119909 (35)
FromProposition 6 we can see that the solution of system(1) and (3) is bounded and so are the derivatives of (119906
119894minus 119906lowast
119894)
(119894 = 1 2 3) by the equations in (1) Applying Lemma 1 weobtain that
lim119905rarrinfin
intΩ
(119906119894minus 119906lowast
119894) 119889119909 = 0 (119894 = 1 2 3) (36)
Recomputing 1198641015840(119905) gives
1198641015840(119905) le minusint
Ω
(1205721198891119906lowast
1
11990621
1003816100381610038161003816nabla119906110038161003816100381610038162
+1198892119906lowast
2
11990622
1003816100381610038161003816nabla119906210038161003816100381610038162
+1205731198893119906lowast
3
11990623
1003816100381610038161003816nabla119906310038161003816100381610038162
)119889119909
le minus119888intΩ
(1003816100381610038161003816nabla1199061
10038161003816100381610038162
+1003816100381610038161003816nabla1199062
10038161003816100381610038162
+1003816100381610038161003816nabla1199063
10038161003816100381610038162
) 119889119909 = minus119892 (119905)
(37)
where 119888 = min1205721198891119906lowast
11198722
1 1198892119906lowast
21198722
2 1205731198893119906lowast
311987223 Using (30)
and (1) we obtain that the derivative of 119892(119905) is bounded in[1 +infin) From Lemma 1 we conclude that 119892(119905) rarr 0 as 119905 rarrinfin Therefore
lim119905rarrinfin
intΩ
(1003816100381610038161003816nabla1199061
10038161003816100381610038162
+1003816100381610038161003816nabla1199062
10038161003816100381610038162
+1003816100381610038161003816nabla1199063
10038161003816100381610038162
) 119889119909 = 0 (38)
Applying the Poincare inequality
intΩ
120582|119906 minus 119906|2119889119909 le int
Ω
|nabla119906|2119889119909 (39)
leads to
lim119905rarrinfin
intΩ
(119906119894minus 119906119894)2
119889119909 = 0 (119894 = 1 2 3) (40)
where 119906119894= (1|Ω|) int
Ω119906119894119889119909 and 120582 is the smallest positive
eigenvalue ofminusΔwith the homogeneousNeumann conditionTherefore
|Ω| (1199061 (119905) minus 119906lowast
1)2
= intΩ
(1199061(119905) minus 119906
lowast
1)2
119889119909
= intΩ
(1199061(119905) minus 119906
1(119905 119909) + 119906
1(119905 119909) minus 119906
lowast
1)2
119889119909
le 2intΩ
(1199061(119905) minus 119906
1(119905 119909))
2
119889119909
+ 2intΩ
(1199061(119905 119909) minus 119906
lowast
1)2
119889119909
(41)
8 Abstract and Applied Analysis
So we have 1199061(119905) rarr 119906
lowast
1as 119905 rarr infin Similarly 119906
2(119905) rarr 119906
lowast
2
and 1199063(119905) rarr 119906
lowast
3as 119905 rarr infin According to (30) there exists
a subsequence 119905119898 and non-negative functions 119908
119894isin 1198622(Ω)
such thatlim119898rarrinfin
1003817100381710038171003817119906119894 (sdot 119905119898) minus 119908119894 (sdot)10038171003817100381710038171198622(Ω) = 0 (119894 = 1 2 3) (42)
Applying (40) and noting that 119906119894(119905) rarr 119906
lowast
119894 we then have119908
119894=
119906lowast
119894 (119894 = 1 2 3) That is
lim119898rarrinfin
1003817100381710038171003817119906119894 (sdot 119905119898) minus 119906lowast
119894
10038171003817100381710038171198622(Ω) = 0 (119894 = 1 2 3) (43)
Furthermore the local stability ofElowast combiningwith (43)gives the following global stability
Theorem 9 Assume that
11986751198861(11988621minus 1198862)
11989811198862
gt 11988611gt11988612Vlowast2
11989821
(44)
1198674and 119866
4hold and then the semi-trivial steady state 119864
2
of system (1)ndash(3) with non-trivial initial functions is globallyasymptotically stable
Proof It is obvious that system (1)ndash(3) always has two non-negative equilibria (119906
1 1199062 1199063) as follows 119864
0= (0 0 0) and
1198641= (119886111988611 0 0) If 119867
1and 119867
4are satisfied system (1) has
the other semitrivial solution denoted by 1198642(Vlowast1 Vlowast2 0) where
1198674 11988631198982gt (11988632minus 1198863) Vlowast
2 (45)
We consider the stability of 1198642under condition 119867
1and
1198674 Equation (1) can be rewritten as
1205971199061
120597119905minus 1198891Δ1199061= 1199061[minus11988611(1199061minus Vlowast
1) minus11988612(1199062minus Vlowast2)
1198981+ 1199061
+11988612Vlowast2(1199061minus Vlowast1)
(1198981+ 1199061) (1198981+ Vlowast1)]
1205971199062
120597119905minus 1198892Δ1199062
= 1199062[intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus Vlowast
1)
(1198981+ 1199061(119904 119910)) (119898
1+ Vlowast1)119889119904 119889119910
minus11988622(1199062minus Vlowast
2) minus
119886231199063
1198982+ 1199062
]
1205971199063
120597119905minus 1198893Δ1199063
= 1199063[intΩ
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321198982(1199062(119904 119910) minus Vlowast
2)
(1198982+ 1199062(119904 119910)) (119898
2+ Vlowast2)119889119904 119889119910
minus119886331199063]
(46)
Similar to the argument ofTheorem 8 we have 1199061(119909 119905) rarr Vlowast
1
and 1199062(119909 119905) rarr Vlowast
2as 119905 rarr infin uniformly on Ω provided that
the following additional condition holds
1198663 119886111198982
1gt 11988612Vlowast
2
1198664 119886221198982
2gt1198862311988632
11988633
+211988612119886211198982
2
1198861111989821minus 11988612Vlowast2
(47)
Next we consider the asymptotic behavior of 1199063(119905 119909) Let
120579 =1198863minus (11988632minus 1198863) Vlowast2
2120575 120575 =
100381610038161003816100381611988632 minus 11988631003816100381610038161003816
(48)
Then there exists 1199051gt 0 such that
Vlowast
2minus 120579 lt 119906
2(119905 119909) lt V
lowast
2+ 120579 forall119905 gt 119905
1 119909 isin Ω (49)
Consider the following two systems
1199083119905minus 1198893Δ1199083= 1199083[minus1198863+11988632(Vlowast2+ 120579)
1198982+ Vlowast2+ 120579
minus 119886331199083]
(119905 119909) isin [1199051 +infin) times Ω
1205971199083
120597]= 0 (119905 119909) isin [119905
1 +infin) times 120597Ω
1199083= 1199063 (119905 119909) isin (minusinfin 119905
1] times Ω
1198823119905minus 1198893Δ1198823= 1198823[minus1198863+11988632(Vlowast2minus 120579)
1198982+ Vlowast2minus 120579
minus 11988633V3]
(119905 119909) isin [1199051 +infin) times Ω
1205971198823
120597]= 0 (119905 119909) isin [119905
1 +infin) times 120597Ω
1198823= 1199063 (119905 119909) isin (minusinfin 119905
1] times Ω
(50)
Combining comparison principle with (50) we obtain that
1198823(119905 119909) le 119906
3(119905 119909) le 119908
3(119905 119909) forall119905 gt 119905
1 119909 isin Ω (51)
By Lemma 4 we obtain
0 = lim119905rarrinfin
1198823(119905 119909) le inf 119906
3(119905 119909) le sup 119906
3(119905 119909)
le lim119905rarrinfin
1199083(119905 119909) = 0 forall119909 isin Ω
(52)
which implies that lim119905rarrinfin
1199063(119905 119909) = 0 uniformly onΩ
Theorem 10 Suppose that1198665and119866
6hold and then the semi-
trivial steady state 1198641of system (1)ndash(3) with non-trivial initial
functions is globally asymptotically stable
Abstract and Applied Analysis 9
Proof We study the stability of the semi-trivial solution 1198641=
(1 0 0) Similarly the equations in (1) can be written as
1205971199061
120597119905minus 1198891Δ1199061= 1199061[minus11988611(1199061minus 1) minus
119886121199062
1198981+ 1199061
]
1205971199062
120597119905minus 1198892Δ1199062
= 1199062[minus1198862+ intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus
1)
(1198981+ 1199061(119904 119910)) (119898
1+ 1)119889119904 119889119910
+119886211
1198981+ 1
minus 119886221199062minus119886231199063
1198982+ 1199062
]
1205971199063
120597119905minus 1198893Δ1199063
= 1199063[minus1198863+ intΩ
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321199062(119904 119910)
1198982+ 1199062(119904 119910)
119889119904 119889119910 minus 119886331199063]
(53)
Define
119881 (119905) = 120572intΩ
[1199061minus 1minus 1log 11990611
] 119889119909 + intΩ
1199062119889119909
+ 120573intΩ
1199063119889119909
(54)
Calculating the derivative of 119881(119905) along 1198641 we get from (54)
that
1198811015840(119905) le minusint
Ω
[120572(11988611minus119886121205981
1198981
)] (1199061minus 1)2
119889119909
+ intΩ
[11988622minus12057211988612
411989811205981
minus11988623
412059821198982
minus11988621
411989811205983
] 1199062
2119889119909
+ intΩ
[12057311988633minus119886231205982
1198982
] 1199062
3119889119909
minus intΩ
[(1198862minus119886211
1198981+ 1
)1199062+ 11988631199063] 119889119909
+119886211205983
1198981
intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times (1199061(119904 119910) minus
1)2
119889119904 119889119910 119889119909
+11988632120573
412059841198982
∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times 1199062
2(119904 119910) 119889119904 119889119910 119889119909
(55)
Define119864 (119905) = 119881 (119905)
+119886211205983
1198981
∬Ω
int
+infin
0
int
119905
119905minus119903
1198701(119909 119910 119903)
times (1199061(119904 119910) minus 119906
lowast
1)2
119889119904 119889119903 119889119910 119889119909
+11988632120573
412059841198982
∬Ω
int
+infin
0
int
119905
119905minus119903
1198702(119909 119910 119903)
times (1199062(119904 119910) minus 119906
lowast
2)2
119889119904 119889119903 119889119910 119889119909
(56)
It is easy to see that
1198641015840(119905) le minus120572119889
11intΩ
1003816100381610038161003816nabla119906110038161003816100381610038162
11990621
119889119909
minus intΩ
[120572(11988611minus119886121205981
1198981
) minus119886211205983
1198981
] (1199061minus 1)2
119889119909
+ intΩ
[11988622minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632
411989821205984
] 1199062
2119889119909
+ intΩ
[12057311988633minus119886231205982
1198982
minus119886321205731205984
1198982
] 1199062
3119889119909
minus intΩ
[(1198862minus119886211
1198981+ 1
)1199062minus 11988631205731199063] 119889119909
(57)
Assume that1198665 1198861(11988621minus 1198862) lt 119898
1119886211988611
1198666 119886221198982
2gt1198862311988632
11988633
+211988612119886211198982
2
1198861111989821
(58)
Let
1198971= 12057211988611minus120572119886121205981
1198981
minus119886211205983
1198981
1198972= 11988622minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632120573
411989821205984
1198973= 11988633120573 minus
119886231205982
1198982
minus120573119886321205984
1198982
(59)
Choose
120572 =11988621
11988612
120573 =11988623
11988632
1205981= 1205983=119886111198981
411988612
1205982= 1205984=119898211988633
411988632
(60)
Then we get 119897119894gt 0 (119894 = 1 2 3) Therefore we have
lim119905rarrinfin
1199061(119905 119909) =
1 (61)
uniformly onΩ
10 Abstract and Applied Analysis
tx
u1
02
46
8
01
23
40
1
2
3
4
(a)
02
46
8
01
23
40
1
2
3
4
t
x
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 2 Global stability of the positive equilibrium 119864lowast with an initial condition (1206011 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos 119909 27 + 05 sin119909)
Next we consider the asymptotic behavior of 1199062(119905 119909) and
1199063(119905 119909) For any 119879 gt 0 integrating (57) over [0 119879] yields
119864 (119879) + 1205721198891
10038171003817100381710038171003817100381710038171003817
10038161003816100381610038161003816100381610038161003816
nabla1199061
1199061
10038161003816100381610038161003816100381610038161003816
10038171003817100381710038171003817100381710038171003817
2
2
+ 1198971
100381710038171003817100381710038161003816100381610038161199061 minus 1
100381610038161003816100381610038171003817100381710038172
2
+ 1198972
100381710038171003817100381710038161003816100381610038161199062100381610038161003816100381610038171003817100381710038172
2+ 1198973
100381710038171003817100381710038161003816100381610038161199063100381610038161003816100381610038171003817100381710038172
2le 119864 (0)
(62)
where |119906119894|2
2= int119879
0intΩ1199062
119894119889119909 119889119905 It implies that |119906
119894|2le 119862119894(119894 =
2 3) for the constant 119862119894which is independent of 119879 Now we
consider the boundedness of |nabla1199062|2and |nabla119906
3|2 From the
Greenrsquos identity we obtain
1198892int
119879
0
intΩ
1003816100381610038161003816nabla1199062 (119905 119909)10038161003816100381610038162
119889119909 119889119905
= minus1198892int
119879
0
intΩ
1199062(119905 119909) Δ119906
2(119905 119909) 119889119909 119889119905
= minusint
119879
0
intΩ
1199062
1205971199062
120597119905119889119909 119889119905 minus 119886
2int
119879
0
intΩ
1199062
2119889119909 119889119905
minus 11988622int
119879
0
intΩ
1199063
2119889119909 119889119905 minus 119886
23int
119879
0
intΩ
1199062
21199063
1198982+ 1199063
119889119909 119889119905
+ 11988621int
119879
0
∬Ω
int
119905
minusinfin
1199061(119904 119910) 119906
2
2(119905 119909)
1198981+ 1199061(119904 119910)
times 1198701(119909 119910 119905 minus 119904) 119889119904 119889119910 119889119909 119889119905
(63)
Note that
int
119879
0
intΩ
1199062(119905 119909)
1205971199062(119905 119909)
120597119905119889119909 119889119905
= int
119879
0
intΩ
1
2
1205971199062
2
120597119905119889119905 119889119909
=1
2intΩ
1199062
2(119879 119909) 119889119909 minus
1
2intΩ
1199062
2(0 119909) 119889119909 le 119872
2
2|Ω|
int
119879
0
intΩ
1199062
21199063119889119909 119889119905 le 119872
3int
119879
0
intΩ
1199062
2119889119909 119889119905
Abstract and Applied Analysis 11
02
46
8
01
23
40
1
2
3
4
tx
u1
(a)
02
46
8
01
23
40
1
2
3
4
tx
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 3 Global stability of the positive equilibrium 1198642with an initial condition (120601
1 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos 119909 27 + 05 sin119909)
int
119879
0
intΩ
1199063
2119889119909 119889119905 le 119872
2int
119879
0
intΩ
1199062
2119889119909 119889119905
int
119879
0
∬Ω
int
119905
minusinfin
1199061(119904 119910) 119906
2
2(119905 119909)
1198981+ 1199061(119904 119910)
1198701(119909 119910 119905 minus 119904) 119889119904 119889119910 119889119909 119889119905
le int
119879
0
intΩ
1199062
2(119905 119909) 119889119909 119889119905
(64)
Thus we get |nabla1199062|2le 1198624 In a similar way we have
|nabla1199063|2le 1198625 Here 119862
4and 119862
5are independent of 119879
It is easy to see that 1199062(119905 119909) 119906
3(119905 119909) isin 119871
2((0infin)
11988212(Ω)) These imply that
lim119905rarrinfin
1003817100381710038171003817119906119894 (sdot 119905)100381710038171003817100381711988212(Ω) = 0 119894 = 2 3 (65)
From the Sobolev compact embedding theorem we know
lim119905rarrinfin
1003817100381710038171003817119906119894 (sdot 119905)1003817100381710038171003817119862(Ω) = 0 119894 = 2 3 (66)
In the end we show that the trivial solution 1198640is an
unstable equilibrium Similarly to the local stability to 119864lowast wecan get the characteristic equation of 119864
0as
(120582 + 1205831198941198631minus 1198861) (120582 + 120583
1198941198632+ 1198862) (120582 + 120583
1198941198633+ 1198863) = 0 (67)
If 119894 = 1 then 1205831= 0 It is easy to see that this equation admits
a positive solution 120582 = 1198861 According to Theorem 51 in [23]
we have the following result
Theorem 11 The trivial equilibrium 1198640is an unstable equilib-
rium of system (1)ndash(3)
4 Numerical Illustrations
In this section we performnumerical simulations to illustratethe theoretical results given in Section 3
12 Abstract and Applied Analysis
u1
02
46
8
01
23
40
1
2
3
4
tx
(a)
02
46
8
01
23
40
1
2
3
4
tx
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 4 Global stability of the positive equilibrium 1198641with an initial condition (120601
1 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos119909 27 + 05 sin119909)
In the following we always take Ω = [0 120587] 119870119894(119909 119910 119905) =
119866119894(119909 119910 119905)119896
119894(119905) where 119896
119894(119905) = (1120591
lowast)119890minus119905120591lowast
(119894 = 1 2) and
1198661(119909 119910 119905) =
1
120587+2
120587
infin
sum
119899=1
119890minus11988921198992
119905 cos 119899119909 cos 119899119910
1198662(119909 119910 119905) =
1
120587+2
120587
infin
sum
119899=1
119890minus11988931198992
119905 cos 119899119909 cos 119899119910
(68)
However it is difficult for us to simulate our results directlybecause of the nonlocal term Similar to [25] the equationsin (1) can be rewritten as follows
1205971199061
120597119905minus 1198891Δ1199061= 1199061(1198861minus 119886111199061minus119886121199062
1198981+ 1199061
)
1205971199062
120597119905minus 1198892Δ1199062= 1199062(minus1198862+ 11988621V1minus 119886221199062minus119886231199063
1198982+ 1199063
)
1205971199063
120597119905minus 1198893Δ1199063= 1199063(minus1198863+ 11988632V2minus 119886331199063)
120597V1
120597119905minus 1198892ΔV1=1
120591lowast(
1199061
1198981+ 1199061
minus V1)
120597V2
120597119905minus 1198893ΔV2=1
120591lowast(
1199062
1198982+ 1199062
minus V2)
(69)
where
V119894= int
120587
0
int
119905
minusinfin
119866119894(119909 119910 119905 minus 119904)
1
120591lowast119890minus(119905minus119904)120591
lowast 119906119894(119904 119910)
119898119894+ 119906119894(119904 119910)
119889119910 119889119904
119894 = 1 2
(70)
Each component is considered with homogeneous Neumannboundary conditions and the initial condition of V
119894is
V119894(0 119909) = int
120587
0
int
0
minusinfin
119866119894(119909 119910 minus119904)
1
120591lowast119890119904120591lowast 119906119894(119904 119910)
119898119894+ 119906119894(119904 119910)
119889119910 119889119904
119894 = 1 2
(71)
Abstract and Applied Analysis 13
In the following examples we fix some coefficients andassume that 119889
1= 1198892= 1198893= 1 119886
1= 3 119886
12= 1 119898
1= 1198982=
1 1198862= 1 119886
22= 12 119886
23= 1 119886
33= 12 and 120591lowast = 1 The
asymptotic behaviors of system (1)ndash(3) are shownby choosingdifferent coefficients 119886
11 11988621 1198863 and 119886
32
Example 12 Let 11988611= 539 119886
21= 8113 119886
3= 113 and
11988632= 14Then it is easy to see that the system admits a unique
positive equilibrium 119864lowast(12 112 112) By Theorem 8 we
see that the positive solution (1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of
system (1)ndash(3) converges to 119864lowast as 119905 rarr infin See Figure 2
Example 13 Let 11988611
= 6 11988621
= 12 1198863= 14 and
11988632= 1 Clearly 119867
2does not hold Hence the positive
steady state is not feasible System (1) admits two semi-trivial steady state 119864
2(04732 02372 0) and 119864
1(12 0 0)
According to Theorem 9 we know that the positive solution(1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of system (1)ndash(3) converges to 119864
2
as 119905 rarr infin See Figure 3
Example 14 Let 11988611= 6 119886
21= 2 119886
3= 14 and 119886
32= 1
Clearly1198671does not hold Hence 119864lowast and 119864
2are not feasible
System (1) has a unique semi-trivial steady state 1198641(12 0 0)
According to Theorem 10 we know that the positive solution(1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of system (1)ndash(3) converges to 119864
1
as 119905 rarr infin See Figure 4
5 Discussion
In this paper we incorporate nonlocal delay into a three-species food chain model with Michaelis-Menten functionalresponse to represent a delay due to the gestation of the preda-tor The conditions under which the spatial homogeneousequilibria are asymptotically stable are given by using theLyapunov functional
We now summarize the ecological meanings of ourtheoretical results Firstly the positive equilibrium 119864
lowast ofsystem (1)ndash(3) exists under the high birth rate of the prey(1198861) and low death rates (119886
2and 119886
3) of predator and top
predator 119864lowast is globally stable if the intraspecific competition11988611is neither too big nor too small and the maximum harvest
rates 11988612 11988623
are small enough Secondly the semi-trivialequilibrium 119864
2of system (1)ndash(3) exists if the birth rate of the
prey (1198861) is high death rate of predator 119886
2is low and the death
rate (1198863) exceeds the conversion rate from predator to top
predator (11988632) 1198642is globally stable if the maximum harvest
rates 11988612 11988623are small and the intra-specific competition 119886
11
is neither too big nor too small Thirdly system (1)ndash(3) hasonly one semi-trivial equilibrium 119864
1when the death rate (119886
2)
exceeds the conversion rate from prey to predator (11988621) 1198641
is globally stable if intra-specific competitions (11988611 11988622 and
11988633) are strong Finally 119864
0is unstable and the non-stability of
trivial equilibrium tells us that not all of the populations goto extinction Furthermore our main results imply that thenonlocal delay is harmless for stabilities of all non-negativesteady states of system (1)ndash(3)
There are still many interesting and challenging problemswith respect to system (1)ndash(3) for example the permanence
and stability of periodic solution or almost periodic solutionThese problems are clearly worthy for further investigations
Acknowledgments
This work is partially supported by PRC Grants NSFC11102076 and 11071209 andNSF of theHigher Education Insti-tutions of Jiangsu Province (12KJD110002 and 12KJD110008)
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] H F Huo andW T Li ldquoPeriodic solution of a delayed predator-prey system with Michaelis-Menten type functional responserdquoJournal of Computational and AppliedMathematics vol 166 no2 pp 453ndash463 2004
[3] Z Lin and M Pedersen ldquoStability in a diffusive food-chainmodel with Michaelis-Menten functional responserdquo NonlinearAnalysis Theory Methods amp Applications vol 57 no 3 pp 421ndash433 2004
[4] B X Dai N Zhang and J Z Zou ldquoPermanence for theMichaelis-Menten type discrete three-species ratio-dependentfood chain model with delayrdquo Journal of Mathematical Analysisand Applications vol 324 no 1 pp 728ndash738 2006
[5] N F Britton ldquoSpatial structures and periodic travelling wavesin an integro-differential reaction-diffusion population modelrdquoSIAM Journal on Applied Mathematics vol 50 no 6 pp 1663ndash1688 1990
[6] S A Gourley and N F Britton ldquoInstability of travelling wavesolutions of a population model with nonlocal effectsrdquo IMAJournal of Applied Mathematics vol 51 no 3 pp 299ndash310 1993
[7] X Zhang and R Xu ldquoTraveling waves of a diffusive predator-prey model with nonlocal delay and stage structurerdquo Journal ofMathematical Analysis and Applications vol 373 no 2 pp 475ndash484 2011
[8] Z C Wang W T Li and S G Ruan ldquoExistence and stability oftraveling wave fronts in reaction advection diffusion equationswith nonlocal delayrdquo Journal of Differential Equations vol 238no 1 pp 153ndash200 2007
[9] S A Gourley and S G Ruan ldquoConvergence and travellingfronts in functional differential equations with nonlocal termsa competition modelrdquo SIAM Journal on Mathematical Analysisvol 35 no 3 pp 806ndash822 2003
[10] X H Lai and T X Yao ldquoExponential stability of impul-sive delayed reaction-diffusion cellular neural networks viaPoincare integral inequalityrdquoAbstract and Applied Analysis vol2013 Article ID 131836 10 pages 2013
[11] R Xu and Z E Ma ldquoGlobal stability of a reaction-diffusionpredator-prey model with a nonlocal delayrdquo Mathematical andComputer Modelling vol 50 no 1-2 pp 194ndash206 2009
[12] Z Li F D Chen and M X He ldquoAsymptotic behavior of thereaction-diffusion model of plankton allelopathy with nonlocaldelaysrdquo Nonlinear Analysis Real World Applications vol 12 no3 pp 1748ndash1758 2011
[13] Y Kuang and H L Smith ldquoGlobal stability in diffusive delayLotka-Volterra systemsrdquoDifferential and Integral Equations vol4 no 1 pp 117ndash128 1991
14 Abstract and Applied Analysis
[14] Z G Lin M Pedersen and L Zhang ldquoA predator-prey systemwith stage-structure for predator and nonlocal delayrdquoNonlinearAnalysis Theory Methods amp Applications vol 72 no 3-4 pp2019ndash2030 2010
[15] R Xu ldquoA reaction-diffusion predator-prey model with stagestructure and nonlocal delayrdquo Applied Mathematics and Com-putation vol 175 no 2 pp 984ndash1006 2006
[16] S S Chen and J P Shi ldquoStability and Hopf bifurcation in adiffusive logistic population model with nonlocal delay effectrdquoJournal of Differential Equations vol 253 no 12 pp 3440ndash34702012
[17] H I Freedman Deterministic Mathematical Models in Popula-tion Ecology vol 57 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1980
[18] P A Abrams and L R Ginzburg ldquoThe nature of predation preydepedent ratio dependent or neitherrdquo Trends in Ecology andEvolution vol 15 no 8 pp 337ndash341 2000
[19] D L DeAngelis R A Goldstein and R V OrsquoNeill ldquoA modelfor trophic iterationrdquo Ecology vol 56 no 4 pp 881ndash892 1975
[20] C V Pao ldquoDynamics of nonlinear parabolic systems with timedelaysrdquo Journal of Mathematical Analysis and Applications vol198 no 3 pp 751ndash779 1996
[21] F Xu Y Hou and Z G Lin ldquoTime delay parabolic system ina predator-prey model with stage structurerdquo Acta MathematicaSinica vol 48 no 6 pp 1121ndash1130 2005 (Chinese)
[22] P Y H Pang and M X Wang ldquoStrategy and stationary patternin a three-species predator-prey modelrdquo Journal of DifferentialEquations vol 200 no 2 pp 245ndash273 2004
[23] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981
[24] K J Brown P C Dunne and R A Gardner ldquoA semilinearparabolic system arising in the theory of superconductivityrdquoJournal of Differential Equations vol 40 no 2 pp 232ndash252 1981
[25] S A Gourley and J W H So ldquoDynamics of a food-limitedpopulation model incorporating nonlocal delays on a finitedomainrdquo Journal of Mathematical Biology vol 44 no 1 pp 49ndash78 2002
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
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OptimizationJournal of
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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4 Abstract and Applied Analysis
L1
L2
L3
025
02
015
01
005
0
0 02 04 06 081
2
Figure 1 Graphs of the equations in (12) 1198713is the boundary
condition (1198863= 11988632V2(1198982+ V2)) and 119871
2corresponds to minus119886
2+
11988621(V1(1198981+ V1)) minus 119886
22V2= 0 while 119871
1corresponds to 119886
1minus 11988611V1minus
11988612V2(1198981+ V1) = 0 Intersection point between 119871
1and 119871
2gives the
positive solution (Vlowast1 Vlowast2) to (12) Parameter values are listed in the
example in Section 4
Let us consider the following equations
1198861minus 11988611V1minus11988612V2
1198981+ V1
= 0
minus 1198862+11988621V1
1198981+ V1
minus 11988622V2= 0
(12)
A direct computation shows that the above equations haveonly one positive solution (Vlowast
1 Vlowast2) if and only if
1198671 1198861(11988621minus 1198862) gt 119898
1119886211988611 (13)
is satisfied Taking1198671into account we consider the equation
1198863= 11988632Vlowast2(1198982+ Vlowast2) which corresponds to 119871
3in Figure 1
It is clear that if 1198863lt 11988632Vlowast2(1198982+ Vlowast2) the intersection
point (Vlowast1 Vlowast2) always lies in 119871
3 Let
1198672 11988631198982lt (11988632minus 1198863) Vlowast
2 (14)
Suppose that1198671and119867
2hold We take 119906
3as a parameter and
consider the following system
1198861minus 119886111199061minus119886121199062
1198981+ 1199061
= 0
minus 1198862+119886211199061
1198981+ 1199061
minus 119886221199062minus119886231199063
1198982+ 1199062
= 0
minus 1198863+119886321199062
1198982+ 1199062
minus 119886331199063= 0
(15)
When the parameter 1199063is sufficiently small the first two
equations in (15) can be approximated by (12) Moreover
by continuously increasing the value of 1199063 1198713goes up and
meanwhile the intersection point between 1198711and 119871
2also
goes up However 1198713goes up faster than 119871
2 while 119871
1keeps
still In other words there exists a critical value 1199063such
that the intersection point lies in 1198713 which implies that
there is a unique positive solution 119864lowast(119906lowast1 119906lowast
2 119906lowast
3) to (15) or
equivalently (1)ndash(3)
Lemma 7 Assume that1198672and 119866
2hold and then the positive
steady state119864lowast of system (1)ndash(3) is locally asymptotically stable
Proof We get the local stability of 119864lowast by performing alinearization and analyzing the corresponding characteristicequation Similarly as in [22] let 0 lt 120583
1lt 1205832lt 1205833lt
be the eigenvalues of minusΔ on Ω with the homogeneousNeumann boundary condition Let 119864(120583
119894) be the eigenspace
corresponding to 120583119894in 1198621(Ω) for 119894 = 1 2 3 Let
X = u = (1199061 1199062 1199063) isin [119862
1(Ω)]3
120597120578u = 0 119909 isin 120597Ω (16)
120593119894119895 119895 = 1 dim119864(120583
119894) be an orthonormal basis of 119864(120583
119894)
and X119894119895= 119888 sdot 120593
119894119895| 119888 isin 119877
3 Then
X119894=
dim119864(120583119894)
⨁
119895=1
X119894119895 X =
infin
⨁
119894=1
X119894 (17)
Let 119863 = diag(1198631 1198632 1198633) and k = (V
1 V2 V3) V119894=
119906119894minus 119906lowast
119894 (119894 = 1 2 3) Then the linearization of (1)
is k119905= 119871k = 119863Δk + Fk(Elowast)k Fk(Elowast)k = 119888
119894119895
where 119888119894119894
= 119887119894119894V119894 119894 = 1 2 3 119888
12= 11988712V2 11988823
=
11988723V3 11988821= 11988721intΩint119905
minusinfinV1(119904 119910)119870
1(119909 119910 119905 minus 119904)119889119904 119889119910 and 119888
32=
11988732intΩint119905
minusinfinV2(119904 119910)119870
2(119909 119910 119905 minus 119904)119889119904 119889119910 The coefficients 119887
119894119895are
defined as follows
11988711= minus11988611119906lowast
1+11988612119906lowast
1119906lowast
2
(1198981+ 119906lowast1)2
11988712= minus
11988612119906lowast
1
1198981+ 119906lowast1
11988713= 0
11988721=119886211198981119906lowast
2
(1198981+ 119906lowast1)2
11988722= minus11988622119906lowast
2+11988623119906lowast
2119906lowast
3
(1198982+ 119906lowast2)2 119887
23= minus
11988623119906lowast
2
1198982+ 119906lowast2
11988731= 0 119887
32=119886321198982119906lowast
3
(1198982+ 119906lowast2)2 119887
33= minus11988633119906lowast
3
(18)
SinceX119894is invariant under the operator 119871 for each 119894 ge 1 then
the operator 119871 on X119894is k119905= 119871k = 119863120583
119894k + Fk(Elowast)k Let V119894 =
119888119894119890120582119905(119894 = 1 2 3) and we can get the characteristic equation
Abstract and Applied Analysis 5
120593119894(120582) = 120582
3+ 1198601198941205822+ 119861119894120582 + 119862
119894= 0 where 119860
119894 119861119894 and 119862
119894are
defined as follows
119860119894= (1198891+ 1198892+ 1198893) 120583119894minus 11988711minus 11988722minus 11988733
119861119894= (11988911198892+ 11988921198893+ 11988931198891) 1205832
119894
+ [1198893(minus11988711minus 11988722) + 1198891(minus11988733minus 11988722)
+1198892(minus11988711minus 11988733)] 120583119894
+ 1198871111988722+ 1198871111988733+ 1198872211988733
minus 1198871211988721int
infin
0
1198961(119904) 119890minus120582119904119889119904 minus 119887
2311988732int
infin
0
1198962(119904) 119890minus120582119904119889119904
119862119894= 1198891119889211988931205833
119894+ [minus1198873311988911198892minus 1198871111988921198893minus 1198872211988911198893] 1205832
119894
+ [11988711119887221198893+ 11988711119887331198892+ 11988722119887331198891+ 11988722119887331198891
minus 11988931198871211988721int
infin
0
1198961(119904) 119890minus120582119904119889119904
minus11988911198872311988732int
infin
0
1198962(119904) 119890minus120582119904119889119904] 120583119894
+ [minus119887111198872211988733+ 119887121198872111988733int
infin
0
1198961(119904) 119890minus120582119904119889119904
+119887111198872311988732int
infin
0
1198962(119904) 119890minus120582119904119889119904]
(19)
It is easy to see that 11988712 11988723 11988733lt 0 and 119887
21 11988732gt 0 It
follows from assumption 1198661and 119866
2that 119886
11lt 0 119886
22lt 0
So 119860119894gt 0 119861
119894gt 0 119862
119894gt 0 and 119860
119894119861119894minus 119862119894gt 0 for 119894 ge 1
from the direct calculation According to the Routh-Hurwitzcriterion the three roots 120582
1198941 1205821198942 1205821198943
of 120593119894(120582) = 0 all have
negative real partsBy continuity of the roots with respect to 120583
119894and Routh-
Hurwitz criterion we can conclude that there exists a positiveconstant 120576 such that
Re 1205821198941 Re 120582
1198942 Re 120582
1198943 le minus120576 119894 ge 1 (20)
Consequently the spectrum of 119871 consisting only of eigen-values lies in Re 120582 le minus120576 It is easy to see that Elowast islocally asymptotically stable and follows fromTheorem 511of [23]
Theorem 8 Assume that
11986731198861(11988621minus 1198862)
11989811198862
gt 11988611gt11988612119906lowast
2
11989821
(21)
1198672and 119866
2hold and the positive steady state 119864lowast of system (1)ndash
(3) with nontrivial initial function is globally asymptoticallystable
Proof It is easy to see that the equations in (1) can be rewrittenas
1205971199061
120597119905minus 1198891Δ1199061= 1199061[ minus 11988611(1199061minus 119906lowast
1)
minus11988612(1199062minus 119906lowast
2)
1198981+ 1199061
+11988612119906lowast
2(1199061minus 119906lowast
1)
(1198981+ 1199061) (1198981+ 119906lowast1)]
1205971199062
120597119905minus 1198892Δ1199062= 1199062[intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus 119906
lowast
1)
(1198981+ 1199061(119904 119910)) (119898
1+ 119906lowast1)119889119904 119889119910
minus 11988622(1199062minus 119906lowast
2) minus11988623(1199063minus 119906lowast
3)
1198982+ 1199062
+11988623119906lowast
3(1199062minus 119906lowast
2)
(1198982+ 1199062) (1198982+ 119906lowast2)]
1205971199063
120597119905minus 1198893Δ1199063= 1199063[intΩ
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321198982(1199062(119904 119910) minus 119906
lowast
2)
(1198982+ 1199062(119904 119910)) (119898
2+ 119906lowast2)119889119904 119889119910
minus11988633(1199063minus 119906lowast
3) ]
(22)
Define
119881 (119905) = 120572intΩ
(1199061minus 119906lowast
1minus 119906lowast
1ln 1199061119906lowast1
)119889119909
+ intΩ
(1199062minus 119906lowast
2minus 119906lowast
2ln 1199062119906lowast2
)119889119909
+ 120573intΩ
(1199062minus 119906lowast
3minus 119906lowast
3ln1199063
119906lowast3
)119889119909
(23)
where 120572 and 120573 are positive constants to be determinedCalculating the derivatives 119881(119905) along the positive solutionto the system (1)ndash(3) yields
1198811015840(119905) = Φ
1(119905) + Φ
2(119905) (24)
where
Φ1(119905) = minusint
Ω
(1205721198891119906lowast
1
11990621
1003816100381610038161003816nabla119906110038161003816100381610038162
+1198892119906lowast
2
11990622
1003816100381610038161003816nabla119906210038161003816100381610038162
+1205731198893119906lowast
3
11990623
1003816100381610038161003816nabla119906310038161003816100381610038162
)119889119909
6 Abstract and Applied Analysis
Φ2(119905) = minusint
Ω
120572[11988611minus
11988612119906lowast
2
(1198981+ 1199061) (1198981+ 119906lowast1)] (1199061minus 119906lowast
1)2
119889119909
minus intΩ
[11988622minus
11988623119906lowast
3
(1198982+ 1199062) (1198982+ 119906lowast2)] (1199062minus 119906lowast
2)2
119889119909
minus intΩ
11988633120573(1199063minus 119906lowast
3)2
119889119909
minus intΩ
11988623(1199063minus 119906lowast
3) (1199062minus 119906lowast
2)
1198982+ 1199062
119889119909
minus intΩ
12057211988612(1199062minus 119906lowast
2) (1199061minus 119906lowast
1)
1198981+ 1199061
119889119909
+∬Ω
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus 119906
lowast
1) (1199062(119905 119909) minus 119906
lowast
2)
(1198981+ 1199061(119904 119910)) (119898
1+ 119906lowast1)
119889119904 119889119910 119889119909
+ 120573∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321198982(1199062(119904 119910) minus 119906
lowast
2) (1199063(119905 119909) minus 119906
lowast
3)
(1198982+ 1199062(119904 119910)) (119898
2+ 119906lowast2)
119889119904 119889119910 119889119909
(25)
Applying the inequality 119886119887 le 1205981198862 + (14120598)1198872 we derive from(25) that
Φ2(119905) le minusint
Ω
120572[11988611minus
11988612119906lowast
2
(1198981+ 1199061) (1198981+ 119906lowast1)] (1199061minus 119906lowast
1)2
119889119909
minus intΩ
11988633120573(1199063minus 119906lowast
3)2
119889119909
minus intΩ
[11988622minus
11988623119906lowast
3
(1198982+ 1199062) (1198982+ 119906lowast2)] (1199062minus 119906lowast
2)2
119889119909
+ intΩ
12057211988612
1198981+ 1199061
[1205981(1199061minus 119906lowast
1)2
+1
41205981
(1199062minus 119906lowast
2)2
] 119889119909
+ intΩ
11988623
1198982+ 1199062
[1
41205982
(1199062minus 119906lowast
2)2
+ 1205982(1199063minus 119906lowast
3)2
] 119889119909
+∬Ω
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981
(1198981+ 1199061(119904 119910)) (119898
1+ 119906lowast1)
times [1205983(1199061(119904 119910) minus 119906
lowast
1)2
+1
41205983
(1199062(119905 119909) minus 119906
lowast
2)2
] 119889119904 119889119910 119889119909
+∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321205731198982
(1198982+ 1199062(119904 119910)) (119898
2+ 119906lowast2)
times [1
41205984
(1199062(119904 119910) minus 119906
lowast
2)2
+1205984(1199063(119905 119909) minus 119906
lowast
3)2
] 119889119904 119889119910 119889119909
(26)
According to the property of the Kernel functions119870119894(119909 119910 119905)
(119894 = 1 2) we know that
Φ2(119905) le minusint
Ω
[12057211988611minus12057211988612119906lowast
2
11989821
minus120572119886121205981
1198981
] (1199061minus 119906lowast
1)2
119889119909
minus intΩ
[11988622minus11988623119906lowast
3
11989822
minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
] (1199062minus 119906lowast
2)2
119889119909
minus intΩ
[11988633120573 minus
119886231205982
1198982
minus119886321205731205984
1198982
] (1199063minus 119906lowast
3)2
119889119909
+119886211205983
1198981
∬Ω
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times (1199061(119904 119910) minus 119906
lowast
1)2
119889119904 119889119910 119889119909
+11988632120573
411989821205984
∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times (1199062(119904 119910) minus 119906
lowast
2)2
119889119904 119889119910 119889119909
(27)
Define a new Lyapunov functional as follows
119864 (119905) = 119881 (119905)
+119886211205983
1198981
∬Ω
int
+infin
0
int
119905
119905minus119903
1198701(119909 119910 119903)
times (1199061(119904 119910) minus 119906
lowast
1)2
119889119904 119889119903 119889119910 119889119909
+11988632120573
411989821205984
∬Ω
int
+infin
0
int
119905
119905minus119903
1198702(119909 119910 119903)
times (1199062(119904 119910) minus 119906
lowast
2)2
119889119904 119889119903 119889119910 119889119909
(28)
Abstract and Applied Analysis 7
It is derived from (27) and (28) that
1198641015840(119905) le minusint
Ω
(1205721198891119906lowast
1
11990621
1003816100381610038161003816nabla119906110038161003816100381610038162
+1198892119906lowast
2
11990622
1003816100381610038161003816nabla119906210038161003816100381610038162
+1205731198893119906lowast
3
11990623
1003816100381610038161003816nabla119906310038161003816100381610038162
)119889119909
minus [12057211988611minus12057211988612119906lowast
2
11989821
minus120572119886121205981
1198981
minus119886211205983
1198981
]intΩ
(1199061minus 119906lowast
1)2
119889119909
minus [11988622minus11988623119906lowast
3
11989822
minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632120573
411989821205984
]intΩ
(1199062minus 119906lowast
2)2
119889119909
minus [11988633120573 minus
119886231205982
1198982
minus120573119886321205984
1198982
]intΩ
(1199063minus 119906lowast
3)2
119889119909
(29)
Since (1199061 1199062 1199063) is the unique positive solution of system (1)
Using Proposition 5 there exists a constant119862which does notdepend on119909 isin Ω or 119905 ge 0 such that 119906
119894(sdot 119905)infinle 119862 (119894 = 1 2 3)
for 119905 ge 0 By Theorem 1198602in [24] we have
1003817100381710038171003817119906119894 (sdot 119905)10038171003817100381710038171198622120572(Ω) le 119862 forall119905 ge 1 (30)
Assume that
1198661 119886111198982
1gt 11988612119906lowast
2
1198662 119886221198982
2gt 11988623119906lowast
3+21198862311988632
11988633
+211988612119886211198982
2
1198861111989821minus 11988612119906lowast2
(31)
and denote
1198971= 12057211988611minus12057211988612119906lowast
2
11989821
minus120572119886121205981
1198981
minus119886211205983
1198981
1198972= 11988622minus11988623119906lowast
3
11989822
minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632120573
411989821205984
1198973= 11988633120573 minus
119886231205982
1198982
minus120573119886321205984
1198982
(32)
Then (29) is transformed into
1198641015840(119905) le minusint
Ω
(1205721198891119906lowast
1
11990621
1003816100381610038161003816nabla119906110038161003816100381610038162
+1198892119906lowast
2
11990622
1003816100381610038161003816nabla119906210038161003816100381610038162
+1205731198893119906lowast
3
11990623
1003816100381610038161003816nabla119906310038161003816100381610038162
)119889119909
minus
3
sum
119894=1
119897119894intΩ
(119906119894minus 119906lowast
119894)2
119889119909
(33)
If we choose
120572 =11988621
11988612
120573 =11988623
11988632
1205981= 1205983=119886111198982
1minus 11988612119906lowast
2
4119898111988612
1205982= 1205984=119898211988633
411988632
(34)
then 119897119894gt 0 (119894 = 1 2 3) Therefore we have
1198641015840(119905) le minus
3
sum
119894=1
119897119894intΩ
(119906119894minus 119906lowast
119894)2
119889119909 (35)
FromProposition 6 we can see that the solution of system(1) and (3) is bounded and so are the derivatives of (119906
119894minus 119906lowast
119894)
(119894 = 1 2 3) by the equations in (1) Applying Lemma 1 weobtain that
lim119905rarrinfin
intΩ
(119906119894minus 119906lowast
119894) 119889119909 = 0 (119894 = 1 2 3) (36)
Recomputing 1198641015840(119905) gives
1198641015840(119905) le minusint
Ω
(1205721198891119906lowast
1
11990621
1003816100381610038161003816nabla119906110038161003816100381610038162
+1198892119906lowast
2
11990622
1003816100381610038161003816nabla119906210038161003816100381610038162
+1205731198893119906lowast
3
11990623
1003816100381610038161003816nabla119906310038161003816100381610038162
)119889119909
le minus119888intΩ
(1003816100381610038161003816nabla1199061
10038161003816100381610038162
+1003816100381610038161003816nabla1199062
10038161003816100381610038162
+1003816100381610038161003816nabla1199063
10038161003816100381610038162
) 119889119909 = minus119892 (119905)
(37)
where 119888 = min1205721198891119906lowast
11198722
1 1198892119906lowast
21198722
2 1205731198893119906lowast
311987223 Using (30)
and (1) we obtain that the derivative of 119892(119905) is bounded in[1 +infin) From Lemma 1 we conclude that 119892(119905) rarr 0 as 119905 rarrinfin Therefore
lim119905rarrinfin
intΩ
(1003816100381610038161003816nabla1199061
10038161003816100381610038162
+1003816100381610038161003816nabla1199062
10038161003816100381610038162
+1003816100381610038161003816nabla1199063
10038161003816100381610038162
) 119889119909 = 0 (38)
Applying the Poincare inequality
intΩ
120582|119906 minus 119906|2119889119909 le int
Ω
|nabla119906|2119889119909 (39)
leads to
lim119905rarrinfin
intΩ
(119906119894minus 119906119894)2
119889119909 = 0 (119894 = 1 2 3) (40)
where 119906119894= (1|Ω|) int
Ω119906119894119889119909 and 120582 is the smallest positive
eigenvalue ofminusΔwith the homogeneousNeumann conditionTherefore
|Ω| (1199061 (119905) minus 119906lowast
1)2
= intΩ
(1199061(119905) minus 119906
lowast
1)2
119889119909
= intΩ
(1199061(119905) minus 119906
1(119905 119909) + 119906
1(119905 119909) minus 119906
lowast
1)2
119889119909
le 2intΩ
(1199061(119905) minus 119906
1(119905 119909))
2
119889119909
+ 2intΩ
(1199061(119905 119909) minus 119906
lowast
1)2
119889119909
(41)
8 Abstract and Applied Analysis
So we have 1199061(119905) rarr 119906
lowast
1as 119905 rarr infin Similarly 119906
2(119905) rarr 119906
lowast
2
and 1199063(119905) rarr 119906
lowast
3as 119905 rarr infin According to (30) there exists
a subsequence 119905119898 and non-negative functions 119908
119894isin 1198622(Ω)
such thatlim119898rarrinfin
1003817100381710038171003817119906119894 (sdot 119905119898) minus 119908119894 (sdot)10038171003817100381710038171198622(Ω) = 0 (119894 = 1 2 3) (42)
Applying (40) and noting that 119906119894(119905) rarr 119906
lowast
119894 we then have119908
119894=
119906lowast
119894 (119894 = 1 2 3) That is
lim119898rarrinfin
1003817100381710038171003817119906119894 (sdot 119905119898) minus 119906lowast
119894
10038171003817100381710038171198622(Ω) = 0 (119894 = 1 2 3) (43)
Furthermore the local stability ofElowast combiningwith (43)gives the following global stability
Theorem 9 Assume that
11986751198861(11988621minus 1198862)
11989811198862
gt 11988611gt11988612Vlowast2
11989821
(44)
1198674and 119866
4hold and then the semi-trivial steady state 119864
2
of system (1)ndash(3) with non-trivial initial functions is globallyasymptotically stable
Proof It is obvious that system (1)ndash(3) always has two non-negative equilibria (119906
1 1199062 1199063) as follows 119864
0= (0 0 0) and
1198641= (119886111988611 0 0) If 119867
1and 119867
4are satisfied system (1) has
the other semitrivial solution denoted by 1198642(Vlowast1 Vlowast2 0) where
1198674 11988631198982gt (11988632minus 1198863) Vlowast
2 (45)
We consider the stability of 1198642under condition 119867
1and
1198674 Equation (1) can be rewritten as
1205971199061
120597119905minus 1198891Δ1199061= 1199061[minus11988611(1199061minus Vlowast
1) minus11988612(1199062minus Vlowast2)
1198981+ 1199061
+11988612Vlowast2(1199061minus Vlowast1)
(1198981+ 1199061) (1198981+ Vlowast1)]
1205971199062
120597119905minus 1198892Δ1199062
= 1199062[intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus Vlowast
1)
(1198981+ 1199061(119904 119910)) (119898
1+ Vlowast1)119889119904 119889119910
minus11988622(1199062minus Vlowast
2) minus
119886231199063
1198982+ 1199062
]
1205971199063
120597119905minus 1198893Δ1199063
= 1199063[intΩ
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321198982(1199062(119904 119910) minus Vlowast
2)
(1198982+ 1199062(119904 119910)) (119898
2+ Vlowast2)119889119904 119889119910
minus119886331199063]
(46)
Similar to the argument ofTheorem 8 we have 1199061(119909 119905) rarr Vlowast
1
and 1199062(119909 119905) rarr Vlowast
2as 119905 rarr infin uniformly on Ω provided that
the following additional condition holds
1198663 119886111198982
1gt 11988612Vlowast
2
1198664 119886221198982
2gt1198862311988632
11988633
+211988612119886211198982
2
1198861111989821minus 11988612Vlowast2
(47)
Next we consider the asymptotic behavior of 1199063(119905 119909) Let
120579 =1198863minus (11988632minus 1198863) Vlowast2
2120575 120575 =
100381610038161003816100381611988632 minus 11988631003816100381610038161003816
(48)
Then there exists 1199051gt 0 such that
Vlowast
2minus 120579 lt 119906
2(119905 119909) lt V
lowast
2+ 120579 forall119905 gt 119905
1 119909 isin Ω (49)
Consider the following two systems
1199083119905minus 1198893Δ1199083= 1199083[minus1198863+11988632(Vlowast2+ 120579)
1198982+ Vlowast2+ 120579
minus 119886331199083]
(119905 119909) isin [1199051 +infin) times Ω
1205971199083
120597]= 0 (119905 119909) isin [119905
1 +infin) times 120597Ω
1199083= 1199063 (119905 119909) isin (minusinfin 119905
1] times Ω
1198823119905minus 1198893Δ1198823= 1198823[minus1198863+11988632(Vlowast2minus 120579)
1198982+ Vlowast2minus 120579
minus 11988633V3]
(119905 119909) isin [1199051 +infin) times Ω
1205971198823
120597]= 0 (119905 119909) isin [119905
1 +infin) times 120597Ω
1198823= 1199063 (119905 119909) isin (minusinfin 119905
1] times Ω
(50)
Combining comparison principle with (50) we obtain that
1198823(119905 119909) le 119906
3(119905 119909) le 119908
3(119905 119909) forall119905 gt 119905
1 119909 isin Ω (51)
By Lemma 4 we obtain
0 = lim119905rarrinfin
1198823(119905 119909) le inf 119906
3(119905 119909) le sup 119906
3(119905 119909)
le lim119905rarrinfin
1199083(119905 119909) = 0 forall119909 isin Ω
(52)
which implies that lim119905rarrinfin
1199063(119905 119909) = 0 uniformly onΩ
Theorem 10 Suppose that1198665and119866
6hold and then the semi-
trivial steady state 1198641of system (1)ndash(3) with non-trivial initial
functions is globally asymptotically stable
Abstract and Applied Analysis 9
Proof We study the stability of the semi-trivial solution 1198641=
(1 0 0) Similarly the equations in (1) can be written as
1205971199061
120597119905minus 1198891Δ1199061= 1199061[minus11988611(1199061minus 1) minus
119886121199062
1198981+ 1199061
]
1205971199062
120597119905minus 1198892Δ1199062
= 1199062[minus1198862+ intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus
1)
(1198981+ 1199061(119904 119910)) (119898
1+ 1)119889119904 119889119910
+119886211
1198981+ 1
minus 119886221199062minus119886231199063
1198982+ 1199062
]
1205971199063
120597119905minus 1198893Δ1199063
= 1199063[minus1198863+ intΩ
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321199062(119904 119910)
1198982+ 1199062(119904 119910)
119889119904 119889119910 minus 119886331199063]
(53)
Define
119881 (119905) = 120572intΩ
[1199061minus 1minus 1log 11990611
] 119889119909 + intΩ
1199062119889119909
+ 120573intΩ
1199063119889119909
(54)
Calculating the derivative of 119881(119905) along 1198641 we get from (54)
that
1198811015840(119905) le minusint
Ω
[120572(11988611minus119886121205981
1198981
)] (1199061minus 1)2
119889119909
+ intΩ
[11988622minus12057211988612
411989811205981
minus11988623
412059821198982
minus11988621
411989811205983
] 1199062
2119889119909
+ intΩ
[12057311988633minus119886231205982
1198982
] 1199062
3119889119909
minus intΩ
[(1198862minus119886211
1198981+ 1
)1199062+ 11988631199063] 119889119909
+119886211205983
1198981
intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times (1199061(119904 119910) minus
1)2
119889119904 119889119910 119889119909
+11988632120573
412059841198982
∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times 1199062
2(119904 119910) 119889119904 119889119910 119889119909
(55)
Define119864 (119905) = 119881 (119905)
+119886211205983
1198981
∬Ω
int
+infin
0
int
119905
119905minus119903
1198701(119909 119910 119903)
times (1199061(119904 119910) minus 119906
lowast
1)2
119889119904 119889119903 119889119910 119889119909
+11988632120573
412059841198982
∬Ω
int
+infin
0
int
119905
119905minus119903
1198702(119909 119910 119903)
times (1199062(119904 119910) minus 119906
lowast
2)2
119889119904 119889119903 119889119910 119889119909
(56)
It is easy to see that
1198641015840(119905) le minus120572119889
11intΩ
1003816100381610038161003816nabla119906110038161003816100381610038162
11990621
119889119909
minus intΩ
[120572(11988611minus119886121205981
1198981
) minus119886211205983
1198981
] (1199061minus 1)2
119889119909
+ intΩ
[11988622minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632
411989821205984
] 1199062
2119889119909
+ intΩ
[12057311988633minus119886231205982
1198982
minus119886321205731205984
1198982
] 1199062
3119889119909
minus intΩ
[(1198862minus119886211
1198981+ 1
)1199062minus 11988631205731199063] 119889119909
(57)
Assume that1198665 1198861(11988621minus 1198862) lt 119898
1119886211988611
1198666 119886221198982
2gt1198862311988632
11988633
+211988612119886211198982
2
1198861111989821
(58)
Let
1198971= 12057211988611minus120572119886121205981
1198981
minus119886211205983
1198981
1198972= 11988622minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632120573
411989821205984
1198973= 11988633120573 minus
119886231205982
1198982
minus120573119886321205984
1198982
(59)
Choose
120572 =11988621
11988612
120573 =11988623
11988632
1205981= 1205983=119886111198981
411988612
1205982= 1205984=119898211988633
411988632
(60)
Then we get 119897119894gt 0 (119894 = 1 2 3) Therefore we have
lim119905rarrinfin
1199061(119905 119909) =
1 (61)
uniformly onΩ
10 Abstract and Applied Analysis
tx
u1
02
46
8
01
23
40
1
2
3
4
(a)
02
46
8
01
23
40
1
2
3
4
t
x
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 2 Global stability of the positive equilibrium 119864lowast with an initial condition (1206011 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos 119909 27 + 05 sin119909)
Next we consider the asymptotic behavior of 1199062(119905 119909) and
1199063(119905 119909) For any 119879 gt 0 integrating (57) over [0 119879] yields
119864 (119879) + 1205721198891
10038171003817100381710038171003817100381710038171003817
10038161003816100381610038161003816100381610038161003816
nabla1199061
1199061
10038161003816100381610038161003816100381610038161003816
10038171003817100381710038171003817100381710038171003817
2
2
+ 1198971
100381710038171003817100381710038161003816100381610038161199061 minus 1
100381610038161003816100381610038171003817100381710038172
2
+ 1198972
100381710038171003817100381710038161003816100381610038161199062100381610038161003816100381610038171003817100381710038172
2+ 1198973
100381710038171003817100381710038161003816100381610038161199063100381610038161003816100381610038171003817100381710038172
2le 119864 (0)
(62)
where |119906119894|2
2= int119879
0intΩ1199062
119894119889119909 119889119905 It implies that |119906
119894|2le 119862119894(119894 =
2 3) for the constant 119862119894which is independent of 119879 Now we
consider the boundedness of |nabla1199062|2and |nabla119906
3|2 From the
Greenrsquos identity we obtain
1198892int
119879
0
intΩ
1003816100381610038161003816nabla1199062 (119905 119909)10038161003816100381610038162
119889119909 119889119905
= minus1198892int
119879
0
intΩ
1199062(119905 119909) Δ119906
2(119905 119909) 119889119909 119889119905
= minusint
119879
0
intΩ
1199062
1205971199062
120597119905119889119909 119889119905 minus 119886
2int
119879
0
intΩ
1199062
2119889119909 119889119905
minus 11988622int
119879
0
intΩ
1199063
2119889119909 119889119905 minus 119886
23int
119879
0
intΩ
1199062
21199063
1198982+ 1199063
119889119909 119889119905
+ 11988621int
119879
0
∬Ω
int
119905
minusinfin
1199061(119904 119910) 119906
2
2(119905 119909)
1198981+ 1199061(119904 119910)
times 1198701(119909 119910 119905 minus 119904) 119889119904 119889119910 119889119909 119889119905
(63)
Note that
int
119879
0
intΩ
1199062(119905 119909)
1205971199062(119905 119909)
120597119905119889119909 119889119905
= int
119879
0
intΩ
1
2
1205971199062
2
120597119905119889119905 119889119909
=1
2intΩ
1199062
2(119879 119909) 119889119909 minus
1
2intΩ
1199062
2(0 119909) 119889119909 le 119872
2
2|Ω|
int
119879
0
intΩ
1199062
21199063119889119909 119889119905 le 119872
3int
119879
0
intΩ
1199062
2119889119909 119889119905
Abstract and Applied Analysis 11
02
46
8
01
23
40
1
2
3
4
tx
u1
(a)
02
46
8
01
23
40
1
2
3
4
tx
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 3 Global stability of the positive equilibrium 1198642with an initial condition (120601
1 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos 119909 27 + 05 sin119909)
int
119879
0
intΩ
1199063
2119889119909 119889119905 le 119872
2int
119879
0
intΩ
1199062
2119889119909 119889119905
int
119879
0
∬Ω
int
119905
minusinfin
1199061(119904 119910) 119906
2
2(119905 119909)
1198981+ 1199061(119904 119910)
1198701(119909 119910 119905 minus 119904) 119889119904 119889119910 119889119909 119889119905
le int
119879
0
intΩ
1199062
2(119905 119909) 119889119909 119889119905
(64)
Thus we get |nabla1199062|2le 1198624 In a similar way we have
|nabla1199063|2le 1198625 Here 119862
4and 119862
5are independent of 119879
It is easy to see that 1199062(119905 119909) 119906
3(119905 119909) isin 119871
2((0infin)
11988212(Ω)) These imply that
lim119905rarrinfin
1003817100381710038171003817119906119894 (sdot 119905)100381710038171003817100381711988212(Ω) = 0 119894 = 2 3 (65)
From the Sobolev compact embedding theorem we know
lim119905rarrinfin
1003817100381710038171003817119906119894 (sdot 119905)1003817100381710038171003817119862(Ω) = 0 119894 = 2 3 (66)
In the end we show that the trivial solution 1198640is an
unstable equilibrium Similarly to the local stability to 119864lowast wecan get the characteristic equation of 119864
0as
(120582 + 1205831198941198631minus 1198861) (120582 + 120583
1198941198632+ 1198862) (120582 + 120583
1198941198633+ 1198863) = 0 (67)
If 119894 = 1 then 1205831= 0 It is easy to see that this equation admits
a positive solution 120582 = 1198861 According to Theorem 51 in [23]
we have the following result
Theorem 11 The trivial equilibrium 1198640is an unstable equilib-
rium of system (1)ndash(3)
4 Numerical Illustrations
In this section we performnumerical simulations to illustratethe theoretical results given in Section 3
12 Abstract and Applied Analysis
u1
02
46
8
01
23
40
1
2
3
4
tx
(a)
02
46
8
01
23
40
1
2
3
4
tx
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 4 Global stability of the positive equilibrium 1198641with an initial condition (120601
1 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos119909 27 + 05 sin119909)
In the following we always take Ω = [0 120587] 119870119894(119909 119910 119905) =
119866119894(119909 119910 119905)119896
119894(119905) where 119896
119894(119905) = (1120591
lowast)119890minus119905120591lowast
(119894 = 1 2) and
1198661(119909 119910 119905) =
1
120587+2
120587
infin
sum
119899=1
119890minus11988921198992
119905 cos 119899119909 cos 119899119910
1198662(119909 119910 119905) =
1
120587+2
120587
infin
sum
119899=1
119890minus11988931198992
119905 cos 119899119909 cos 119899119910
(68)
However it is difficult for us to simulate our results directlybecause of the nonlocal term Similar to [25] the equationsin (1) can be rewritten as follows
1205971199061
120597119905minus 1198891Δ1199061= 1199061(1198861minus 119886111199061minus119886121199062
1198981+ 1199061
)
1205971199062
120597119905minus 1198892Δ1199062= 1199062(minus1198862+ 11988621V1minus 119886221199062minus119886231199063
1198982+ 1199063
)
1205971199063
120597119905minus 1198893Δ1199063= 1199063(minus1198863+ 11988632V2minus 119886331199063)
120597V1
120597119905minus 1198892ΔV1=1
120591lowast(
1199061
1198981+ 1199061
minus V1)
120597V2
120597119905minus 1198893ΔV2=1
120591lowast(
1199062
1198982+ 1199062
minus V2)
(69)
where
V119894= int
120587
0
int
119905
minusinfin
119866119894(119909 119910 119905 minus 119904)
1
120591lowast119890minus(119905minus119904)120591
lowast 119906119894(119904 119910)
119898119894+ 119906119894(119904 119910)
119889119910 119889119904
119894 = 1 2
(70)
Each component is considered with homogeneous Neumannboundary conditions and the initial condition of V
119894is
V119894(0 119909) = int
120587
0
int
0
minusinfin
119866119894(119909 119910 minus119904)
1
120591lowast119890119904120591lowast 119906119894(119904 119910)
119898119894+ 119906119894(119904 119910)
119889119910 119889119904
119894 = 1 2
(71)
Abstract and Applied Analysis 13
In the following examples we fix some coefficients andassume that 119889
1= 1198892= 1198893= 1 119886
1= 3 119886
12= 1 119898
1= 1198982=
1 1198862= 1 119886
22= 12 119886
23= 1 119886
33= 12 and 120591lowast = 1 The
asymptotic behaviors of system (1)ndash(3) are shownby choosingdifferent coefficients 119886
11 11988621 1198863 and 119886
32
Example 12 Let 11988611= 539 119886
21= 8113 119886
3= 113 and
11988632= 14Then it is easy to see that the system admits a unique
positive equilibrium 119864lowast(12 112 112) By Theorem 8 we
see that the positive solution (1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of
system (1)ndash(3) converges to 119864lowast as 119905 rarr infin See Figure 2
Example 13 Let 11988611
= 6 11988621
= 12 1198863= 14 and
11988632= 1 Clearly 119867
2does not hold Hence the positive
steady state is not feasible System (1) admits two semi-trivial steady state 119864
2(04732 02372 0) and 119864
1(12 0 0)
According to Theorem 9 we know that the positive solution(1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of system (1)ndash(3) converges to 119864
2
as 119905 rarr infin See Figure 3
Example 14 Let 11988611= 6 119886
21= 2 119886
3= 14 and 119886
32= 1
Clearly1198671does not hold Hence 119864lowast and 119864
2are not feasible
System (1) has a unique semi-trivial steady state 1198641(12 0 0)
According to Theorem 10 we know that the positive solution(1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of system (1)ndash(3) converges to 119864
1
as 119905 rarr infin See Figure 4
5 Discussion
In this paper we incorporate nonlocal delay into a three-species food chain model with Michaelis-Menten functionalresponse to represent a delay due to the gestation of the preda-tor The conditions under which the spatial homogeneousequilibria are asymptotically stable are given by using theLyapunov functional
We now summarize the ecological meanings of ourtheoretical results Firstly the positive equilibrium 119864
lowast ofsystem (1)ndash(3) exists under the high birth rate of the prey(1198861) and low death rates (119886
2and 119886
3) of predator and top
predator 119864lowast is globally stable if the intraspecific competition11988611is neither too big nor too small and the maximum harvest
rates 11988612 11988623
are small enough Secondly the semi-trivialequilibrium 119864
2of system (1)ndash(3) exists if the birth rate of the
prey (1198861) is high death rate of predator 119886
2is low and the death
rate (1198863) exceeds the conversion rate from predator to top
predator (11988632) 1198642is globally stable if the maximum harvest
rates 11988612 11988623are small and the intra-specific competition 119886
11
is neither too big nor too small Thirdly system (1)ndash(3) hasonly one semi-trivial equilibrium 119864
1when the death rate (119886
2)
exceeds the conversion rate from prey to predator (11988621) 1198641
is globally stable if intra-specific competitions (11988611 11988622 and
11988633) are strong Finally 119864
0is unstable and the non-stability of
trivial equilibrium tells us that not all of the populations goto extinction Furthermore our main results imply that thenonlocal delay is harmless for stabilities of all non-negativesteady states of system (1)ndash(3)
There are still many interesting and challenging problemswith respect to system (1)ndash(3) for example the permanence
and stability of periodic solution or almost periodic solutionThese problems are clearly worthy for further investigations
Acknowledgments
This work is partially supported by PRC Grants NSFC11102076 and 11071209 andNSF of theHigher Education Insti-tutions of Jiangsu Province (12KJD110002 and 12KJD110008)
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] H F Huo andW T Li ldquoPeriodic solution of a delayed predator-prey system with Michaelis-Menten type functional responserdquoJournal of Computational and AppliedMathematics vol 166 no2 pp 453ndash463 2004
[3] Z Lin and M Pedersen ldquoStability in a diffusive food-chainmodel with Michaelis-Menten functional responserdquo NonlinearAnalysis Theory Methods amp Applications vol 57 no 3 pp 421ndash433 2004
[4] B X Dai N Zhang and J Z Zou ldquoPermanence for theMichaelis-Menten type discrete three-species ratio-dependentfood chain model with delayrdquo Journal of Mathematical Analysisand Applications vol 324 no 1 pp 728ndash738 2006
[5] N F Britton ldquoSpatial structures and periodic travelling wavesin an integro-differential reaction-diffusion population modelrdquoSIAM Journal on Applied Mathematics vol 50 no 6 pp 1663ndash1688 1990
[6] S A Gourley and N F Britton ldquoInstability of travelling wavesolutions of a population model with nonlocal effectsrdquo IMAJournal of Applied Mathematics vol 51 no 3 pp 299ndash310 1993
[7] X Zhang and R Xu ldquoTraveling waves of a diffusive predator-prey model with nonlocal delay and stage structurerdquo Journal ofMathematical Analysis and Applications vol 373 no 2 pp 475ndash484 2011
[8] Z C Wang W T Li and S G Ruan ldquoExistence and stability oftraveling wave fronts in reaction advection diffusion equationswith nonlocal delayrdquo Journal of Differential Equations vol 238no 1 pp 153ndash200 2007
[9] S A Gourley and S G Ruan ldquoConvergence and travellingfronts in functional differential equations with nonlocal termsa competition modelrdquo SIAM Journal on Mathematical Analysisvol 35 no 3 pp 806ndash822 2003
[10] X H Lai and T X Yao ldquoExponential stability of impul-sive delayed reaction-diffusion cellular neural networks viaPoincare integral inequalityrdquoAbstract and Applied Analysis vol2013 Article ID 131836 10 pages 2013
[11] R Xu and Z E Ma ldquoGlobal stability of a reaction-diffusionpredator-prey model with a nonlocal delayrdquo Mathematical andComputer Modelling vol 50 no 1-2 pp 194ndash206 2009
[12] Z Li F D Chen and M X He ldquoAsymptotic behavior of thereaction-diffusion model of plankton allelopathy with nonlocaldelaysrdquo Nonlinear Analysis Real World Applications vol 12 no3 pp 1748ndash1758 2011
[13] Y Kuang and H L Smith ldquoGlobal stability in diffusive delayLotka-Volterra systemsrdquoDifferential and Integral Equations vol4 no 1 pp 117ndash128 1991
14 Abstract and Applied Analysis
[14] Z G Lin M Pedersen and L Zhang ldquoA predator-prey systemwith stage-structure for predator and nonlocal delayrdquoNonlinearAnalysis Theory Methods amp Applications vol 72 no 3-4 pp2019ndash2030 2010
[15] R Xu ldquoA reaction-diffusion predator-prey model with stagestructure and nonlocal delayrdquo Applied Mathematics and Com-putation vol 175 no 2 pp 984ndash1006 2006
[16] S S Chen and J P Shi ldquoStability and Hopf bifurcation in adiffusive logistic population model with nonlocal delay effectrdquoJournal of Differential Equations vol 253 no 12 pp 3440ndash34702012
[17] H I Freedman Deterministic Mathematical Models in Popula-tion Ecology vol 57 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1980
[18] P A Abrams and L R Ginzburg ldquoThe nature of predation preydepedent ratio dependent or neitherrdquo Trends in Ecology andEvolution vol 15 no 8 pp 337ndash341 2000
[19] D L DeAngelis R A Goldstein and R V OrsquoNeill ldquoA modelfor trophic iterationrdquo Ecology vol 56 no 4 pp 881ndash892 1975
[20] C V Pao ldquoDynamics of nonlinear parabolic systems with timedelaysrdquo Journal of Mathematical Analysis and Applications vol198 no 3 pp 751ndash779 1996
[21] F Xu Y Hou and Z G Lin ldquoTime delay parabolic system ina predator-prey model with stage structurerdquo Acta MathematicaSinica vol 48 no 6 pp 1121ndash1130 2005 (Chinese)
[22] P Y H Pang and M X Wang ldquoStrategy and stationary patternin a three-species predator-prey modelrdquo Journal of DifferentialEquations vol 200 no 2 pp 245ndash273 2004
[23] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981
[24] K J Brown P C Dunne and R A Gardner ldquoA semilinearparabolic system arising in the theory of superconductivityrdquoJournal of Differential Equations vol 40 no 2 pp 232ndash252 1981
[25] S A Gourley and J W H So ldquoDynamics of a food-limitedpopulation model incorporating nonlocal delays on a finitedomainrdquo Journal of Mathematical Biology vol 44 no 1 pp 49ndash78 2002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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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
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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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
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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
120593119894(120582) = 120582
3+ 1198601198941205822+ 119861119894120582 + 119862
119894= 0 where 119860
119894 119861119894 and 119862
119894are
defined as follows
119860119894= (1198891+ 1198892+ 1198893) 120583119894minus 11988711minus 11988722minus 11988733
119861119894= (11988911198892+ 11988921198893+ 11988931198891) 1205832
119894
+ [1198893(minus11988711minus 11988722) + 1198891(minus11988733minus 11988722)
+1198892(minus11988711minus 11988733)] 120583119894
+ 1198871111988722+ 1198871111988733+ 1198872211988733
minus 1198871211988721int
infin
0
1198961(119904) 119890minus120582119904119889119904 minus 119887
2311988732int
infin
0
1198962(119904) 119890minus120582119904119889119904
119862119894= 1198891119889211988931205833
119894+ [minus1198873311988911198892minus 1198871111988921198893minus 1198872211988911198893] 1205832
119894
+ [11988711119887221198893+ 11988711119887331198892+ 11988722119887331198891+ 11988722119887331198891
minus 11988931198871211988721int
infin
0
1198961(119904) 119890minus120582119904119889119904
minus11988911198872311988732int
infin
0
1198962(119904) 119890minus120582119904119889119904] 120583119894
+ [minus119887111198872211988733+ 119887121198872111988733int
infin
0
1198961(119904) 119890minus120582119904119889119904
+119887111198872311988732int
infin
0
1198962(119904) 119890minus120582119904119889119904]
(19)
It is easy to see that 11988712 11988723 11988733lt 0 and 119887
21 11988732gt 0 It
follows from assumption 1198661and 119866
2that 119886
11lt 0 119886
22lt 0
So 119860119894gt 0 119861
119894gt 0 119862
119894gt 0 and 119860
119894119861119894minus 119862119894gt 0 for 119894 ge 1
from the direct calculation According to the Routh-Hurwitzcriterion the three roots 120582
1198941 1205821198942 1205821198943
of 120593119894(120582) = 0 all have
negative real partsBy continuity of the roots with respect to 120583
119894and Routh-
Hurwitz criterion we can conclude that there exists a positiveconstant 120576 such that
Re 1205821198941 Re 120582
1198942 Re 120582
1198943 le minus120576 119894 ge 1 (20)
Consequently the spectrum of 119871 consisting only of eigen-values lies in Re 120582 le minus120576 It is easy to see that Elowast islocally asymptotically stable and follows fromTheorem 511of [23]
Theorem 8 Assume that
11986731198861(11988621minus 1198862)
11989811198862
gt 11988611gt11988612119906lowast
2
11989821
(21)
1198672and 119866
2hold and the positive steady state 119864lowast of system (1)ndash
(3) with nontrivial initial function is globally asymptoticallystable
Proof It is easy to see that the equations in (1) can be rewrittenas
1205971199061
120597119905minus 1198891Δ1199061= 1199061[ minus 11988611(1199061minus 119906lowast
1)
minus11988612(1199062minus 119906lowast
2)
1198981+ 1199061
+11988612119906lowast
2(1199061minus 119906lowast
1)
(1198981+ 1199061) (1198981+ 119906lowast1)]
1205971199062
120597119905minus 1198892Δ1199062= 1199062[intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus 119906
lowast
1)
(1198981+ 1199061(119904 119910)) (119898
1+ 119906lowast1)119889119904 119889119910
minus 11988622(1199062minus 119906lowast
2) minus11988623(1199063minus 119906lowast
3)
1198982+ 1199062
+11988623119906lowast
3(1199062minus 119906lowast
2)
(1198982+ 1199062) (1198982+ 119906lowast2)]
1205971199063
120597119905minus 1198893Δ1199063= 1199063[intΩ
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321198982(1199062(119904 119910) minus 119906
lowast
2)
(1198982+ 1199062(119904 119910)) (119898
2+ 119906lowast2)119889119904 119889119910
minus11988633(1199063minus 119906lowast
3) ]
(22)
Define
119881 (119905) = 120572intΩ
(1199061minus 119906lowast
1minus 119906lowast
1ln 1199061119906lowast1
)119889119909
+ intΩ
(1199062minus 119906lowast
2minus 119906lowast
2ln 1199062119906lowast2
)119889119909
+ 120573intΩ
(1199062minus 119906lowast
3minus 119906lowast
3ln1199063
119906lowast3
)119889119909
(23)
where 120572 and 120573 are positive constants to be determinedCalculating the derivatives 119881(119905) along the positive solutionto the system (1)ndash(3) yields
1198811015840(119905) = Φ
1(119905) + Φ
2(119905) (24)
where
Φ1(119905) = minusint
Ω
(1205721198891119906lowast
1
11990621
1003816100381610038161003816nabla119906110038161003816100381610038162
+1198892119906lowast
2
11990622
1003816100381610038161003816nabla119906210038161003816100381610038162
+1205731198893119906lowast
3
11990623
1003816100381610038161003816nabla119906310038161003816100381610038162
)119889119909
6 Abstract and Applied Analysis
Φ2(119905) = minusint
Ω
120572[11988611minus
11988612119906lowast
2
(1198981+ 1199061) (1198981+ 119906lowast1)] (1199061minus 119906lowast
1)2
119889119909
minus intΩ
[11988622minus
11988623119906lowast
3
(1198982+ 1199062) (1198982+ 119906lowast2)] (1199062minus 119906lowast
2)2
119889119909
minus intΩ
11988633120573(1199063minus 119906lowast
3)2
119889119909
minus intΩ
11988623(1199063minus 119906lowast
3) (1199062minus 119906lowast
2)
1198982+ 1199062
119889119909
minus intΩ
12057211988612(1199062minus 119906lowast
2) (1199061minus 119906lowast
1)
1198981+ 1199061
119889119909
+∬Ω
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus 119906
lowast
1) (1199062(119905 119909) minus 119906
lowast
2)
(1198981+ 1199061(119904 119910)) (119898
1+ 119906lowast1)
119889119904 119889119910 119889119909
+ 120573∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321198982(1199062(119904 119910) minus 119906
lowast
2) (1199063(119905 119909) minus 119906
lowast
3)
(1198982+ 1199062(119904 119910)) (119898
2+ 119906lowast2)
119889119904 119889119910 119889119909
(25)
Applying the inequality 119886119887 le 1205981198862 + (14120598)1198872 we derive from(25) that
Φ2(119905) le minusint
Ω
120572[11988611minus
11988612119906lowast
2
(1198981+ 1199061) (1198981+ 119906lowast1)] (1199061minus 119906lowast
1)2
119889119909
minus intΩ
11988633120573(1199063minus 119906lowast
3)2
119889119909
minus intΩ
[11988622minus
11988623119906lowast
3
(1198982+ 1199062) (1198982+ 119906lowast2)] (1199062minus 119906lowast
2)2
119889119909
+ intΩ
12057211988612
1198981+ 1199061
[1205981(1199061minus 119906lowast
1)2
+1
41205981
(1199062minus 119906lowast
2)2
] 119889119909
+ intΩ
11988623
1198982+ 1199062
[1
41205982
(1199062minus 119906lowast
2)2
+ 1205982(1199063minus 119906lowast
3)2
] 119889119909
+∬Ω
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981
(1198981+ 1199061(119904 119910)) (119898
1+ 119906lowast1)
times [1205983(1199061(119904 119910) minus 119906
lowast
1)2
+1
41205983
(1199062(119905 119909) minus 119906
lowast
2)2
] 119889119904 119889119910 119889119909
+∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321205731198982
(1198982+ 1199062(119904 119910)) (119898
2+ 119906lowast2)
times [1
41205984
(1199062(119904 119910) minus 119906
lowast
2)2
+1205984(1199063(119905 119909) minus 119906
lowast
3)2
] 119889119904 119889119910 119889119909
(26)
According to the property of the Kernel functions119870119894(119909 119910 119905)
(119894 = 1 2) we know that
Φ2(119905) le minusint
Ω
[12057211988611minus12057211988612119906lowast
2
11989821
minus120572119886121205981
1198981
] (1199061minus 119906lowast
1)2
119889119909
minus intΩ
[11988622minus11988623119906lowast
3
11989822
minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
] (1199062minus 119906lowast
2)2
119889119909
minus intΩ
[11988633120573 minus
119886231205982
1198982
minus119886321205731205984
1198982
] (1199063minus 119906lowast
3)2
119889119909
+119886211205983
1198981
∬Ω
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times (1199061(119904 119910) minus 119906
lowast
1)2
119889119904 119889119910 119889119909
+11988632120573
411989821205984
∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times (1199062(119904 119910) minus 119906
lowast
2)2
119889119904 119889119910 119889119909
(27)
Define a new Lyapunov functional as follows
119864 (119905) = 119881 (119905)
+119886211205983
1198981
∬Ω
int
+infin
0
int
119905
119905minus119903
1198701(119909 119910 119903)
times (1199061(119904 119910) minus 119906
lowast
1)2
119889119904 119889119903 119889119910 119889119909
+11988632120573
411989821205984
∬Ω
int
+infin
0
int
119905
119905minus119903
1198702(119909 119910 119903)
times (1199062(119904 119910) minus 119906
lowast
2)2
119889119904 119889119903 119889119910 119889119909
(28)
Abstract and Applied Analysis 7
It is derived from (27) and (28) that
1198641015840(119905) le minusint
Ω
(1205721198891119906lowast
1
11990621
1003816100381610038161003816nabla119906110038161003816100381610038162
+1198892119906lowast
2
11990622
1003816100381610038161003816nabla119906210038161003816100381610038162
+1205731198893119906lowast
3
11990623
1003816100381610038161003816nabla119906310038161003816100381610038162
)119889119909
minus [12057211988611minus12057211988612119906lowast
2
11989821
minus120572119886121205981
1198981
minus119886211205983
1198981
]intΩ
(1199061minus 119906lowast
1)2
119889119909
minus [11988622minus11988623119906lowast
3
11989822
minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632120573
411989821205984
]intΩ
(1199062minus 119906lowast
2)2
119889119909
minus [11988633120573 minus
119886231205982
1198982
minus120573119886321205984
1198982
]intΩ
(1199063minus 119906lowast
3)2
119889119909
(29)
Since (1199061 1199062 1199063) is the unique positive solution of system (1)
Using Proposition 5 there exists a constant119862which does notdepend on119909 isin Ω or 119905 ge 0 such that 119906
119894(sdot 119905)infinle 119862 (119894 = 1 2 3)
for 119905 ge 0 By Theorem 1198602in [24] we have
1003817100381710038171003817119906119894 (sdot 119905)10038171003817100381710038171198622120572(Ω) le 119862 forall119905 ge 1 (30)
Assume that
1198661 119886111198982
1gt 11988612119906lowast
2
1198662 119886221198982
2gt 11988623119906lowast
3+21198862311988632
11988633
+211988612119886211198982
2
1198861111989821minus 11988612119906lowast2
(31)
and denote
1198971= 12057211988611minus12057211988612119906lowast
2
11989821
minus120572119886121205981
1198981
minus119886211205983
1198981
1198972= 11988622minus11988623119906lowast
3
11989822
minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632120573
411989821205984
1198973= 11988633120573 minus
119886231205982
1198982
minus120573119886321205984
1198982
(32)
Then (29) is transformed into
1198641015840(119905) le minusint
Ω
(1205721198891119906lowast
1
11990621
1003816100381610038161003816nabla119906110038161003816100381610038162
+1198892119906lowast
2
11990622
1003816100381610038161003816nabla119906210038161003816100381610038162
+1205731198893119906lowast
3
11990623
1003816100381610038161003816nabla119906310038161003816100381610038162
)119889119909
minus
3
sum
119894=1
119897119894intΩ
(119906119894minus 119906lowast
119894)2
119889119909
(33)
If we choose
120572 =11988621
11988612
120573 =11988623
11988632
1205981= 1205983=119886111198982
1minus 11988612119906lowast
2
4119898111988612
1205982= 1205984=119898211988633
411988632
(34)
then 119897119894gt 0 (119894 = 1 2 3) Therefore we have
1198641015840(119905) le minus
3
sum
119894=1
119897119894intΩ
(119906119894minus 119906lowast
119894)2
119889119909 (35)
FromProposition 6 we can see that the solution of system(1) and (3) is bounded and so are the derivatives of (119906
119894minus 119906lowast
119894)
(119894 = 1 2 3) by the equations in (1) Applying Lemma 1 weobtain that
lim119905rarrinfin
intΩ
(119906119894minus 119906lowast
119894) 119889119909 = 0 (119894 = 1 2 3) (36)
Recomputing 1198641015840(119905) gives
1198641015840(119905) le minusint
Ω
(1205721198891119906lowast
1
11990621
1003816100381610038161003816nabla119906110038161003816100381610038162
+1198892119906lowast
2
11990622
1003816100381610038161003816nabla119906210038161003816100381610038162
+1205731198893119906lowast
3
11990623
1003816100381610038161003816nabla119906310038161003816100381610038162
)119889119909
le minus119888intΩ
(1003816100381610038161003816nabla1199061
10038161003816100381610038162
+1003816100381610038161003816nabla1199062
10038161003816100381610038162
+1003816100381610038161003816nabla1199063
10038161003816100381610038162
) 119889119909 = minus119892 (119905)
(37)
where 119888 = min1205721198891119906lowast
11198722
1 1198892119906lowast
21198722
2 1205731198893119906lowast
311987223 Using (30)
and (1) we obtain that the derivative of 119892(119905) is bounded in[1 +infin) From Lemma 1 we conclude that 119892(119905) rarr 0 as 119905 rarrinfin Therefore
lim119905rarrinfin
intΩ
(1003816100381610038161003816nabla1199061
10038161003816100381610038162
+1003816100381610038161003816nabla1199062
10038161003816100381610038162
+1003816100381610038161003816nabla1199063
10038161003816100381610038162
) 119889119909 = 0 (38)
Applying the Poincare inequality
intΩ
120582|119906 minus 119906|2119889119909 le int
Ω
|nabla119906|2119889119909 (39)
leads to
lim119905rarrinfin
intΩ
(119906119894minus 119906119894)2
119889119909 = 0 (119894 = 1 2 3) (40)
where 119906119894= (1|Ω|) int
Ω119906119894119889119909 and 120582 is the smallest positive
eigenvalue ofminusΔwith the homogeneousNeumann conditionTherefore
|Ω| (1199061 (119905) minus 119906lowast
1)2
= intΩ
(1199061(119905) minus 119906
lowast
1)2
119889119909
= intΩ
(1199061(119905) minus 119906
1(119905 119909) + 119906
1(119905 119909) minus 119906
lowast
1)2
119889119909
le 2intΩ
(1199061(119905) minus 119906
1(119905 119909))
2
119889119909
+ 2intΩ
(1199061(119905 119909) minus 119906
lowast
1)2
119889119909
(41)
8 Abstract and Applied Analysis
So we have 1199061(119905) rarr 119906
lowast
1as 119905 rarr infin Similarly 119906
2(119905) rarr 119906
lowast
2
and 1199063(119905) rarr 119906
lowast
3as 119905 rarr infin According to (30) there exists
a subsequence 119905119898 and non-negative functions 119908
119894isin 1198622(Ω)
such thatlim119898rarrinfin
1003817100381710038171003817119906119894 (sdot 119905119898) minus 119908119894 (sdot)10038171003817100381710038171198622(Ω) = 0 (119894 = 1 2 3) (42)
Applying (40) and noting that 119906119894(119905) rarr 119906
lowast
119894 we then have119908
119894=
119906lowast
119894 (119894 = 1 2 3) That is
lim119898rarrinfin
1003817100381710038171003817119906119894 (sdot 119905119898) minus 119906lowast
119894
10038171003817100381710038171198622(Ω) = 0 (119894 = 1 2 3) (43)
Furthermore the local stability ofElowast combiningwith (43)gives the following global stability
Theorem 9 Assume that
11986751198861(11988621minus 1198862)
11989811198862
gt 11988611gt11988612Vlowast2
11989821
(44)
1198674and 119866
4hold and then the semi-trivial steady state 119864
2
of system (1)ndash(3) with non-trivial initial functions is globallyasymptotically stable
Proof It is obvious that system (1)ndash(3) always has two non-negative equilibria (119906
1 1199062 1199063) as follows 119864
0= (0 0 0) and
1198641= (119886111988611 0 0) If 119867
1and 119867
4are satisfied system (1) has
the other semitrivial solution denoted by 1198642(Vlowast1 Vlowast2 0) where
1198674 11988631198982gt (11988632minus 1198863) Vlowast
2 (45)
We consider the stability of 1198642under condition 119867
1and
1198674 Equation (1) can be rewritten as
1205971199061
120597119905minus 1198891Δ1199061= 1199061[minus11988611(1199061minus Vlowast
1) minus11988612(1199062minus Vlowast2)
1198981+ 1199061
+11988612Vlowast2(1199061minus Vlowast1)
(1198981+ 1199061) (1198981+ Vlowast1)]
1205971199062
120597119905minus 1198892Δ1199062
= 1199062[intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus Vlowast
1)
(1198981+ 1199061(119904 119910)) (119898
1+ Vlowast1)119889119904 119889119910
minus11988622(1199062minus Vlowast
2) minus
119886231199063
1198982+ 1199062
]
1205971199063
120597119905minus 1198893Δ1199063
= 1199063[intΩ
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321198982(1199062(119904 119910) minus Vlowast
2)
(1198982+ 1199062(119904 119910)) (119898
2+ Vlowast2)119889119904 119889119910
minus119886331199063]
(46)
Similar to the argument ofTheorem 8 we have 1199061(119909 119905) rarr Vlowast
1
and 1199062(119909 119905) rarr Vlowast
2as 119905 rarr infin uniformly on Ω provided that
the following additional condition holds
1198663 119886111198982
1gt 11988612Vlowast
2
1198664 119886221198982
2gt1198862311988632
11988633
+211988612119886211198982
2
1198861111989821minus 11988612Vlowast2
(47)
Next we consider the asymptotic behavior of 1199063(119905 119909) Let
120579 =1198863minus (11988632minus 1198863) Vlowast2
2120575 120575 =
100381610038161003816100381611988632 minus 11988631003816100381610038161003816
(48)
Then there exists 1199051gt 0 such that
Vlowast
2minus 120579 lt 119906
2(119905 119909) lt V
lowast
2+ 120579 forall119905 gt 119905
1 119909 isin Ω (49)
Consider the following two systems
1199083119905minus 1198893Δ1199083= 1199083[minus1198863+11988632(Vlowast2+ 120579)
1198982+ Vlowast2+ 120579
minus 119886331199083]
(119905 119909) isin [1199051 +infin) times Ω
1205971199083
120597]= 0 (119905 119909) isin [119905
1 +infin) times 120597Ω
1199083= 1199063 (119905 119909) isin (minusinfin 119905
1] times Ω
1198823119905minus 1198893Δ1198823= 1198823[minus1198863+11988632(Vlowast2minus 120579)
1198982+ Vlowast2minus 120579
minus 11988633V3]
(119905 119909) isin [1199051 +infin) times Ω
1205971198823
120597]= 0 (119905 119909) isin [119905
1 +infin) times 120597Ω
1198823= 1199063 (119905 119909) isin (minusinfin 119905
1] times Ω
(50)
Combining comparison principle with (50) we obtain that
1198823(119905 119909) le 119906
3(119905 119909) le 119908
3(119905 119909) forall119905 gt 119905
1 119909 isin Ω (51)
By Lemma 4 we obtain
0 = lim119905rarrinfin
1198823(119905 119909) le inf 119906
3(119905 119909) le sup 119906
3(119905 119909)
le lim119905rarrinfin
1199083(119905 119909) = 0 forall119909 isin Ω
(52)
which implies that lim119905rarrinfin
1199063(119905 119909) = 0 uniformly onΩ
Theorem 10 Suppose that1198665and119866
6hold and then the semi-
trivial steady state 1198641of system (1)ndash(3) with non-trivial initial
functions is globally asymptotically stable
Abstract and Applied Analysis 9
Proof We study the stability of the semi-trivial solution 1198641=
(1 0 0) Similarly the equations in (1) can be written as
1205971199061
120597119905minus 1198891Δ1199061= 1199061[minus11988611(1199061minus 1) minus
119886121199062
1198981+ 1199061
]
1205971199062
120597119905minus 1198892Δ1199062
= 1199062[minus1198862+ intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus
1)
(1198981+ 1199061(119904 119910)) (119898
1+ 1)119889119904 119889119910
+119886211
1198981+ 1
minus 119886221199062minus119886231199063
1198982+ 1199062
]
1205971199063
120597119905minus 1198893Δ1199063
= 1199063[minus1198863+ intΩ
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321199062(119904 119910)
1198982+ 1199062(119904 119910)
119889119904 119889119910 minus 119886331199063]
(53)
Define
119881 (119905) = 120572intΩ
[1199061minus 1minus 1log 11990611
] 119889119909 + intΩ
1199062119889119909
+ 120573intΩ
1199063119889119909
(54)
Calculating the derivative of 119881(119905) along 1198641 we get from (54)
that
1198811015840(119905) le minusint
Ω
[120572(11988611minus119886121205981
1198981
)] (1199061minus 1)2
119889119909
+ intΩ
[11988622minus12057211988612
411989811205981
minus11988623
412059821198982
minus11988621
411989811205983
] 1199062
2119889119909
+ intΩ
[12057311988633minus119886231205982
1198982
] 1199062
3119889119909
minus intΩ
[(1198862minus119886211
1198981+ 1
)1199062+ 11988631199063] 119889119909
+119886211205983
1198981
intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times (1199061(119904 119910) minus
1)2
119889119904 119889119910 119889119909
+11988632120573
412059841198982
∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times 1199062
2(119904 119910) 119889119904 119889119910 119889119909
(55)
Define119864 (119905) = 119881 (119905)
+119886211205983
1198981
∬Ω
int
+infin
0
int
119905
119905minus119903
1198701(119909 119910 119903)
times (1199061(119904 119910) minus 119906
lowast
1)2
119889119904 119889119903 119889119910 119889119909
+11988632120573
412059841198982
∬Ω
int
+infin
0
int
119905
119905minus119903
1198702(119909 119910 119903)
times (1199062(119904 119910) minus 119906
lowast
2)2
119889119904 119889119903 119889119910 119889119909
(56)
It is easy to see that
1198641015840(119905) le minus120572119889
11intΩ
1003816100381610038161003816nabla119906110038161003816100381610038162
11990621
119889119909
minus intΩ
[120572(11988611minus119886121205981
1198981
) minus119886211205983
1198981
] (1199061minus 1)2
119889119909
+ intΩ
[11988622minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632
411989821205984
] 1199062
2119889119909
+ intΩ
[12057311988633minus119886231205982
1198982
minus119886321205731205984
1198982
] 1199062
3119889119909
minus intΩ
[(1198862minus119886211
1198981+ 1
)1199062minus 11988631205731199063] 119889119909
(57)
Assume that1198665 1198861(11988621minus 1198862) lt 119898
1119886211988611
1198666 119886221198982
2gt1198862311988632
11988633
+211988612119886211198982
2
1198861111989821
(58)
Let
1198971= 12057211988611minus120572119886121205981
1198981
minus119886211205983
1198981
1198972= 11988622minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632120573
411989821205984
1198973= 11988633120573 minus
119886231205982
1198982
minus120573119886321205984
1198982
(59)
Choose
120572 =11988621
11988612
120573 =11988623
11988632
1205981= 1205983=119886111198981
411988612
1205982= 1205984=119898211988633
411988632
(60)
Then we get 119897119894gt 0 (119894 = 1 2 3) Therefore we have
lim119905rarrinfin
1199061(119905 119909) =
1 (61)
uniformly onΩ
10 Abstract and Applied Analysis
tx
u1
02
46
8
01
23
40
1
2
3
4
(a)
02
46
8
01
23
40
1
2
3
4
t
x
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 2 Global stability of the positive equilibrium 119864lowast with an initial condition (1206011 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos 119909 27 + 05 sin119909)
Next we consider the asymptotic behavior of 1199062(119905 119909) and
1199063(119905 119909) For any 119879 gt 0 integrating (57) over [0 119879] yields
119864 (119879) + 1205721198891
10038171003817100381710038171003817100381710038171003817
10038161003816100381610038161003816100381610038161003816
nabla1199061
1199061
10038161003816100381610038161003816100381610038161003816
10038171003817100381710038171003817100381710038171003817
2
2
+ 1198971
100381710038171003817100381710038161003816100381610038161199061 minus 1
100381610038161003816100381610038171003817100381710038172
2
+ 1198972
100381710038171003817100381710038161003816100381610038161199062100381610038161003816100381610038171003817100381710038172
2+ 1198973
100381710038171003817100381710038161003816100381610038161199063100381610038161003816100381610038171003817100381710038172
2le 119864 (0)
(62)
where |119906119894|2
2= int119879
0intΩ1199062
119894119889119909 119889119905 It implies that |119906
119894|2le 119862119894(119894 =
2 3) for the constant 119862119894which is independent of 119879 Now we
consider the boundedness of |nabla1199062|2and |nabla119906
3|2 From the
Greenrsquos identity we obtain
1198892int
119879
0
intΩ
1003816100381610038161003816nabla1199062 (119905 119909)10038161003816100381610038162
119889119909 119889119905
= minus1198892int
119879
0
intΩ
1199062(119905 119909) Δ119906
2(119905 119909) 119889119909 119889119905
= minusint
119879
0
intΩ
1199062
1205971199062
120597119905119889119909 119889119905 minus 119886
2int
119879
0
intΩ
1199062
2119889119909 119889119905
minus 11988622int
119879
0
intΩ
1199063
2119889119909 119889119905 minus 119886
23int
119879
0
intΩ
1199062
21199063
1198982+ 1199063
119889119909 119889119905
+ 11988621int
119879
0
∬Ω
int
119905
minusinfin
1199061(119904 119910) 119906
2
2(119905 119909)
1198981+ 1199061(119904 119910)
times 1198701(119909 119910 119905 minus 119904) 119889119904 119889119910 119889119909 119889119905
(63)
Note that
int
119879
0
intΩ
1199062(119905 119909)
1205971199062(119905 119909)
120597119905119889119909 119889119905
= int
119879
0
intΩ
1
2
1205971199062
2
120597119905119889119905 119889119909
=1
2intΩ
1199062
2(119879 119909) 119889119909 minus
1
2intΩ
1199062
2(0 119909) 119889119909 le 119872
2
2|Ω|
int
119879
0
intΩ
1199062
21199063119889119909 119889119905 le 119872
3int
119879
0
intΩ
1199062
2119889119909 119889119905
Abstract and Applied Analysis 11
02
46
8
01
23
40
1
2
3
4
tx
u1
(a)
02
46
8
01
23
40
1
2
3
4
tx
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 3 Global stability of the positive equilibrium 1198642with an initial condition (120601
1 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos 119909 27 + 05 sin119909)
int
119879
0
intΩ
1199063
2119889119909 119889119905 le 119872
2int
119879
0
intΩ
1199062
2119889119909 119889119905
int
119879
0
∬Ω
int
119905
minusinfin
1199061(119904 119910) 119906
2
2(119905 119909)
1198981+ 1199061(119904 119910)
1198701(119909 119910 119905 minus 119904) 119889119904 119889119910 119889119909 119889119905
le int
119879
0
intΩ
1199062
2(119905 119909) 119889119909 119889119905
(64)
Thus we get |nabla1199062|2le 1198624 In a similar way we have
|nabla1199063|2le 1198625 Here 119862
4and 119862
5are independent of 119879
It is easy to see that 1199062(119905 119909) 119906
3(119905 119909) isin 119871
2((0infin)
11988212(Ω)) These imply that
lim119905rarrinfin
1003817100381710038171003817119906119894 (sdot 119905)100381710038171003817100381711988212(Ω) = 0 119894 = 2 3 (65)
From the Sobolev compact embedding theorem we know
lim119905rarrinfin
1003817100381710038171003817119906119894 (sdot 119905)1003817100381710038171003817119862(Ω) = 0 119894 = 2 3 (66)
In the end we show that the trivial solution 1198640is an
unstable equilibrium Similarly to the local stability to 119864lowast wecan get the characteristic equation of 119864
0as
(120582 + 1205831198941198631minus 1198861) (120582 + 120583
1198941198632+ 1198862) (120582 + 120583
1198941198633+ 1198863) = 0 (67)
If 119894 = 1 then 1205831= 0 It is easy to see that this equation admits
a positive solution 120582 = 1198861 According to Theorem 51 in [23]
we have the following result
Theorem 11 The trivial equilibrium 1198640is an unstable equilib-
rium of system (1)ndash(3)
4 Numerical Illustrations
In this section we performnumerical simulations to illustratethe theoretical results given in Section 3
12 Abstract and Applied Analysis
u1
02
46
8
01
23
40
1
2
3
4
tx
(a)
02
46
8
01
23
40
1
2
3
4
tx
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 4 Global stability of the positive equilibrium 1198641with an initial condition (120601
1 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos119909 27 + 05 sin119909)
In the following we always take Ω = [0 120587] 119870119894(119909 119910 119905) =
119866119894(119909 119910 119905)119896
119894(119905) where 119896
119894(119905) = (1120591
lowast)119890minus119905120591lowast
(119894 = 1 2) and
1198661(119909 119910 119905) =
1
120587+2
120587
infin
sum
119899=1
119890minus11988921198992
119905 cos 119899119909 cos 119899119910
1198662(119909 119910 119905) =
1
120587+2
120587
infin
sum
119899=1
119890minus11988931198992
119905 cos 119899119909 cos 119899119910
(68)
However it is difficult for us to simulate our results directlybecause of the nonlocal term Similar to [25] the equationsin (1) can be rewritten as follows
1205971199061
120597119905minus 1198891Δ1199061= 1199061(1198861minus 119886111199061minus119886121199062
1198981+ 1199061
)
1205971199062
120597119905minus 1198892Δ1199062= 1199062(minus1198862+ 11988621V1minus 119886221199062minus119886231199063
1198982+ 1199063
)
1205971199063
120597119905minus 1198893Δ1199063= 1199063(minus1198863+ 11988632V2minus 119886331199063)
120597V1
120597119905minus 1198892ΔV1=1
120591lowast(
1199061
1198981+ 1199061
minus V1)
120597V2
120597119905minus 1198893ΔV2=1
120591lowast(
1199062
1198982+ 1199062
minus V2)
(69)
where
V119894= int
120587
0
int
119905
minusinfin
119866119894(119909 119910 119905 minus 119904)
1
120591lowast119890minus(119905minus119904)120591
lowast 119906119894(119904 119910)
119898119894+ 119906119894(119904 119910)
119889119910 119889119904
119894 = 1 2
(70)
Each component is considered with homogeneous Neumannboundary conditions and the initial condition of V
119894is
V119894(0 119909) = int
120587
0
int
0
minusinfin
119866119894(119909 119910 minus119904)
1
120591lowast119890119904120591lowast 119906119894(119904 119910)
119898119894+ 119906119894(119904 119910)
119889119910 119889119904
119894 = 1 2
(71)
Abstract and Applied Analysis 13
In the following examples we fix some coefficients andassume that 119889
1= 1198892= 1198893= 1 119886
1= 3 119886
12= 1 119898
1= 1198982=
1 1198862= 1 119886
22= 12 119886
23= 1 119886
33= 12 and 120591lowast = 1 The
asymptotic behaviors of system (1)ndash(3) are shownby choosingdifferent coefficients 119886
11 11988621 1198863 and 119886
32
Example 12 Let 11988611= 539 119886
21= 8113 119886
3= 113 and
11988632= 14Then it is easy to see that the system admits a unique
positive equilibrium 119864lowast(12 112 112) By Theorem 8 we
see that the positive solution (1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of
system (1)ndash(3) converges to 119864lowast as 119905 rarr infin See Figure 2
Example 13 Let 11988611
= 6 11988621
= 12 1198863= 14 and
11988632= 1 Clearly 119867
2does not hold Hence the positive
steady state is not feasible System (1) admits two semi-trivial steady state 119864
2(04732 02372 0) and 119864
1(12 0 0)
According to Theorem 9 we know that the positive solution(1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of system (1)ndash(3) converges to 119864
2
as 119905 rarr infin See Figure 3
Example 14 Let 11988611= 6 119886
21= 2 119886
3= 14 and 119886
32= 1
Clearly1198671does not hold Hence 119864lowast and 119864
2are not feasible
System (1) has a unique semi-trivial steady state 1198641(12 0 0)
According to Theorem 10 we know that the positive solution(1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of system (1)ndash(3) converges to 119864
1
as 119905 rarr infin See Figure 4
5 Discussion
In this paper we incorporate nonlocal delay into a three-species food chain model with Michaelis-Menten functionalresponse to represent a delay due to the gestation of the preda-tor The conditions under which the spatial homogeneousequilibria are asymptotically stable are given by using theLyapunov functional
We now summarize the ecological meanings of ourtheoretical results Firstly the positive equilibrium 119864
lowast ofsystem (1)ndash(3) exists under the high birth rate of the prey(1198861) and low death rates (119886
2and 119886
3) of predator and top
predator 119864lowast is globally stable if the intraspecific competition11988611is neither too big nor too small and the maximum harvest
rates 11988612 11988623
are small enough Secondly the semi-trivialequilibrium 119864
2of system (1)ndash(3) exists if the birth rate of the
prey (1198861) is high death rate of predator 119886
2is low and the death
rate (1198863) exceeds the conversion rate from predator to top
predator (11988632) 1198642is globally stable if the maximum harvest
rates 11988612 11988623are small and the intra-specific competition 119886
11
is neither too big nor too small Thirdly system (1)ndash(3) hasonly one semi-trivial equilibrium 119864
1when the death rate (119886
2)
exceeds the conversion rate from prey to predator (11988621) 1198641
is globally stable if intra-specific competitions (11988611 11988622 and
11988633) are strong Finally 119864
0is unstable and the non-stability of
trivial equilibrium tells us that not all of the populations goto extinction Furthermore our main results imply that thenonlocal delay is harmless for stabilities of all non-negativesteady states of system (1)ndash(3)
There are still many interesting and challenging problemswith respect to system (1)ndash(3) for example the permanence
and stability of periodic solution or almost periodic solutionThese problems are clearly worthy for further investigations
Acknowledgments
This work is partially supported by PRC Grants NSFC11102076 and 11071209 andNSF of theHigher Education Insti-tutions of Jiangsu Province (12KJD110002 and 12KJD110008)
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] H F Huo andW T Li ldquoPeriodic solution of a delayed predator-prey system with Michaelis-Menten type functional responserdquoJournal of Computational and AppliedMathematics vol 166 no2 pp 453ndash463 2004
[3] Z Lin and M Pedersen ldquoStability in a diffusive food-chainmodel with Michaelis-Menten functional responserdquo NonlinearAnalysis Theory Methods amp Applications vol 57 no 3 pp 421ndash433 2004
[4] B X Dai N Zhang and J Z Zou ldquoPermanence for theMichaelis-Menten type discrete three-species ratio-dependentfood chain model with delayrdquo Journal of Mathematical Analysisand Applications vol 324 no 1 pp 728ndash738 2006
[5] N F Britton ldquoSpatial structures and periodic travelling wavesin an integro-differential reaction-diffusion population modelrdquoSIAM Journal on Applied Mathematics vol 50 no 6 pp 1663ndash1688 1990
[6] S A Gourley and N F Britton ldquoInstability of travelling wavesolutions of a population model with nonlocal effectsrdquo IMAJournal of Applied Mathematics vol 51 no 3 pp 299ndash310 1993
[7] X Zhang and R Xu ldquoTraveling waves of a diffusive predator-prey model with nonlocal delay and stage structurerdquo Journal ofMathematical Analysis and Applications vol 373 no 2 pp 475ndash484 2011
[8] Z C Wang W T Li and S G Ruan ldquoExistence and stability oftraveling wave fronts in reaction advection diffusion equationswith nonlocal delayrdquo Journal of Differential Equations vol 238no 1 pp 153ndash200 2007
[9] S A Gourley and S G Ruan ldquoConvergence and travellingfronts in functional differential equations with nonlocal termsa competition modelrdquo SIAM Journal on Mathematical Analysisvol 35 no 3 pp 806ndash822 2003
[10] X H Lai and T X Yao ldquoExponential stability of impul-sive delayed reaction-diffusion cellular neural networks viaPoincare integral inequalityrdquoAbstract and Applied Analysis vol2013 Article ID 131836 10 pages 2013
[11] R Xu and Z E Ma ldquoGlobal stability of a reaction-diffusionpredator-prey model with a nonlocal delayrdquo Mathematical andComputer Modelling vol 50 no 1-2 pp 194ndash206 2009
[12] Z Li F D Chen and M X He ldquoAsymptotic behavior of thereaction-diffusion model of plankton allelopathy with nonlocaldelaysrdquo Nonlinear Analysis Real World Applications vol 12 no3 pp 1748ndash1758 2011
[13] Y Kuang and H L Smith ldquoGlobal stability in diffusive delayLotka-Volterra systemsrdquoDifferential and Integral Equations vol4 no 1 pp 117ndash128 1991
14 Abstract and Applied Analysis
[14] Z G Lin M Pedersen and L Zhang ldquoA predator-prey systemwith stage-structure for predator and nonlocal delayrdquoNonlinearAnalysis Theory Methods amp Applications vol 72 no 3-4 pp2019ndash2030 2010
[15] R Xu ldquoA reaction-diffusion predator-prey model with stagestructure and nonlocal delayrdquo Applied Mathematics and Com-putation vol 175 no 2 pp 984ndash1006 2006
[16] S S Chen and J P Shi ldquoStability and Hopf bifurcation in adiffusive logistic population model with nonlocal delay effectrdquoJournal of Differential Equations vol 253 no 12 pp 3440ndash34702012
[17] H I Freedman Deterministic Mathematical Models in Popula-tion Ecology vol 57 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1980
[18] P A Abrams and L R Ginzburg ldquoThe nature of predation preydepedent ratio dependent or neitherrdquo Trends in Ecology andEvolution vol 15 no 8 pp 337ndash341 2000
[19] D L DeAngelis R A Goldstein and R V OrsquoNeill ldquoA modelfor trophic iterationrdquo Ecology vol 56 no 4 pp 881ndash892 1975
[20] C V Pao ldquoDynamics of nonlinear parabolic systems with timedelaysrdquo Journal of Mathematical Analysis and Applications vol198 no 3 pp 751ndash779 1996
[21] F Xu Y Hou and Z G Lin ldquoTime delay parabolic system ina predator-prey model with stage structurerdquo Acta MathematicaSinica vol 48 no 6 pp 1121ndash1130 2005 (Chinese)
[22] P Y H Pang and M X Wang ldquoStrategy and stationary patternin a three-species predator-prey modelrdquo Journal of DifferentialEquations vol 200 no 2 pp 245ndash273 2004
[23] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981
[24] K J Brown P C Dunne and R A Gardner ldquoA semilinearparabolic system arising in the theory of superconductivityrdquoJournal of Differential Equations vol 40 no 2 pp 232ndash252 1981
[25] S A Gourley and J W H So ldquoDynamics of a food-limitedpopulation model incorporating nonlocal delays on a finitedomainrdquo Journal of Mathematical Biology vol 44 no 1 pp 49ndash78 2002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
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Operations ResearchAdvances in
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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
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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
Φ2(119905) = minusint
Ω
120572[11988611minus
11988612119906lowast
2
(1198981+ 1199061) (1198981+ 119906lowast1)] (1199061minus 119906lowast
1)2
119889119909
minus intΩ
[11988622minus
11988623119906lowast
3
(1198982+ 1199062) (1198982+ 119906lowast2)] (1199062minus 119906lowast
2)2
119889119909
minus intΩ
11988633120573(1199063minus 119906lowast
3)2
119889119909
minus intΩ
11988623(1199063minus 119906lowast
3) (1199062minus 119906lowast
2)
1198982+ 1199062
119889119909
minus intΩ
12057211988612(1199062minus 119906lowast
2) (1199061minus 119906lowast
1)
1198981+ 1199061
119889119909
+∬Ω
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus 119906
lowast
1) (1199062(119905 119909) minus 119906
lowast
2)
(1198981+ 1199061(119904 119910)) (119898
1+ 119906lowast1)
119889119904 119889119910 119889119909
+ 120573∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321198982(1199062(119904 119910) minus 119906
lowast
2) (1199063(119905 119909) minus 119906
lowast
3)
(1198982+ 1199062(119904 119910)) (119898
2+ 119906lowast2)
119889119904 119889119910 119889119909
(25)
Applying the inequality 119886119887 le 1205981198862 + (14120598)1198872 we derive from(25) that
Φ2(119905) le minusint
Ω
120572[11988611minus
11988612119906lowast
2
(1198981+ 1199061) (1198981+ 119906lowast1)] (1199061minus 119906lowast
1)2
119889119909
minus intΩ
11988633120573(1199063minus 119906lowast
3)2
119889119909
minus intΩ
[11988622minus
11988623119906lowast
3
(1198982+ 1199062) (1198982+ 119906lowast2)] (1199062minus 119906lowast
2)2
119889119909
+ intΩ
12057211988612
1198981+ 1199061
[1205981(1199061minus 119906lowast
1)2
+1
41205981
(1199062minus 119906lowast
2)2
] 119889119909
+ intΩ
11988623
1198982+ 1199062
[1
41205982
(1199062minus 119906lowast
2)2
+ 1205982(1199063minus 119906lowast
3)2
] 119889119909
+∬Ω
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981
(1198981+ 1199061(119904 119910)) (119898
1+ 119906lowast1)
times [1205983(1199061(119904 119910) minus 119906
lowast
1)2
+1
41205983
(1199062(119905 119909) minus 119906
lowast
2)2
] 119889119904 119889119910 119889119909
+∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321205731198982
(1198982+ 1199062(119904 119910)) (119898
2+ 119906lowast2)
times [1
41205984
(1199062(119904 119910) minus 119906
lowast
2)2
+1205984(1199063(119905 119909) minus 119906
lowast
3)2
] 119889119904 119889119910 119889119909
(26)
According to the property of the Kernel functions119870119894(119909 119910 119905)
(119894 = 1 2) we know that
Φ2(119905) le minusint
Ω
[12057211988611minus12057211988612119906lowast
2
11989821
minus120572119886121205981
1198981
] (1199061minus 119906lowast
1)2
119889119909
minus intΩ
[11988622minus11988623119906lowast
3
11989822
minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
] (1199062minus 119906lowast
2)2
119889119909
minus intΩ
[11988633120573 minus
119886231205982
1198982
minus119886321205731205984
1198982
] (1199063minus 119906lowast
3)2
119889119909
+119886211205983
1198981
∬Ω
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times (1199061(119904 119910) minus 119906
lowast
1)2
119889119904 119889119910 119889119909
+11988632120573
411989821205984
∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times (1199062(119904 119910) minus 119906
lowast
2)2
119889119904 119889119910 119889119909
(27)
Define a new Lyapunov functional as follows
119864 (119905) = 119881 (119905)
+119886211205983
1198981
∬Ω
int
+infin
0
int
119905
119905minus119903
1198701(119909 119910 119903)
times (1199061(119904 119910) minus 119906
lowast
1)2
119889119904 119889119903 119889119910 119889119909
+11988632120573
411989821205984
∬Ω
int
+infin
0
int
119905
119905minus119903
1198702(119909 119910 119903)
times (1199062(119904 119910) minus 119906
lowast
2)2
119889119904 119889119903 119889119910 119889119909
(28)
Abstract and Applied Analysis 7
It is derived from (27) and (28) that
1198641015840(119905) le minusint
Ω
(1205721198891119906lowast
1
11990621
1003816100381610038161003816nabla119906110038161003816100381610038162
+1198892119906lowast
2
11990622
1003816100381610038161003816nabla119906210038161003816100381610038162
+1205731198893119906lowast
3
11990623
1003816100381610038161003816nabla119906310038161003816100381610038162
)119889119909
minus [12057211988611minus12057211988612119906lowast
2
11989821
minus120572119886121205981
1198981
minus119886211205983
1198981
]intΩ
(1199061minus 119906lowast
1)2
119889119909
minus [11988622minus11988623119906lowast
3
11989822
minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632120573
411989821205984
]intΩ
(1199062minus 119906lowast
2)2
119889119909
minus [11988633120573 minus
119886231205982
1198982
minus120573119886321205984
1198982
]intΩ
(1199063minus 119906lowast
3)2
119889119909
(29)
Since (1199061 1199062 1199063) is the unique positive solution of system (1)
Using Proposition 5 there exists a constant119862which does notdepend on119909 isin Ω or 119905 ge 0 such that 119906
119894(sdot 119905)infinle 119862 (119894 = 1 2 3)
for 119905 ge 0 By Theorem 1198602in [24] we have
1003817100381710038171003817119906119894 (sdot 119905)10038171003817100381710038171198622120572(Ω) le 119862 forall119905 ge 1 (30)
Assume that
1198661 119886111198982
1gt 11988612119906lowast
2
1198662 119886221198982
2gt 11988623119906lowast
3+21198862311988632
11988633
+211988612119886211198982
2
1198861111989821minus 11988612119906lowast2
(31)
and denote
1198971= 12057211988611minus12057211988612119906lowast
2
11989821
minus120572119886121205981
1198981
minus119886211205983
1198981
1198972= 11988622minus11988623119906lowast
3
11989822
minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632120573
411989821205984
1198973= 11988633120573 minus
119886231205982
1198982
minus120573119886321205984
1198982
(32)
Then (29) is transformed into
1198641015840(119905) le minusint
Ω
(1205721198891119906lowast
1
11990621
1003816100381610038161003816nabla119906110038161003816100381610038162
+1198892119906lowast
2
11990622
1003816100381610038161003816nabla119906210038161003816100381610038162
+1205731198893119906lowast
3
11990623
1003816100381610038161003816nabla119906310038161003816100381610038162
)119889119909
minus
3
sum
119894=1
119897119894intΩ
(119906119894minus 119906lowast
119894)2
119889119909
(33)
If we choose
120572 =11988621
11988612
120573 =11988623
11988632
1205981= 1205983=119886111198982
1minus 11988612119906lowast
2
4119898111988612
1205982= 1205984=119898211988633
411988632
(34)
then 119897119894gt 0 (119894 = 1 2 3) Therefore we have
1198641015840(119905) le minus
3
sum
119894=1
119897119894intΩ
(119906119894minus 119906lowast
119894)2
119889119909 (35)
FromProposition 6 we can see that the solution of system(1) and (3) is bounded and so are the derivatives of (119906
119894minus 119906lowast
119894)
(119894 = 1 2 3) by the equations in (1) Applying Lemma 1 weobtain that
lim119905rarrinfin
intΩ
(119906119894minus 119906lowast
119894) 119889119909 = 0 (119894 = 1 2 3) (36)
Recomputing 1198641015840(119905) gives
1198641015840(119905) le minusint
Ω
(1205721198891119906lowast
1
11990621
1003816100381610038161003816nabla119906110038161003816100381610038162
+1198892119906lowast
2
11990622
1003816100381610038161003816nabla119906210038161003816100381610038162
+1205731198893119906lowast
3
11990623
1003816100381610038161003816nabla119906310038161003816100381610038162
)119889119909
le minus119888intΩ
(1003816100381610038161003816nabla1199061
10038161003816100381610038162
+1003816100381610038161003816nabla1199062
10038161003816100381610038162
+1003816100381610038161003816nabla1199063
10038161003816100381610038162
) 119889119909 = minus119892 (119905)
(37)
where 119888 = min1205721198891119906lowast
11198722
1 1198892119906lowast
21198722
2 1205731198893119906lowast
311987223 Using (30)
and (1) we obtain that the derivative of 119892(119905) is bounded in[1 +infin) From Lemma 1 we conclude that 119892(119905) rarr 0 as 119905 rarrinfin Therefore
lim119905rarrinfin
intΩ
(1003816100381610038161003816nabla1199061
10038161003816100381610038162
+1003816100381610038161003816nabla1199062
10038161003816100381610038162
+1003816100381610038161003816nabla1199063
10038161003816100381610038162
) 119889119909 = 0 (38)
Applying the Poincare inequality
intΩ
120582|119906 minus 119906|2119889119909 le int
Ω
|nabla119906|2119889119909 (39)
leads to
lim119905rarrinfin
intΩ
(119906119894minus 119906119894)2
119889119909 = 0 (119894 = 1 2 3) (40)
where 119906119894= (1|Ω|) int
Ω119906119894119889119909 and 120582 is the smallest positive
eigenvalue ofminusΔwith the homogeneousNeumann conditionTherefore
|Ω| (1199061 (119905) minus 119906lowast
1)2
= intΩ
(1199061(119905) minus 119906
lowast
1)2
119889119909
= intΩ
(1199061(119905) minus 119906
1(119905 119909) + 119906
1(119905 119909) minus 119906
lowast
1)2
119889119909
le 2intΩ
(1199061(119905) minus 119906
1(119905 119909))
2
119889119909
+ 2intΩ
(1199061(119905 119909) minus 119906
lowast
1)2
119889119909
(41)
8 Abstract and Applied Analysis
So we have 1199061(119905) rarr 119906
lowast
1as 119905 rarr infin Similarly 119906
2(119905) rarr 119906
lowast
2
and 1199063(119905) rarr 119906
lowast
3as 119905 rarr infin According to (30) there exists
a subsequence 119905119898 and non-negative functions 119908
119894isin 1198622(Ω)
such thatlim119898rarrinfin
1003817100381710038171003817119906119894 (sdot 119905119898) minus 119908119894 (sdot)10038171003817100381710038171198622(Ω) = 0 (119894 = 1 2 3) (42)
Applying (40) and noting that 119906119894(119905) rarr 119906
lowast
119894 we then have119908
119894=
119906lowast
119894 (119894 = 1 2 3) That is
lim119898rarrinfin
1003817100381710038171003817119906119894 (sdot 119905119898) minus 119906lowast
119894
10038171003817100381710038171198622(Ω) = 0 (119894 = 1 2 3) (43)
Furthermore the local stability ofElowast combiningwith (43)gives the following global stability
Theorem 9 Assume that
11986751198861(11988621minus 1198862)
11989811198862
gt 11988611gt11988612Vlowast2
11989821
(44)
1198674and 119866
4hold and then the semi-trivial steady state 119864
2
of system (1)ndash(3) with non-trivial initial functions is globallyasymptotically stable
Proof It is obvious that system (1)ndash(3) always has two non-negative equilibria (119906
1 1199062 1199063) as follows 119864
0= (0 0 0) and
1198641= (119886111988611 0 0) If 119867
1and 119867
4are satisfied system (1) has
the other semitrivial solution denoted by 1198642(Vlowast1 Vlowast2 0) where
1198674 11988631198982gt (11988632minus 1198863) Vlowast
2 (45)
We consider the stability of 1198642under condition 119867
1and
1198674 Equation (1) can be rewritten as
1205971199061
120597119905minus 1198891Δ1199061= 1199061[minus11988611(1199061minus Vlowast
1) minus11988612(1199062minus Vlowast2)
1198981+ 1199061
+11988612Vlowast2(1199061minus Vlowast1)
(1198981+ 1199061) (1198981+ Vlowast1)]
1205971199062
120597119905minus 1198892Δ1199062
= 1199062[intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus Vlowast
1)
(1198981+ 1199061(119904 119910)) (119898
1+ Vlowast1)119889119904 119889119910
minus11988622(1199062minus Vlowast
2) minus
119886231199063
1198982+ 1199062
]
1205971199063
120597119905minus 1198893Δ1199063
= 1199063[intΩ
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321198982(1199062(119904 119910) minus Vlowast
2)
(1198982+ 1199062(119904 119910)) (119898
2+ Vlowast2)119889119904 119889119910
minus119886331199063]
(46)
Similar to the argument ofTheorem 8 we have 1199061(119909 119905) rarr Vlowast
1
and 1199062(119909 119905) rarr Vlowast
2as 119905 rarr infin uniformly on Ω provided that
the following additional condition holds
1198663 119886111198982
1gt 11988612Vlowast
2
1198664 119886221198982
2gt1198862311988632
11988633
+211988612119886211198982
2
1198861111989821minus 11988612Vlowast2
(47)
Next we consider the asymptotic behavior of 1199063(119905 119909) Let
120579 =1198863minus (11988632minus 1198863) Vlowast2
2120575 120575 =
100381610038161003816100381611988632 minus 11988631003816100381610038161003816
(48)
Then there exists 1199051gt 0 such that
Vlowast
2minus 120579 lt 119906
2(119905 119909) lt V
lowast
2+ 120579 forall119905 gt 119905
1 119909 isin Ω (49)
Consider the following two systems
1199083119905minus 1198893Δ1199083= 1199083[minus1198863+11988632(Vlowast2+ 120579)
1198982+ Vlowast2+ 120579
minus 119886331199083]
(119905 119909) isin [1199051 +infin) times Ω
1205971199083
120597]= 0 (119905 119909) isin [119905
1 +infin) times 120597Ω
1199083= 1199063 (119905 119909) isin (minusinfin 119905
1] times Ω
1198823119905minus 1198893Δ1198823= 1198823[minus1198863+11988632(Vlowast2minus 120579)
1198982+ Vlowast2minus 120579
minus 11988633V3]
(119905 119909) isin [1199051 +infin) times Ω
1205971198823
120597]= 0 (119905 119909) isin [119905
1 +infin) times 120597Ω
1198823= 1199063 (119905 119909) isin (minusinfin 119905
1] times Ω
(50)
Combining comparison principle with (50) we obtain that
1198823(119905 119909) le 119906
3(119905 119909) le 119908
3(119905 119909) forall119905 gt 119905
1 119909 isin Ω (51)
By Lemma 4 we obtain
0 = lim119905rarrinfin
1198823(119905 119909) le inf 119906
3(119905 119909) le sup 119906
3(119905 119909)
le lim119905rarrinfin
1199083(119905 119909) = 0 forall119909 isin Ω
(52)
which implies that lim119905rarrinfin
1199063(119905 119909) = 0 uniformly onΩ
Theorem 10 Suppose that1198665and119866
6hold and then the semi-
trivial steady state 1198641of system (1)ndash(3) with non-trivial initial
functions is globally asymptotically stable
Abstract and Applied Analysis 9
Proof We study the stability of the semi-trivial solution 1198641=
(1 0 0) Similarly the equations in (1) can be written as
1205971199061
120597119905minus 1198891Δ1199061= 1199061[minus11988611(1199061minus 1) minus
119886121199062
1198981+ 1199061
]
1205971199062
120597119905minus 1198892Δ1199062
= 1199062[minus1198862+ intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus
1)
(1198981+ 1199061(119904 119910)) (119898
1+ 1)119889119904 119889119910
+119886211
1198981+ 1
minus 119886221199062minus119886231199063
1198982+ 1199062
]
1205971199063
120597119905minus 1198893Δ1199063
= 1199063[minus1198863+ intΩ
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321199062(119904 119910)
1198982+ 1199062(119904 119910)
119889119904 119889119910 minus 119886331199063]
(53)
Define
119881 (119905) = 120572intΩ
[1199061minus 1minus 1log 11990611
] 119889119909 + intΩ
1199062119889119909
+ 120573intΩ
1199063119889119909
(54)
Calculating the derivative of 119881(119905) along 1198641 we get from (54)
that
1198811015840(119905) le minusint
Ω
[120572(11988611minus119886121205981
1198981
)] (1199061minus 1)2
119889119909
+ intΩ
[11988622minus12057211988612
411989811205981
minus11988623
412059821198982
minus11988621
411989811205983
] 1199062
2119889119909
+ intΩ
[12057311988633minus119886231205982
1198982
] 1199062
3119889119909
minus intΩ
[(1198862minus119886211
1198981+ 1
)1199062+ 11988631199063] 119889119909
+119886211205983
1198981
intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times (1199061(119904 119910) minus
1)2
119889119904 119889119910 119889119909
+11988632120573
412059841198982
∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times 1199062
2(119904 119910) 119889119904 119889119910 119889119909
(55)
Define119864 (119905) = 119881 (119905)
+119886211205983
1198981
∬Ω
int
+infin
0
int
119905
119905minus119903
1198701(119909 119910 119903)
times (1199061(119904 119910) minus 119906
lowast
1)2
119889119904 119889119903 119889119910 119889119909
+11988632120573
412059841198982
∬Ω
int
+infin
0
int
119905
119905minus119903
1198702(119909 119910 119903)
times (1199062(119904 119910) minus 119906
lowast
2)2
119889119904 119889119903 119889119910 119889119909
(56)
It is easy to see that
1198641015840(119905) le minus120572119889
11intΩ
1003816100381610038161003816nabla119906110038161003816100381610038162
11990621
119889119909
minus intΩ
[120572(11988611minus119886121205981
1198981
) minus119886211205983
1198981
] (1199061minus 1)2
119889119909
+ intΩ
[11988622minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632
411989821205984
] 1199062
2119889119909
+ intΩ
[12057311988633minus119886231205982
1198982
minus119886321205731205984
1198982
] 1199062
3119889119909
minus intΩ
[(1198862minus119886211
1198981+ 1
)1199062minus 11988631205731199063] 119889119909
(57)
Assume that1198665 1198861(11988621minus 1198862) lt 119898
1119886211988611
1198666 119886221198982
2gt1198862311988632
11988633
+211988612119886211198982
2
1198861111989821
(58)
Let
1198971= 12057211988611minus120572119886121205981
1198981
minus119886211205983
1198981
1198972= 11988622minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632120573
411989821205984
1198973= 11988633120573 minus
119886231205982
1198982
minus120573119886321205984
1198982
(59)
Choose
120572 =11988621
11988612
120573 =11988623
11988632
1205981= 1205983=119886111198981
411988612
1205982= 1205984=119898211988633
411988632
(60)
Then we get 119897119894gt 0 (119894 = 1 2 3) Therefore we have
lim119905rarrinfin
1199061(119905 119909) =
1 (61)
uniformly onΩ
10 Abstract and Applied Analysis
tx
u1
02
46
8
01
23
40
1
2
3
4
(a)
02
46
8
01
23
40
1
2
3
4
t
x
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 2 Global stability of the positive equilibrium 119864lowast with an initial condition (1206011 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos 119909 27 + 05 sin119909)
Next we consider the asymptotic behavior of 1199062(119905 119909) and
1199063(119905 119909) For any 119879 gt 0 integrating (57) over [0 119879] yields
119864 (119879) + 1205721198891
10038171003817100381710038171003817100381710038171003817
10038161003816100381610038161003816100381610038161003816
nabla1199061
1199061
10038161003816100381610038161003816100381610038161003816
10038171003817100381710038171003817100381710038171003817
2
2
+ 1198971
100381710038171003817100381710038161003816100381610038161199061 minus 1
100381610038161003816100381610038171003817100381710038172
2
+ 1198972
100381710038171003817100381710038161003816100381610038161199062100381610038161003816100381610038171003817100381710038172
2+ 1198973
100381710038171003817100381710038161003816100381610038161199063100381610038161003816100381610038171003817100381710038172
2le 119864 (0)
(62)
where |119906119894|2
2= int119879
0intΩ1199062
119894119889119909 119889119905 It implies that |119906
119894|2le 119862119894(119894 =
2 3) for the constant 119862119894which is independent of 119879 Now we
consider the boundedness of |nabla1199062|2and |nabla119906
3|2 From the
Greenrsquos identity we obtain
1198892int
119879
0
intΩ
1003816100381610038161003816nabla1199062 (119905 119909)10038161003816100381610038162
119889119909 119889119905
= minus1198892int
119879
0
intΩ
1199062(119905 119909) Δ119906
2(119905 119909) 119889119909 119889119905
= minusint
119879
0
intΩ
1199062
1205971199062
120597119905119889119909 119889119905 minus 119886
2int
119879
0
intΩ
1199062
2119889119909 119889119905
minus 11988622int
119879
0
intΩ
1199063
2119889119909 119889119905 minus 119886
23int
119879
0
intΩ
1199062
21199063
1198982+ 1199063
119889119909 119889119905
+ 11988621int
119879
0
∬Ω
int
119905
minusinfin
1199061(119904 119910) 119906
2
2(119905 119909)
1198981+ 1199061(119904 119910)
times 1198701(119909 119910 119905 minus 119904) 119889119904 119889119910 119889119909 119889119905
(63)
Note that
int
119879
0
intΩ
1199062(119905 119909)
1205971199062(119905 119909)
120597119905119889119909 119889119905
= int
119879
0
intΩ
1
2
1205971199062
2
120597119905119889119905 119889119909
=1
2intΩ
1199062
2(119879 119909) 119889119909 minus
1
2intΩ
1199062
2(0 119909) 119889119909 le 119872
2
2|Ω|
int
119879
0
intΩ
1199062
21199063119889119909 119889119905 le 119872
3int
119879
0
intΩ
1199062
2119889119909 119889119905
Abstract and Applied Analysis 11
02
46
8
01
23
40
1
2
3
4
tx
u1
(a)
02
46
8
01
23
40
1
2
3
4
tx
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 3 Global stability of the positive equilibrium 1198642with an initial condition (120601
1 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos 119909 27 + 05 sin119909)
int
119879
0
intΩ
1199063
2119889119909 119889119905 le 119872
2int
119879
0
intΩ
1199062
2119889119909 119889119905
int
119879
0
∬Ω
int
119905
minusinfin
1199061(119904 119910) 119906
2
2(119905 119909)
1198981+ 1199061(119904 119910)
1198701(119909 119910 119905 minus 119904) 119889119904 119889119910 119889119909 119889119905
le int
119879
0
intΩ
1199062
2(119905 119909) 119889119909 119889119905
(64)
Thus we get |nabla1199062|2le 1198624 In a similar way we have
|nabla1199063|2le 1198625 Here 119862
4and 119862
5are independent of 119879
It is easy to see that 1199062(119905 119909) 119906
3(119905 119909) isin 119871
2((0infin)
11988212(Ω)) These imply that
lim119905rarrinfin
1003817100381710038171003817119906119894 (sdot 119905)100381710038171003817100381711988212(Ω) = 0 119894 = 2 3 (65)
From the Sobolev compact embedding theorem we know
lim119905rarrinfin
1003817100381710038171003817119906119894 (sdot 119905)1003817100381710038171003817119862(Ω) = 0 119894 = 2 3 (66)
In the end we show that the trivial solution 1198640is an
unstable equilibrium Similarly to the local stability to 119864lowast wecan get the characteristic equation of 119864
0as
(120582 + 1205831198941198631minus 1198861) (120582 + 120583
1198941198632+ 1198862) (120582 + 120583
1198941198633+ 1198863) = 0 (67)
If 119894 = 1 then 1205831= 0 It is easy to see that this equation admits
a positive solution 120582 = 1198861 According to Theorem 51 in [23]
we have the following result
Theorem 11 The trivial equilibrium 1198640is an unstable equilib-
rium of system (1)ndash(3)
4 Numerical Illustrations
In this section we performnumerical simulations to illustratethe theoretical results given in Section 3
12 Abstract and Applied Analysis
u1
02
46
8
01
23
40
1
2
3
4
tx
(a)
02
46
8
01
23
40
1
2
3
4
tx
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 4 Global stability of the positive equilibrium 1198641with an initial condition (120601
1 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos119909 27 + 05 sin119909)
In the following we always take Ω = [0 120587] 119870119894(119909 119910 119905) =
119866119894(119909 119910 119905)119896
119894(119905) where 119896
119894(119905) = (1120591
lowast)119890minus119905120591lowast
(119894 = 1 2) and
1198661(119909 119910 119905) =
1
120587+2
120587
infin
sum
119899=1
119890minus11988921198992
119905 cos 119899119909 cos 119899119910
1198662(119909 119910 119905) =
1
120587+2
120587
infin
sum
119899=1
119890minus11988931198992
119905 cos 119899119909 cos 119899119910
(68)
However it is difficult for us to simulate our results directlybecause of the nonlocal term Similar to [25] the equationsin (1) can be rewritten as follows
1205971199061
120597119905minus 1198891Δ1199061= 1199061(1198861minus 119886111199061minus119886121199062
1198981+ 1199061
)
1205971199062
120597119905minus 1198892Δ1199062= 1199062(minus1198862+ 11988621V1minus 119886221199062minus119886231199063
1198982+ 1199063
)
1205971199063
120597119905minus 1198893Δ1199063= 1199063(minus1198863+ 11988632V2minus 119886331199063)
120597V1
120597119905minus 1198892ΔV1=1
120591lowast(
1199061
1198981+ 1199061
minus V1)
120597V2
120597119905minus 1198893ΔV2=1
120591lowast(
1199062
1198982+ 1199062
minus V2)
(69)
where
V119894= int
120587
0
int
119905
minusinfin
119866119894(119909 119910 119905 minus 119904)
1
120591lowast119890minus(119905minus119904)120591
lowast 119906119894(119904 119910)
119898119894+ 119906119894(119904 119910)
119889119910 119889119904
119894 = 1 2
(70)
Each component is considered with homogeneous Neumannboundary conditions and the initial condition of V
119894is
V119894(0 119909) = int
120587
0
int
0
minusinfin
119866119894(119909 119910 minus119904)
1
120591lowast119890119904120591lowast 119906119894(119904 119910)
119898119894+ 119906119894(119904 119910)
119889119910 119889119904
119894 = 1 2
(71)
Abstract and Applied Analysis 13
In the following examples we fix some coefficients andassume that 119889
1= 1198892= 1198893= 1 119886
1= 3 119886
12= 1 119898
1= 1198982=
1 1198862= 1 119886
22= 12 119886
23= 1 119886
33= 12 and 120591lowast = 1 The
asymptotic behaviors of system (1)ndash(3) are shownby choosingdifferent coefficients 119886
11 11988621 1198863 and 119886
32
Example 12 Let 11988611= 539 119886
21= 8113 119886
3= 113 and
11988632= 14Then it is easy to see that the system admits a unique
positive equilibrium 119864lowast(12 112 112) By Theorem 8 we
see that the positive solution (1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of
system (1)ndash(3) converges to 119864lowast as 119905 rarr infin See Figure 2
Example 13 Let 11988611
= 6 11988621
= 12 1198863= 14 and
11988632= 1 Clearly 119867
2does not hold Hence the positive
steady state is not feasible System (1) admits two semi-trivial steady state 119864
2(04732 02372 0) and 119864
1(12 0 0)
According to Theorem 9 we know that the positive solution(1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of system (1)ndash(3) converges to 119864
2
as 119905 rarr infin See Figure 3
Example 14 Let 11988611= 6 119886
21= 2 119886
3= 14 and 119886
32= 1
Clearly1198671does not hold Hence 119864lowast and 119864
2are not feasible
System (1) has a unique semi-trivial steady state 1198641(12 0 0)
According to Theorem 10 we know that the positive solution(1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of system (1)ndash(3) converges to 119864
1
as 119905 rarr infin See Figure 4
5 Discussion
In this paper we incorporate nonlocal delay into a three-species food chain model with Michaelis-Menten functionalresponse to represent a delay due to the gestation of the preda-tor The conditions under which the spatial homogeneousequilibria are asymptotically stable are given by using theLyapunov functional
We now summarize the ecological meanings of ourtheoretical results Firstly the positive equilibrium 119864
lowast ofsystem (1)ndash(3) exists under the high birth rate of the prey(1198861) and low death rates (119886
2and 119886
3) of predator and top
predator 119864lowast is globally stable if the intraspecific competition11988611is neither too big nor too small and the maximum harvest
rates 11988612 11988623
are small enough Secondly the semi-trivialequilibrium 119864
2of system (1)ndash(3) exists if the birth rate of the
prey (1198861) is high death rate of predator 119886
2is low and the death
rate (1198863) exceeds the conversion rate from predator to top
predator (11988632) 1198642is globally stable if the maximum harvest
rates 11988612 11988623are small and the intra-specific competition 119886
11
is neither too big nor too small Thirdly system (1)ndash(3) hasonly one semi-trivial equilibrium 119864
1when the death rate (119886
2)
exceeds the conversion rate from prey to predator (11988621) 1198641
is globally stable if intra-specific competitions (11988611 11988622 and
11988633) are strong Finally 119864
0is unstable and the non-stability of
trivial equilibrium tells us that not all of the populations goto extinction Furthermore our main results imply that thenonlocal delay is harmless for stabilities of all non-negativesteady states of system (1)ndash(3)
There are still many interesting and challenging problemswith respect to system (1)ndash(3) for example the permanence
and stability of periodic solution or almost periodic solutionThese problems are clearly worthy for further investigations
Acknowledgments
This work is partially supported by PRC Grants NSFC11102076 and 11071209 andNSF of theHigher Education Insti-tutions of Jiangsu Province (12KJD110002 and 12KJD110008)
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] H F Huo andW T Li ldquoPeriodic solution of a delayed predator-prey system with Michaelis-Menten type functional responserdquoJournal of Computational and AppliedMathematics vol 166 no2 pp 453ndash463 2004
[3] Z Lin and M Pedersen ldquoStability in a diffusive food-chainmodel with Michaelis-Menten functional responserdquo NonlinearAnalysis Theory Methods amp Applications vol 57 no 3 pp 421ndash433 2004
[4] B X Dai N Zhang and J Z Zou ldquoPermanence for theMichaelis-Menten type discrete three-species ratio-dependentfood chain model with delayrdquo Journal of Mathematical Analysisand Applications vol 324 no 1 pp 728ndash738 2006
[5] N F Britton ldquoSpatial structures and periodic travelling wavesin an integro-differential reaction-diffusion population modelrdquoSIAM Journal on Applied Mathematics vol 50 no 6 pp 1663ndash1688 1990
[6] S A Gourley and N F Britton ldquoInstability of travelling wavesolutions of a population model with nonlocal effectsrdquo IMAJournal of Applied Mathematics vol 51 no 3 pp 299ndash310 1993
[7] X Zhang and R Xu ldquoTraveling waves of a diffusive predator-prey model with nonlocal delay and stage structurerdquo Journal ofMathematical Analysis and Applications vol 373 no 2 pp 475ndash484 2011
[8] Z C Wang W T Li and S G Ruan ldquoExistence and stability oftraveling wave fronts in reaction advection diffusion equationswith nonlocal delayrdquo Journal of Differential Equations vol 238no 1 pp 153ndash200 2007
[9] S A Gourley and S G Ruan ldquoConvergence and travellingfronts in functional differential equations with nonlocal termsa competition modelrdquo SIAM Journal on Mathematical Analysisvol 35 no 3 pp 806ndash822 2003
[10] X H Lai and T X Yao ldquoExponential stability of impul-sive delayed reaction-diffusion cellular neural networks viaPoincare integral inequalityrdquoAbstract and Applied Analysis vol2013 Article ID 131836 10 pages 2013
[11] R Xu and Z E Ma ldquoGlobal stability of a reaction-diffusionpredator-prey model with a nonlocal delayrdquo Mathematical andComputer Modelling vol 50 no 1-2 pp 194ndash206 2009
[12] Z Li F D Chen and M X He ldquoAsymptotic behavior of thereaction-diffusion model of plankton allelopathy with nonlocaldelaysrdquo Nonlinear Analysis Real World Applications vol 12 no3 pp 1748ndash1758 2011
[13] Y Kuang and H L Smith ldquoGlobal stability in diffusive delayLotka-Volterra systemsrdquoDifferential and Integral Equations vol4 no 1 pp 117ndash128 1991
14 Abstract and Applied Analysis
[14] Z G Lin M Pedersen and L Zhang ldquoA predator-prey systemwith stage-structure for predator and nonlocal delayrdquoNonlinearAnalysis Theory Methods amp Applications vol 72 no 3-4 pp2019ndash2030 2010
[15] R Xu ldquoA reaction-diffusion predator-prey model with stagestructure and nonlocal delayrdquo Applied Mathematics and Com-putation vol 175 no 2 pp 984ndash1006 2006
[16] S S Chen and J P Shi ldquoStability and Hopf bifurcation in adiffusive logistic population model with nonlocal delay effectrdquoJournal of Differential Equations vol 253 no 12 pp 3440ndash34702012
[17] H I Freedman Deterministic Mathematical Models in Popula-tion Ecology vol 57 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1980
[18] P A Abrams and L R Ginzburg ldquoThe nature of predation preydepedent ratio dependent or neitherrdquo Trends in Ecology andEvolution vol 15 no 8 pp 337ndash341 2000
[19] D L DeAngelis R A Goldstein and R V OrsquoNeill ldquoA modelfor trophic iterationrdquo Ecology vol 56 no 4 pp 881ndash892 1975
[20] C V Pao ldquoDynamics of nonlinear parabolic systems with timedelaysrdquo Journal of Mathematical Analysis and Applications vol198 no 3 pp 751ndash779 1996
[21] F Xu Y Hou and Z G Lin ldquoTime delay parabolic system ina predator-prey model with stage structurerdquo Acta MathematicaSinica vol 48 no 6 pp 1121ndash1130 2005 (Chinese)
[22] P Y H Pang and M X Wang ldquoStrategy and stationary patternin a three-species predator-prey modelrdquo Journal of DifferentialEquations vol 200 no 2 pp 245ndash273 2004
[23] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981
[24] K J Brown P C Dunne and R A Gardner ldquoA semilinearparabolic system arising in the theory of superconductivityrdquoJournal of Differential Equations vol 40 no 2 pp 232ndash252 1981
[25] S A Gourley and J W H So ldquoDynamics of a food-limitedpopulation model incorporating nonlocal delays on a finitedomainrdquo Journal of Mathematical Biology vol 44 no 1 pp 49ndash78 2002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
It is derived from (27) and (28) that
1198641015840(119905) le minusint
Ω
(1205721198891119906lowast
1
11990621
1003816100381610038161003816nabla119906110038161003816100381610038162
+1198892119906lowast
2
11990622
1003816100381610038161003816nabla119906210038161003816100381610038162
+1205731198893119906lowast
3
11990623
1003816100381610038161003816nabla119906310038161003816100381610038162
)119889119909
minus [12057211988611minus12057211988612119906lowast
2
11989821
minus120572119886121205981
1198981
minus119886211205983
1198981
]intΩ
(1199061minus 119906lowast
1)2
119889119909
minus [11988622minus11988623119906lowast
3
11989822
minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632120573
411989821205984
]intΩ
(1199062minus 119906lowast
2)2
119889119909
minus [11988633120573 minus
119886231205982
1198982
minus120573119886321205984
1198982
]intΩ
(1199063minus 119906lowast
3)2
119889119909
(29)
Since (1199061 1199062 1199063) is the unique positive solution of system (1)
Using Proposition 5 there exists a constant119862which does notdepend on119909 isin Ω or 119905 ge 0 such that 119906
119894(sdot 119905)infinle 119862 (119894 = 1 2 3)
for 119905 ge 0 By Theorem 1198602in [24] we have
1003817100381710038171003817119906119894 (sdot 119905)10038171003817100381710038171198622120572(Ω) le 119862 forall119905 ge 1 (30)
Assume that
1198661 119886111198982
1gt 11988612119906lowast
2
1198662 119886221198982
2gt 11988623119906lowast
3+21198862311988632
11988633
+211988612119886211198982
2
1198861111989821minus 11988612119906lowast2
(31)
and denote
1198971= 12057211988611minus12057211988612119906lowast
2
11989821
minus120572119886121205981
1198981
minus119886211205983
1198981
1198972= 11988622minus11988623119906lowast
3
11989822
minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632120573
411989821205984
1198973= 11988633120573 minus
119886231205982
1198982
minus120573119886321205984
1198982
(32)
Then (29) is transformed into
1198641015840(119905) le minusint
Ω
(1205721198891119906lowast
1
11990621
1003816100381610038161003816nabla119906110038161003816100381610038162
+1198892119906lowast
2
11990622
1003816100381610038161003816nabla119906210038161003816100381610038162
+1205731198893119906lowast
3
11990623
1003816100381610038161003816nabla119906310038161003816100381610038162
)119889119909
minus
3
sum
119894=1
119897119894intΩ
(119906119894minus 119906lowast
119894)2
119889119909
(33)
If we choose
120572 =11988621
11988612
120573 =11988623
11988632
1205981= 1205983=119886111198982
1minus 11988612119906lowast
2
4119898111988612
1205982= 1205984=119898211988633
411988632
(34)
then 119897119894gt 0 (119894 = 1 2 3) Therefore we have
1198641015840(119905) le minus
3
sum
119894=1
119897119894intΩ
(119906119894minus 119906lowast
119894)2
119889119909 (35)
FromProposition 6 we can see that the solution of system(1) and (3) is bounded and so are the derivatives of (119906
119894minus 119906lowast
119894)
(119894 = 1 2 3) by the equations in (1) Applying Lemma 1 weobtain that
lim119905rarrinfin
intΩ
(119906119894minus 119906lowast
119894) 119889119909 = 0 (119894 = 1 2 3) (36)
Recomputing 1198641015840(119905) gives
1198641015840(119905) le minusint
Ω
(1205721198891119906lowast
1
11990621
1003816100381610038161003816nabla119906110038161003816100381610038162
+1198892119906lowast
2
11990622
1003816100381610038161003816nabla119906210038161003816100381610038162
+1205731198893119906lowast
3
11990623
1003816100381610038161003816nabla119906310038161003816100381610038162
)119889119909
le minus119888intΩ
(1003816100381610038161003816nabla1199061
10038161003816100381610038162
+1003816100381610038161003816nabla1199062
10038161003816100381610038162
+1003816100381610038161003816nabla1199063
10038161003816100381610038162
) 119889119909 = minus119892 (119905)
(37)
where 119888 = min1205721198891119906lowast
11198722
1 1198892119906lowast
21198722
2 1205731198893119906lowast
311987223 Using (30)
and (1) we obtain that the derivative of 119892(119905) is bounded in[1 +infin) From Lemma 1 we conclude that 119892(119905) rarr 0 as 119905 rarrinfin Therefore
lim119905rarrinfin
intΩ
(1003816100381610038161003816nabla1199061
10038161003816100381610038162
+1003816100381610038161003816nabla1199062
10038161003816100381610038162
+1003816100381610038161003816nabla1199063
10038161003816100381610038162
) 119889119909 = 0 (38)
Applying the Poincare inequality
intΩ
120582|119906 minus 119906|2119889119909 le int
Ω
|nabla119906|2119889119909 (39)
leads to
lim119905rarrinfin
intΩ
(119906119894minus 119906119894)2
119889119909 = 0 (119894 = 1 2 3) (40)
where 119906119894= (1|Ω|) int
Ω119906119894119889119909 and 120582 is the smallest positive
eigenvalue ofminusΔwith the homogeneousNeumann conditionTherefore
|Ω| (1199061 (119905) minus 119906lowast
1)2
= intΩ
(1199061(119905) minus 119906
lowast
1)2
119889119909
= intΩ
(1199061(119905) minus 119906
1(119905 119909) + 119906
1(119905 119909) minus 119906
lowast
1)2
119889119909
le 2intΩ
(1199061(119905) minus 119906
1(119905 119909))
2
119889119909
+ 2intΩ
(1199061(119905 119909) minus 119906
lowast
1)2
119889119909
(41)
8 Abstract and Applied Analysis
So we have 1199061(119905) rarr 119906
lowast
1as 119905 rarr infin Similarly 119906
2(119905) rarr 119906
lowast
2
and 1199063(119905) rarr 119906
lowast
3as 119905 rarr infin According to (30) there exists
a subsequence 119905119898 and non-negative functions 119908
119894isin 1198622(Ω)
such thatlim119898rarrinfin
1003817100381710038171003817119906119894 (sdot 119905119898) minus 119908119894 (sdot)10038171003817100381710038171198622(Ω) = 0 (119894 = 1 2 3) (42)
Applying (40) and noting that 119906119894(119905) rarr 119906
lowast
119894 we then have119908
119894=
119906lowast
119894 (119894 = 1 2 3) That is
lim119898rarrinfin
1003817100381710038171003817119906119894 (sdot 119905119898) minus 119906lowast
119894
10038171003817100381710038171198622(Ω) = 0 (119894 = 1 2 3) (43)
Furthermore the local stability ofElowast combiningwith (43)gives the following global stability
Theorem 9 Assume that
11986751198861(11988621minus 1198862)
11989811198862
gt 11988611gt11988612Vlowast2
11989821
(44)
1198674and 119866
4hold and then the semi-trivial steady state 119864
2
of system (1)ndash(3) with non-trivial initial functions is globallyasymptotically stable
Proof It is obvious that system (1)ndash(3) always has two non-negative equilibria (119906
1 1199062 1199063) as follows 119864
0= (0 0 0) and
1198641= (119886111988611 0 0) If 119867
1and 119867
4are satisfied system (1) has
the other semitrivial solution denoted by 1198642(Vlowast1 Vlowast2 0) where
1198674 11988631198982gt (11988632minus 1198863) Vlowast
2 (45)
We consider the stability of 1198642under condition 119867
1and
1198674 Equation (1) can be rewritten as
1205971199061
120597119905minus 1198891Δ1199061= 1199061[minus11988611(1199061minus Vlowast
1) minus11988612(1199062minus Vlowast2)
1198981+ 1199061
+11988612Vlowast2(1199061minus Vlowast1)
(1198981+ 1199061) (1198981+ Vlowast1)]
1205971199062
120597119905minus 1198892Δ1199062
= 1199062[intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus Vlowast
1)
(1198981+ 1199061(119904 119910)) (119898
1+ Vlowast1)119889119904 119889119910
minus11988622(1199062minus Vlowast
2) minus
119886231199063
1198982+ 1199062
]
1205971199063
120597119905minus 1198893Δ1199063
= 1199063[intΩ
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321198982(1199062(119904 119910) minus Vlowast
2)
(1198982+ 1199062(119904 119910)) (119898
2+ Vlowast2)119889119904 119889119910
minus119886331199063]
(46)
Similar to the argument ofTheorem 8 we have 1199061(119909 119905) rarr Vlowast
1
and 1199062(119909 119905) rarr Vlowast
2as 119905 rarr infin uniformly on Ω provided that
the following additional condition holds
1198663 119886111198982
1gt 11988612Vlowast
2
1198664 119886221198982
2gt1198862311988632
11988633
+211988612119886211198982
2
1198861111989821minus 11988612Vlowast2
(47)
Next we consider the asymptotic behavior of 1199063(119905 119909) Let
120579 =1198863minus (11988632minus 1198863) Vlowast2
2120575 120575 =
100381610038161003816100381611988632 minus 11988631003816100381610038161003816
(48)
Then there exists 1199051gt 0 such that
Vlowast
2minus 120579 lt 119906
2(119905 119909) lt V
lowast
2+ 120579 forall119905 gt 119905
1 119909 isin Ω (49)
Consider the following two systems
1199083119905minus 1198893Δ1199083= 1199083[minus1198863+11988632(Vlowast2+ 120579)
1198982+ Vlowast2+ 120579
minus 119886331199083]
(119905 119909) isin [1199051 +infin) times Ω
1205971199083
120597]= 0 (119905 119909) isin [119905
1 +infin) times 120597Ω
1199083= 1199063 (119905 119909) isin (minusinfin 119905
1] times Ω
1198823119905minus 1198893Δ1198823= 1198823[minus1198863+11988632(Vlowast2minus 120579)
1198982+ Vlowast2minus 120579
minus 11988633V3]
(119905 119909) isin [1199051 +infin) times Ω
1205971198823
120597]= 0 (119905 119909) isin [119905
1 +infin) times 120597Ω
1198823= 1199063 (119905 119909) isin (minusinfin 119905
1] times Ω
(50)
Combining comparison principle with (50) we obtain that
1198823(119905 119909) le 119906
3(119905 119909) le 119908
3(119905 119909) forall119905 gt 119905
1 119909 isin Ω (51)
By Lemma 4 we obtain
0 = lim119905rarrinfin
1198823(119905 119909) le inf 119906
3(119905 119909) le sup 119906
3(119905 119909)
le lim119905rarrinfin
1199083(119905 119909) = 0 forall119909 isin Ω
(52)
which implies that lim119905rarrinfin
1199063(119905 119909) = 0 uniformly onΩ
Theorem 10 Suppose that1198665and119866
6hold and then the semi-
trivial steady state 1198641of system (1)ndash(3) with non-trivial initial
functions is globally asymptotically stable
Abstract and Applied Analysis 9
Proof We study the stability of the semi-trivial solution 1198641=
(1 0 0) Similarly the equations in (1) can be written as
1205971199061
120597119905minus 1198891Δ1199061= 1199061[minus11988611(1199061minus 1) minus
119886121199062
1198981+ 1199061
]
1205971199062
120597119905minus 1198892Δ1199062
= 1199062[minus1198862+ intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus
1)
(1198981+ 1199061(119904 119910)) (119898
1+ 1)119889119904 119889119910
+119886211
1198981+ 1
minus 119886221199062minus119886231199063
1198982+ 1199062
]
1205971199063
120597119905minus 1198893Δ1199063
= 1199063[minus1198863+ intΩ
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321199062(119904 119910)
1198982+ 1199062(119904 119910)
119889119904 119889119910 minus 119886331199063]
(53)
Define
119881 (119905) = 120572intΩ
[1199061minus 1minus 1log 11990611
] 119889119909 + intΩ
1199062119889119909
+ 120573intΩ
1199063119889119909
(54)
Calculating the derivative of 119881(119905) along 1198641 we get from (54)
that
1198811015840(119905) le minusint
Ω
[120572(11988611minus119886121205981
1198981
)] (1199061minus 1)2
119889119909
+ intΩ
[11988622minus12057211988612
411989811205981
minus11988623
412059821198982
minus11988621
411989811205983
] 1199062
2119889119909
+ intΩ
[12057311988633minus119886231205982
1198982
] 1199062
3119889119909
minus intΩ
[(1198862minus119886211
1198981+ 1
)1199062+ 11988631199063] 119889119909
+119886211205983
1198981
intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times (1199061(119904 119910) minus
1)2
119889119904 119889119910 119889119909
+11988632120573
412059841198982
∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times 1199062
2(119904 119910) 119889119904 119889119910 119889119909
(55)
Define119864 (119905) = 119881 (119905)
+119886211205983
1198981
∬Ω
int
+infin
0
int
119905
119905minus119903
1198701(119909 119910 119903)
times (1199061(119904 119910) minus 119906
lowast
1)2
119889119904 119889119903 119889119910 119889119909
+11988632120573
412059841198982
∬Ω
int
+infin
0
int
119905
119905minus119903
1198702(119909 119910 119903)
times (1199062(119904 119910) minus 119906
lowast
2)2
119889119904 119889119903 119889119910 119889119909
(56)
It is easy to see that
1198641015840(119905) le minus120572119889
11intΩ
1003816100381610038161003816nabla119906110038161003816100381610038162
11990621
119889119909
minus intΩ
[120572(11988611minus119886121205981
1198981
) minus119886211205983
1198981
] (1199061minus 1)2
119889119909
+ intΩ
[11988622minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632
411989821205984
] 1199062
2119889119909
+ intΩ
[12057311988633minus119886231205982
1198982
minus119886321205731205984
1198982
] 1199062
3119889119909
minus intΩ
[(1198862minus119886211
1198981+ 1
)1199062minus 11988631205731199063] 119889119909
(57)
Assume that1198665 1198861(11988621minus 1198862) lt 119898
1119886211988611
1198666 119886221198982
2gt1198862311988632
11988633
+211988612119886211198982
2
1198861111989821
(58)
Let
1198971= 12057211988611minus120572119886121205981
1198981
minus119886211205983
1198981
1198972= 11988622minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632120573
411989821205984
1198973= 11988633120573 minus
119886231205982
1198982
minus120573119886321205984
1198982
(59)
Choose
120572 =11988621
11988612
120573 =11988623
11988632
1205981= 1205983=119886111198981
411988612
1205982= 1205984=119898211988633
411988632
(60)
Then we get 119897119894gt 0 (119894 = 1 2 3) Therefore we have
lim119905rarrinfin
1199061(119905 119909) =
1 (61)
uniformly onΩ
10 Abstract and Applied Analysis
tx
u1
02
46
8
01
23
40
1
2
3
4
(a)
02
46
8
01
23
40
1
2
3
4
t
x
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 2 Global stability of the positive equilibrium 119864lowast with an initial condition (1206011 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos 119909 27 + 05 sin119909)
Next we consider the asymptotic behavior of 1199062(119905 119909) and
1199063(119905 119909) For any 119879 gt 0 integrating (57) over [0 119879] yields
119864 (119879) + 1205721198891
10038171003817100381710038171003817100381710038171003817
10038161003816100381610038161003816100381610038161003816
nabla1199061
1199061
10038161003816100381610038161003816100381610038161003816
10038171003817100381710038171003817100381710038171003817
2
2
+ 1198971
100381710038171003817100381710038161003816100381610038161199061 minus 1
100381610038161003816100381610038171003817100381710038172
2
+ 1198972
100381710038171003817100381710038161003816100381610038161199062100381610038161003816100381610038171003817100381710038172
2+ 1198973
100381710038171003817100381710038161003816100381610038161199063100381610038161003816100381610038171003817100381710038172
2le 119864 (0)
(62)
where |119906119894|2
2= int119879
0intΩ1199062
119894119889119909 119889119905 It implies that |119906
119894|2le 119862119894(119894 =
2 3) for the constant 119862119894which is independent of 119879 Now we
consider the boundedness of |nabla1199062|2and |nabla119906
3|2 From the
Greenrsquos identity we obtain
1198892int
119879
0
intΩ
1003816100381610038161003816nabla1199062 (119905 119909)10038161003816100381610038162
119889119909 119889119905
= minus1198892int
119879
0
intΩ
1199062(119905 119909) Δ119906
2(119905 119909) 119889119909 119889119905
= minusint
119879
0
intΩ
1199062
1205971199062
120597119905119889119909 119889119905 minus 119886
2int
119879
0
intΩ
1199062
2119889119909 119889119905
minus 11988622int
119879
0
intΩ
1199063
2119889119909 119889119905 minus 119886
23int
119879
0
intΩ
1199062
21199063
1198982+ 1199063
119889119909 119889119905
+ 11988621int
119879
0
∬Ω
int
119905
minusinfin
1199061(119904 119910) 119906
2
2(119905 119909)
1198981+ 1199061(119904 119910)
times 1198701(119909 119910 119905 minus 119904) 119889119904 119889119910 119889119909 119889119905
(63)
Note that
int
119879
0
intΩ
1199062(119905 119909)
1205971199062(119905 119909)
120597119905119889119909 119889119905
= int
119879
0
intΩ
1
2
1205971199062
2
120597119905119889119905 119889119909
=1
2intΩ
1199062
2(119879 119909) 119889119909 minus
1
2intΩ
1199062
2(0 119909) 119889119909 le 119872
2
2|Ω|
int
119879
0
intΩ
1199062
21199063119889119909 119889119905 le 119872
3int
119879
0
intΩ
1199062
2119889119909 119889119905
Abstract and Applied Analysis 11
02
46
8
01
23
40
1
2
3
4
tx
u1
(a)
02
46
8
01
23
40
1
2
3
4
tx
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 3 Global stability of the positive equilibrium 1198642with an initial condition (120601
1 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos 119909 27 + 05 sin119909)
int
119879
0
intΩ
1199063
2119889119909 119889119905 le 119872
2int
119879
0
intΩ
1199062
2119889119909 119889119905
int
119879
0
∬Ω
int
119905
minusinfin
1199061(119904 119910) 119906
2
2(119905 119909)
1198981+ 1199061(119904 119910)
1198701(119909 119910 119905 minus 119904) 119889119904 119889119910 119889119909 119889119905
le int
119879
0
intΩ
1199062
2(119905 119909) 119889119909 119889119905
(64)
Thus we get |nabla1199062|2le 1198624 In a similar way we have
|nabla1199063|2le 1198625 Here 119862
4and 119862
5are independent of 119879
It is easy to see that 1199062(119905 119909) 119906
3(119905 119909) isin 119871
2((0infin)
11988212(Ω)) These imply that
lim119905rarrinfin
1003817100381710038171003817119906119894 (sdot 119905)100381710038171003817100381711988212(Ω) = 0 119894 = 2 3 (65)
From the Sobolev compact embedding theorem we know
lim119905rarrinfin
1003817100381710038171003817119906119894 (sdot 119905)1003817100381710038171003817119862(Ω) = 0 119894 = 2 3 (66)
In the end we show that the trivial solution 1198640is an
unstable equilibrium Similarly to the local stability to 119864lowast wecan get the characteristic equation of 119864
0as
(120582 + 1205831198941198631minus 1198861) (120582 + 120583
1198941198632+ 1198862) (120582 + 120583
1198941198633+ 1198863) = 0 (67)
If 119894 = 1 then 1205831= 0 It is easy to see that this equation admits
a positive solution 120582 = 1198861 According to Theorem 51 in [23]
we have the following result
Theorem 11 The trivial equilibrium 1198640is an unstable equilib-
rium of system (1)ndash(3)
4 Numerical Illustrations
In this section we performnumerical simulations to illustratethe theoretical results given in Section 3
12 Abstract and Applied Analysis
u1
02
46
8
01
23
40
1
2
3
4
tx
(a)
02
46
8
01
23
40
1
2
3
4
tx
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 4 Global stability of the positive equilibrium 1198641with an initial condition (120601
1 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos119909 27 + 05 sin119909)
In the following we always take Ω = [0 120587] 119870119894(119909 119910 119905) =
119866119894(119909 119910 119905)119896
119894(119905) where 119896
119894(119905) = (1120591
lowast)119890minus119905120591lowast
(119894 = 1 2) and
1198661(119909 119910 119905) =
1
120587+2
120587
infin
sum
119899=1
119890minus11988921198992
119905 cos 119899119909 cos 119899119910
1198662(119909 119910 119905) =
1
120587+2
120587
infin
sum
119899=1
119890minus11988931198992
119905 cos 119899119909 cos 119899119910
(68)
However it is difficult for us to simulate our results directlybecause of the nonlocal term Similar to [25] the equationsin (1) can be rewritten as follows
1205971199061
120597119905minus 1198891Δ1199061= 1199061(1198861minus 119886111199061minus119886121199062
1198981+ 1199061
)
1205971199062
120597119905minus 1198892Δ1199062= 1199062(minus1198862+ 11988621V1minus 119886221199062minus119886231199063
1198982+ 1199063
)
1205971199063
120597119905minus 1198893Δ1199063= 1199063(minus1198863+ 11988632V2minus 119886331199063)
120597V1
120597119905minus 1198892ΔV1=1
120591lowast(
1199061
1198981+ 1199061
minus V1)
120597V2
120597119905minus 1198893ΔV2=1
120591lowast(
1199062
1198982+ 1199062
minus V2)
(69)
where
V119894= int
120587
0
int
119905
minusinfin
119866119894(119909 119910 119905 minus 119904)
1
120591lowast119890minus(119905minus119904)120591
lowast 119906119894(119904 119910)
119898119894+ 119906119894(119904 119910)
119889119910 119889119904
119894 = 1 2
(70)
Each component is considered with homogeneous Neumannboundary conditions and the initial condition of V
119894is
V119894(0 119909) = int
120587
0
int
0
minusinfin
119866119894(119909 119910 minus119904)
1
120591lowast119890119904120591lowast 119906119894(119904 119910)
119898119894+ 119906119894(119904 119910)
119889119910 119889119904
119894 = 1 2
(71)
Abstract and Applied Analysis 13
In the following examples we fix some coefficients andassume that 119889
1= 1198892= 1198893= 1 119886
1= 3 119886
12= 1 119898
1= 1198982=
1 1198862= 1 119886
22= 12 119886
23= 1 119886
33= 12 and 120591lowast = 1 The
asymptotic behaviors of system (1)ndash(3) are shownby choosingdifferent coefficients 119886
11 11988621 1198863 and 119886
32
Example 12 Let 11988611= 539 119886
21= 8113 119886
3= 113 and
11988632= 14Then it is easy to see that the system admits a unique
positive equilibrium 119864lowast(12 112 112) By Theorem 8 we
see that the positive solution (1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of
system (1)ndash(3) converges to 119864lowast as 119905 rarr infin See Figure 2
Example 13 Let 11988611
= 6 11988621
= 12 1198863= 14 and
11988632= 1 Clearly 119867
2does not hold Hence the positive
steady state is not feasible System (1) admits two semi-trivial steady state 119864
2(04732 02372 0) and 119864
1(12 0 0)
According to Theorem 9 we know that the positive solution(1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of system (1)ndash(3) converges to 119864
2
as 119905 rarr infin See Figure 3
Example 14 Let 11988611= 6 119886
21= 2 119886
3= 14 and 119886
32= 1
Clearly1198671does not hold Hence 119864lowast and 119864
2are not feasible
System (1) has a unique semi-trivial steady state 1198641(12 0 0)
According to Theorem 10 we know that the positive solution(1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of system (1)ndash(3) converges to 119864
1
as 119905 rarr infin See Figure 4
5 Discussion
In this paper we incorporate nonlocal delay into a three-species food chain model with Michaelis-Menten functionalresponse to represent a delay due to the gestation of the preda-tor The conditions under which the spatial homogeneousequilibria are asymptotically stable are given by using theLyapunov functional
We now summarize the ecological meanings of ourtheoretical results Firstly the positive equilibrium 119864
lowast ofsystem (1)ndash(3) exists under the high birth rate of the prey(1198861) and low death rates (119886
2and 119886
3) of predator and top
predator 119864lowast is globally stable if the intraspecific competition11988611is neither too big nor too small and the maximum harvest
rates 11988612 11988623
are small enough Secondly the semi-trivialequilibrium 119864
2of system (1)ndash(3) exists if the birth rate of the
prey (1198861) is high death rate of predator 119886
2is low and the death
rate (1198863) exceeds the conversion rate from predator to top
predator (11988632) 1198642is globally stable if the maximum harvest
rates 11988612 11988623are small and the intra-specific competition 119886
11
is neither too big nor too small Thirdly system (1)ndash(3) hasonly one semi-trivial equilibrium 119864
1when the death rate (119886
2)
exceeds the conversion rate from prey to predator (11988621) 1198641
is globally stable if intra-specific competitions (11988611 11988622 and
11988633) are strong Finally 119864
0is unstable and the non-stability of
trivial equilibrium tells us that not all of the populations goto extinction Furthermore our main results imply that thenonlocal delay is harmless for stabilities of all non-negativesteady states of system (1)ndash(3)
There are still many interesting and challenging problemswith respect to system (1)ndash(3) for example the permanence
and stability of periodic solution or almost periodic solutionThese problems are clearly worthy for further investigations
Acknowledgments
This work is partially supported by PRC Grants NSFC11102076 and 11071209 andNSF of theHigher Education Insti-tutions of Jiangsu Province (12KJD110002 and 12KJD110008)
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] H F Huo andW T Li ldquoPeriodic solution of a delayed predator-prey system with Michaelis-Menten type functional responserdquoJournal of Computational and AppliedMathematics vol 166 no2 pp 453ndash463 2004
[3] Z Lin and M Pedersen ldquoStability in a diffusive food-chainmodel with Michaelis-Menten functional responserdquo NonlinearAnalysis Theory Methods amp Applications vol 57 no 3 pp 421ndash433 2004
[4] B X Dai N Zhang and J Z Zou ldquoPermanence for theMichaelis-Menten type discrete three-species ratio-dependentfood chain model with delayrdquo Journal of Mathematical Analysisand Applications vol 324 no 1 pp 728ndash738 2006
[5] N F Britton ldquoSpatial structures and periodic travelling wavesin an integro-differential reaction-diffusion population modelrdquoSIAM Journal on Applied Mathematics vol 50 no 6 pp 1663ndash1688 1990
[6] S A Gourley and N F Britton ldquoInstability of travelling wavesolutions of a population model with nonlocal effectsrdquo IMAJournal of Applied Mathematics vol 51 no 3 pp 299ndash310 1993
[7] X Zhang and R Xu ldquoTraveling waves of a diffusive predator-prey model with nonlocal delay and stage structurerdquo Journal ofMathematical Analysis and Applications vol 373 no 2 pp 475ndash484 2011
[8] Z C Wang W T Li and S G Ruan ldquoExistence and stability oftraveling wave fronts in reaction advection diffusion equationswith nonlocal delayrdquo Journal of Differential Equations vol 238no 1 pp 153ndash200 2007
[9] S A Gourley and S G Ruan ldquoConvergence and travellingfronts in functional differential equations with nonlocal termsa competition modelrdquo SIAM Journal on Mathematical Analysisvol 35 no 3 pp 806ndash822 2003
[10] X H Lai and T X Yao ldquoExponential stability of impul-sive delayed reaction-diffusion cellular neural networks viaPoincare integral inequalityrdquoAbstract and Applied Analysis vol2013 Article ID 131836 10 pages 2013
[11] R Xu and Z E Ma ldquoGlobal stability of a reaction-diffusionpredator-prey model with a nonlocal delayrdquo Mathematical andComputer Modelling vol 50 no 1-2 pp 194ndash206 2009
[12] Z Li F D Chen and M X He ldquoAsymptotic behavior of thereaction-diffusion model of plankton allelopathy with nonlocaldelaysrdquo Nonlinear Analysis Real World Applications vol 12 no3 pp 1748ndash1758 2011
[13] Y Kuang and H L Smith ldquoGlobal stability in diffusive delayLotka-Volterra systemsrdquoDifferential and Integral Equations vol4 no 1 pp 117ndash128 1991
14 Abstract and Applied Analysis
[14] Z G Lin M Pedersen and L Zhang ldquoA predator-prey systemwith stage-structure for predator and nonlocal delayrdquoNonlinearAnalysis Theory Methods amp Applications vol 72 no 3-4 pp2019ndash2030 2010
[15] R Xu ldquoA reaction-diffusion predator-prey model with stagestructure and nonlocal delayrdquo Applied Mathematics and Com-putation vol 175 no 2 pp 984ndash1006 2006
[16] S S Chen and J P Shi ldquoStability and Hopf bifurcation in adiffusive logistic population model with nonlocal delay effectrdquoJournal of Differential Equations vol 253 no 12 pp 3440ndash34702012
[17] H I Freedman Deterministic Mathematical Models in Popula-tion Ecology vol 57 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1980
[18] P A Abrams and L R Ginzburg ldquoThe nature of predation preydepedent ratio dependent or neitherrdquo Trends in Ecology andEvolution vol 15 no 8 pp 337ndash341 2000
[19] D L DeAngelis R A Goldstein and R V OrsquoNeill ldquoA modelfor trophic iterationrdquo Ecology vol 56 no 4 pp 881ndash892 1975
[20] C V Pao ldquoDynamics of nonlinear parabolic systems with timedelaysrdquo Journal of Mathematical Analysis and Applications vol198 no 3 pp 751ndash779 1996
[21] F Xu Y Hou and Z G Lin ldquoTime delay parabolic system ina predator-prey model with stage structurerdquo Acta MathematicaSinica vol 48 no 6 pp 1121ndash1130 2005 (Chinese)
[22] P Y H Pang and M X Wang ldquoStrategy and stationary patternin a three-species predator-prey modelrdquo Journal of DifferentialEquations vol 200 no 2 pp 245ndash273 2004
[23] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981
[24] K J Brown P C Dunne and R A Gardner ldquoA semilinearparabolic system arising in the theory of superconductivityrdquoJournal of Differential Equations vol 40 no 2 pp 232ndash252 1981
[25] S A Gourley and J W H So ldquoDynamics of a food-limitedpopulation model incorporating nonlocal delays on a finitedomainrdquo Journal of Mathematical Biology vol 44 no 1 pp 49ndash78 2002
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
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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
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Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
So we have 1199061(119905) rarr 119906
lowast
1as 119905 rarr infin Similarly 119906
2(119905) rarr 119906
lowast
2
and 1199063(119905) rarr 119906
lowast
3as 119905 rarr infin According to (30) there exists
a subsequence 119905119898 and non-negative functions 119908
119894isin 1198622(Ω)
such thatlim119898rarrinfin
1003817100381710038171003817119906119894 (sdot 119905119898) minus 119908119894 (sdot)10038171003817100381710038171198622(Ω) = 0 (119894 = 1 2 3) (42)
Applying (40) and noting that 119906119894(119905) rarr 119906
lowast
119894 we then have119908
119894=
119906lowast
119894 (119894 = 1 2 3) That is
lim119898rarrinfin
1003817100381710038171003817119906119894 (sdot 119905119898) minus 119906lowast
119894
10038171003817100381710038171198622(Ω) = 0 (119894 = 1 2 3) (43)
Furthermore the local stability ofElowast combiningwith (43)gives the following global stability
Theorem 9 Assume that
11986751198861(11988621minus 1198862)
11989811198862
gt 11988611gt11988612Vlowast2
11989821
(44)
1198674and 119866
4hold and then the semi-trivial steady state 119864
2
of system (1)ndash(3) with non-trivial initial functions is globallyasymptotically stable
Proof It is obvious that system (1)ndash(3) always has two non-negative equilibria (119906
1 1199062 1199063) as follows 119864
0= (0 0 0) and
1198641= (119886111988611 0 0) If 119867
1and 119867
4are satisfied system (1) has
the other semitrivial solution denoted by 1198642(Vlowast1 Vlowast2 0) where
1198674 11988631198982gt (11988632minus 1198863) Vlowast
2 (45)
We consider the stability of 1198642under condition 119867
1and
1198674 Equation (1) can be rewritten as
1205971199061
120597119905minus 1198891Δ1199061= 1199061[minus11988611(1199061minus Vlowast
1) minus11988612(1199062minus Vlowast2)
1198981+ 1199061
+11988612Vlowast2(1199061minus Vlowast1)
(1198981+ 1199061) (1198981+ Vlowast1)]
1205971199062
120597119905minus 1198892Δ1199062
= 1199062[intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus Vlowast
1)
(1198981+ 1199061(119904 119910)) (119898
1+ Vlowast1)119889119904 119889119910
minus11988622(1199062minus Vlowast
2) minus
119886231199063
1198982+ 1199062
]
1205971199063
120597119905minus 1198893Δ1199063
= 1199063[intΩ
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321198982(1199062(119904 119910) minus Vlowast
2)
(1198982+ 1199062(119904 119910)) (119898
2+ Vlowast2)119889119904 119889119910
minus119886331199063]
(46)
Similar to the argument ofTheorem 8 we have 1199061(119909 119905) rarr Vlowast
1
and 1199062(119909 119905) rarr Vlowast
2as 119905 rarr infin uniformly on Ω provided that
the following additional condition holds
1198663 119886111198982
1gt 11988612Vlowast
2
1198664 119886221198982
2gt1198862311988632
11988633
+211988612119886211198982
2
1198861111989821minus 11988612Vlowast2
(47)
Next we consider the asymptotic behavior of 1199063(119905 119909) Let
120579 =1198863minus (11988632minus 1198863) Vlowast2
2120575 120575 =
100381610038161003816100381611988632 minus 11988631003816100381610038161003816
(48)
Then there exists 1199051gt 0 such that
Vlowast
2minus 120579 lt 119906
2(119905 119909) lt V
lowast
2+ 120579 forall119905 gt 119905
1 119909 isin Ω (49)
Consider the following two systems
1199083119905minus 1198893Δ1199083= 1199083[minus1198863+11988632(Vlowast2+ 120579)
1198982+ Vlowast2+ 120579
minus 119886331199083]
(119905 119909) isin [1199051 +infin) times Ω
1205971199083
120597]= 0 (119905 119909) isin [119905
1 +infin) times 120597Ω
1199083= 1199063 (119905 119909) isin (minusinfin 119905
1] times Ω
1198823119905minus 1198893Δ1198823= 1198823[minus1198863+11988632(Vlowast2minus 120579)
1198982+ Vlowast2minus 120579
minus 11988633V3]
(119905 119909) isin [1199051 +infin) times Ω
1205971198823
120597]= 0 (119905 119909) isin [119905
1 +infin) times 120597Ω
1198823= 1199063 (119905 119909) isin (minusinfin 119905
1] times Ω
(50)
Combining comparison principle with (50) we obtain that
1198823(119905 119909) le 119906
3(119905 119909) le 119908
3(119905 119909) forall119905 gt 119905
1 119909 isin Ω (51)
By Lemma 4 we obtain
0 = lim119905rarrinfin
1198823(119905 119909) le inf 119906
3(119905 119909) le sup 119906
3(119905 119909)
le lim119905rarrinfin
1199083(119905 119909) = 0 forall119909 isin Ω
(52)
which implies that lim119905rarrinfin
1199063(119905 119909) = 0 uniformly onΩ
Theorem 10 Suppose that1198665and119866
6hold and then the semi-
trivial steady state 1198641of system (1)ndash(3) with non-trivial initial
functions is globally asymptotically stable
Abstract and Applied Analysis 9
Proof We study the stability of the semi-trivial solution 1198641=
(1 0 0) Similarly the equations in (1) can be written as
1205971199061
120597119905minus 1198891Δ1199061= 1199061[minus11988611(1199061minus 1) minus
119886121199062
1198981+ 1199061
]
1205971199062
120597119905minus 1198892Δ1199062
= 1199062[minus1198862+ intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus
1)
(1198981+ 1199061(119904 119910)) (119898
1+ 1)119889119904 119889119910
+119886211
1198981+ 1
minus 119886221199062minus119886231199063
1198982+ 1199062
]
1205971199063
120597119905minus 1198893Δ1199063
= 1199063[minus1198863+ intΩ
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321199062(119904 119910)
1198982+ 1199062(119904 119910)
119889119904 119889119910 minus 119886331199063]
(53)
Define
119881 (119905) = 120572intΩ
[1199061minus 1minus 1log 11990611
] 119889119909 + intΩ
1199062119889119909
+ 120573intΩ
1199063119889119909
(54)
Calculating the derivative of 119881(119905) along 1198641 we get from (54)
that
1198811015840(119905) le minusint
Ω
[120572(11988611minus119886121205981
1198981
)] (1199061minus 1)2
119889119909
+ intΩ
[11988622minus12057211988612
411989811205981
minus11988623
412059821198982
minus11988621
411989811205983
] 1199062
2119889119909
+ intΩ
[12057311988633minus119886231205982
1198982
] 1199062
3119889119909
minus intΩ
[(1198862minus119886211
1198981+ 1
)1199062+ 11988631199063] 119889119909
+119886211205983
1198981
intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times (1199061(119904 119910) minus
1)2
119889119904 119889119910 119889119909
+11988632120573
412059841198982
∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times 1199062
2(119904 119910) 119889119904 119889119910 119889119909
(55)
Define119864 (119905) = 119881 (119905)
+119886211205983
1198981
∬Ω
int
+infin
0
int
119905
119905minus119903
1198701(119909 119910 119903)
times (1199061(119904 119910) minus 119906
lowast
1)2
119889119904 119889119903 119889119910 119889119909
+11988632120573
412059841198982
∬Ω
int
+infin
0
int
119905
119905minus119903
1198702(119909 119910 119903)
times (1199062(119904 119910) minus 119906
lowast
2)2
119889119904 119889119903 119889119910 119889119909
(56)
It is easy to see that
1198641015840(119905) le minus120572119889
11intΩ
1003816100381610038161003816nabla119906110038161003816100381610038162
11990621
119889119909
minus intΩ
[120572(11988611minus119886121205981
1198981
) minus119886211205983
1198981
] (1199061minus 1)2
119889119909
+ intΩ
[11988622minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632
411989821205984
] 1199062
2119889119909
+ intΩ
[12057311988633minus119886231205982
1198982
minus119886321205731205984
1198982
] 1199062
3119889119909
minus intΩ
[(1198862minus119886211
1198981+ 1
)1199062minus 11988631205731199063] 119889119909
(57)
Assume that1198665 1198861(11988621minus 1198862) lt 119898
1119886211988611
1198666 119886221198982
2gt1198862311988632
11988633
+211988612119886211198982
2
1198861111989821
(58)
Let
1198971= 12057211988611minus120572119886121205981
1198981
minus119886211205983
1198981
1198972= 11988622minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632120573
411989821205984
1198973= 11988633120573 minus
119886231205982
1198982
minus120573119886321205984
1198982
(59)
Choose
120572 =11988621
11988612
120573 =11988623
11988632
1205981= 1205983=119886111198981
411988612
1205982= 1205984=119898211988633
411988632
(60)
Then we get 119897119894gt 0 (119894 = 1 2 3) Therefore we have
lim119905rarrinfin
1199061(119905 119909) =
1 (61)
uniformly onΩ
10 Abstract and Applied Analysis
tx
u1
02
46
8
01
23
40
1
2
3
4
(a)
02
46
8
01
23
40
1
2
3
4
t
x
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 2 Global stability of the positive equilibrium 119864lowast with an initial condition (1206011 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos 119909 27 + 05 sin119909)
Next we consider the asymptotic behavior of 1199062(119905 119909) and
1199063(119905 119909) For any 119879 gt 0 integrating (57) over [0 119879] yields
119864 (119879) + 1205721198891
10038171003817100381710038171003817100381710038171003817
10038161003816100381610038161003816100381610038161003816
nabla1199061
1199061
10038161003816100381610038161003816100381610038161003816
10038171003817100381710038171003817100381710038171003817
2
2
+ 1198971
100381710038171003817100381710038161003816100381610038161199061 minus 1
100381610038161003816100381610038171003817100381710038172
2
+ 1198972
100381710038171003817100381710038161003816100381610038161199062100381610038161003816100381610038171003817100381710038172
2+ 1198973
100381710038171003817100381710038161003816100381610038161199063100381610038161003816100381610038171003817100381710038172
2le 119864 (0)
(62)
where |119906119894|2
2= int119879
0intΩ1199062
119894119889119909 119889119905 It implies that |119906
119894|2le 119862119894(119894 =
2 3) for the constant 119862119894which is independent of 119879 Now we
consider the boundedness of |nabla1199062|2and |nabla119906
3|2 From the
Greenrsquos identity we obtain
1198892int
119879
0
intΩ
1003816100381610038161003816nabla1199062 (119905 119909)10038161003816100381610038162
119889119909 119889119905
= minus1198892int
119879
0
intΩ
1199062(119905 119909) Δ119906
2(119905 119909) 119889119909 119889119905
= minusint
119879
0
intΩ
1199062
1205971199062
120597119905119889119909 119889119905 minus 119886
2int
119879
0
intΩ
1199062
2119889119909 119889119905
minus 11988622int
119879
0
intΩ
1199063
2119889119909 119889119905 minus 119886
23int
119879
0
intΩ
1199062
21199063
1198982+ 1199063
119889119909 119889119905
+ 11988621int
119879
0
∬Ω
int
119905
minusinfin
1199061(119904 119910) 119906
2
2(119905 119909)
1198981+ 1199061(119904 119910)
times 1198701(119909 119910 119905 minus 119904) 119889119904 119889119910 119889119909 119889119905
(63)
Note that
int
119879
0
intΩ
1199062(119905 119909)
1205971199062(119905 119909)
120597119905119889119909 119889119905
= int
119879
0
intΩ
1
2
1205971199062
2
120597119905119889119905 119889119909
=1
2intΩ
1199062
2(119879 119909) 119889119909 minus
1
2intΩ
1199062
2(0 119909) 119889119909 le 119872
2
2|Ω|
int
119879
0
intΩ
1199062
21199063119889119909 119889119905 le 119872
3int
119879
0
intΩ
1199062
2119889119909 119889119905
Abstract and Applied Analysis 11
02
46
8
01
23
40
1
2
3
4
tx
u1
(a)
02
46
8
01
23
40
1
2
3
4
tx
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 3 Global stability of the positive equilibrium 1198642with an initial condition (120601
1 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos 119909 27 + 05 sin119909)
int
119879
0
intΩ
1199063
2119889119909 119889119905 le 119872
2int
119879
0
intΩ
1199062
2119889119909 119889119905
int
119879
0
∬Ω
int
119905
minusinfin
1199061(119904 119910) 119906
2
2(119905 119909)
1198981+ 1199061(119904 119910)
1198701(119909 119910 119905 minus 119904) 119889119904 119889119910 119889119909 119889119905
le int
119879
0
intΩ
1199062
2(119905 119909) 119889119909 119889119905
(64)
Thus we get |nabla1199062|2le 1198624 In a similar way we have
|nabla1199063|2le 1198625 Here 119862
4and 119862
5are independent of 119879
It is easy to see that 1199062(119905 119909) 119906
3(119905 119909) isin 119871
2((0infin)
11988212(Ω)) These imply that
lim119905rarrinfin
1003817100381710038171003817119906119894 (sdot 119905)100381710038171003817100381711988212(Ω) = 0 119894 = 2 3 (65)
From the Sobolev compact embedding theorem we know
lim119905rarrinfin
1003817100381710038171003817119906119894 (sdot 119905)1003817100381710038171003817119862(Ω) = 0 119894 = 2 3 (66)
In the end we show that the trivial solution 1198640is an
unstable equilibrium Similarly to the local stability to 119864lowast wecan get the characteristic equation of 119864
0as
(120582 + 1205831198941198631minus 1198861) (120582 + 120583
1198941198632+ 1198862) (120582 + 120583
1198941198633+ 1198863) = 0 (67)
If 119894 = 1 then 1205831= 0 It is easy to see that this equation admits
a positive solution 120582 = 1198861 According to Theorem 51 in [23]
we have the following result
Theorem 11 The trivial equilibrium 1198640is an unstable equilib-
rium of system (1)ndash(3)
4 Numerical Illustrations
In this section we performnumerical simulations to illustratethe theoretical results given in Section 3
12 Abstract and Applied Analysis
u1
02
46
8
01
23
40
1
2
3
4
tx
(a)
02
46
8
01
23
40
1
2
3
4
tx
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 4 Global stability of the positive equilibrium 1198641with an initial condition (120601
1 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos119909 27 + 05 sin119909)
In the following we always take Ω = [0 120587] 119870119894(119909 119910 119905) =
119866119894(119909 119910 119905)119896
119894(119905) where 119896
119894(119905) = (1120591
lowast)119890minus119905120591lowast
(119894 = 1 2) and
1198661(119909 119910 119905) =
1
120587+2
120587
infin
sum
119899=1
119890minus11988921198992
119905 cos 119899119909 cos 119899119910
1198662(119909 119910 119905) =
1
120587+2
120587
infin
sum
119899=1
119890minus11988931198992
119905 cos 119899119909 cos 119899119910
(68)
However it is difficult for us to simulate our results directlybecause of the nonlocal term Similar to [25] the equationsin (1) can be rewritten as follows
1205971199061
120597119905minus 1198891Δ1199061= 1199061(1198861minus 119886111199061minus119886121199062
1198981+ 1199061
)
1205971199062
120597119905minus 1198892Δ1199062= 1199062(minus1198862+ 11988621V1minus 119886221199062minus119886231199063
1198982+ 1199063
)
1205971199063
120597119905minus 1198893Δ1199063= 1199063(minus1198863+ 11988632V2minus 119886331199063)
120597V1
120597119905minus 1198892ΔV1=1
120591lowast(
1199061
1198981+ 1199061
minus V1)
120597V2
120597119905minus 1198893ΔV2=1
120591lowast(
1199062
1198982+ 1199062
minus V2)
(69)
where
V119894= int
120587
0
int
119905
minusinfin
119866119894(119909 119910 119905 minus 119904)
1
120591lowast119890minus(119905minus119904)120591
lowast 119906119894(119904 119910)
119898119894+ 119906119894(119904 119910)
119889119910 119889119904
119894 = 1 2
(70)
Each component is considered with homogeneous Neumannboundary conditions and the initial condition of V
119894is
V119894(0 119909) = int
120587
0
int
0
minusinfin
119866119894(119909 119910 minus119904)
1
120591lowast119890119904120591lowast 119906119894(119904 119910)
119898119894+ 119906119894(119904 119910)
119889119910 119889119904
119894 = 1 2
(71)
Abstract and Applied Analysis 13
In the following examples we fix some coefficients andassume that 119889
1= 1198892= 1198893= 1 119886
1= 3 119886
12= 1 119898
1= 1198982=
1 1198862= 1 119886
22= 12 119886
23= 1 119886
33= 12 and 120591lowast = 1 The
asymptotic behaviors of system (1)ndash(3) are shownby choosingdifferent coefficients 119886
11 11988621 1198863 and 119886
32
Example 12 Let 11988611= 539 119886
21= 8113 119886
3= 113 and
11988632= 14Then it is easy to see that the system admits a unique
positive equilibrium 119864lowast(12 112 112) By Theorem 8 we
see that the positive solution (1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of
system (1)ndash(3) converges to 119864lowast as 119905 rarr infin See Figure 2
Example 13 Let 11988611
= 6 11988621
= 12 1198863= 14 and
11988632= 1 Clearly 119867
2does not hold Hence the positive
steady state is not feasible System (1) admits two semi-trivial steady state 119864
2(04732 02372 0) and 119864
1(12 0 0)
According to Theorem 9 we know that the positive solution(1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of system (1)ndash(3) converges to 119864
2
as 119905 rarr infin See Figure 3
Example 14 Let 11988611= 6 119886
21= 2 119886
3= 14 and 119886
32= 1
Clearly1198671does not hold Hence 119864lowast and 119864
2are not feasible
System (1) has a unique semi-trivial steady state 1198641(12 0 0)
According to Theorem 10 we know that the positive solution(1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of system (1)ndash(3) converges to 119864
1
as 119905 rarr infin See Figure 4
5 Discussion
In this paper we incorporate nonlocal delay into a three-species food chain model with Michaelis-Menten functionalresponse to represent a delay due to the gestation of the preda-tor The conditions under which the spatial homogeneousequilibria are asymptotically stable are given by using theLyapunov functional
We now summarize the ecological meanings of ourtheoretical results Firstly the positive equilibrium 119864
lowast ofsystem (1)ndash(3) exists under the high birth rate of the prey(1198861) and low death rates (119886
2and 119886
3) of predator and top
predator 119864lowast is globally stable if the intraspecific competition11988611is neither too big nor too small and the maximum harvest
rates 11988612 11988623
are small enough Secondly the semi-trivialequilibrium 119864
2of system (1)ndash(3) exists if the birth rate of the
prey (1198861) is high death rate of predator 119886
2is low and the death
rate (1198863) exceeds the conversion rate from predator to top
predator (11988632) 1198642is globally stable if the maximum harvest
rates 11988612 11988623are small and the intra-specific competition 119886
11
is neither too big nor too small Thirdly system (1)ndash(3) hasonly one semi-trivial equilibrium 119864
1when the death rate (119886
2)
exceeds the conversion rate from prey to predator (11988621) 1198641
is globally stable if intra-specific competitions (11988611 11988622 and
11988633) are strong Finally 119864
0is unstable and the non-stability of
trivial equilibrium tells us that not all of the populations goto extinction Furthermore our main results imply that thenonlocal delay is harmless for stabilities of all non-negativesteady states of system (1)ndash(3)
There are still many interesting and challenging problemswith respect to system (1)ndash(3) for example the permanence
and stability of periodic solution or almost periodic solutionThese problems are clearly worthy for further investigations
Acknowledgments
This work is partially supported by PRC Grants NSFC11102076 and 11071209 andNSF of theHigher Education Insti-tutions of Jiangsu Province (12KJD110002 and 12KJD110008)
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] H F Huo andW T Li ldquoPeriodic solution of a delayed predator-prey system with Michaelis-Menten type functional responserdquoJournal of Computational and AppliedMathematics vol 166 no2 pp 453ndash463 2004
[3] Z Lin and M Pedersen ldquoStability in a diffusive food-chainmodel with Michaelis-Menten functional responserdquo NonlinearAnalysis Theory Methods amp Applications vol 57 no 3 pp 421ndash433 2004
[4] B X Dai N Zhang and J Z Zou ldquoPermanence for theMichaelis-Menten type discrete three-species ratio-dependentfood chain model with delayrdquo Journal of Mathematical Analysisand Applications vol 324 no 1 pp 728ndash738 2006
[5] N F Britton ldquoSpatial structures and periodic travelling wavesin an integro-differential reaction-diffusion population modelrdquoSIAM Journal on Applied Mathematics vol 50 no 6 pp 1663ndash1688 1990
[6] S A Gourley and N F Britton ldquoInstability of travelling wavesolutions of a population model with nonlocal effectsrdquo IMAJournal of Applied Mathematics vol 51 no 3 pp 299ndash310 1993
[7] X Zhang and R Xu ldquoTraveling waves of a diffusive predator-prey model with nonlocal delay and stage structurerdquo Journal ofMathematical Analysis and Applications vol 373 no 2 pp 475ndash484 2011
[8] Z C Wang W T Li and S G Ruan ldquoExistence and stability oftraveling wave fronts in reaction advection diffusion equationswith nonlocal delayrdquo Journal of Differential Equations vol 238no 1 pp 153ndash200 2007
[9] S A Gourley and S G Ruan ldquoConvergence and travellingfronts in functional differential equations with nonlocal termsa competition modelrdquo SIAM Journal on Mathematical Analysisvol 35 no 3 pp 806ndash822 2003
[10] X H Lai and T X Yao ldquoExponential stability of impul-sive delayed reaction-diffusion cellular neural networks viaPoincare integral inequalityrdquoAbstract and Applied Analysis vol2013 Article ID 131836 10 pages 2013
[11] R Xu and Z E Ma ldquoGlobal stability of a reaction-diffusionpredator-prey model with a nonlocal delayrdquo Mathematical andComputer Modelling vol 50 no 1-2 pp 194ndash206 2009
[12] Z Li F D Chen and M X He ldquoAsymptotic behavior of thereaction-diffusion model of plankton allelopathy with nonlocaldelaysrdquo Nonlinear Analysis Real World Applications vol 12 no3 pp 1748ndash1758 2011
[13] Y Kuang and H L Smith ldquoGlobal stability in diffusive delayLotka-Volterra systemsrdquoDifferential and Integral Equations vol4 no 1 pp 117ndash128 1991
14 Abstract and Applied Analysis
[14] Z G Lin M Pedersen and L Zhang ldquoA predator-prey systemwith stage-structure for predator and nonlocal delayrdquoNonlinearAnalysis Theory Methods amp Applications vol 72 no 3-4 pp2019ndash2030 2010
[15] R Xu ldquoA reaction-diffusion predator-prey model with stagestructure and nonlocal delayrdquo Applied Mathematics and Com-putation vol 175 no 2 pp 984ndash1006 2006
[16] S S Chen and J P Shi ldquoStability and Hopf bifurcation in adiffusive logistic population model with nonlocal delay effectrdquoJournal of Differential Equations vol 253 no 12 pp 3440ndash34702012
[17] H I Freedman Deterministic Mathematical Models in Popula-tion Ecology vol 57 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1980
[18] P A Abrams and L R Ginzburg ldquoThe nature of predation preydepedent ratio dependent or neitherrdquo Trends in Ecology andEvolution vol 15 no 8 pp 337ndash341 2000
[19] D L DeAngelis R A Goldstein and R V OrsquoNeill ldquoA modelfor trophic iterationrdquo Ecology vol 56 no 4 pp 881ndash892 1975
[20] C V Pao ldquoDynamics of nonlinear parabolic systems with timedelaysrdquo Journal of Mathematical Analysis and Applications vol198 no 3 pp 751ndash779 1996
[21] F Xu Y Hou and Z G Lin ldquoTime delay parabolic system ina predator-prey model with stage structurerdquo Acta MathematicaSinica vol 48 no 6 pp 1121ndash1130 2005 (Chinese)
[22] P Y H Pang and M X Wang ldquoStrategy and stationary patternin a three-species predator-prey modelrdquo Journal of DifferentialEquations vol 200 no 2 pp 245ndash273 2004
[23] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981
[24] K J Brown P C Dunne and R A Gardner ldquoA semilinearparabolic system arising in the theory of superconductivityrdquoJournal of Differential Equations vol 40 no 2 pp 232ndash252 1981
[25] S A Gourley and J W H So ldquoDynamics of a food-limitedpopulation model incorporating nonlocal delays on a finitedomainrdquo Journal of Mathematical Biology vol 44 no 1 pp 49ndash78 2002
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
Proof We study the stability of the semi-trivial solution 1198641=
(1 0 0) Similarly the equations in (1) can be written as
1205971199061
120597119905minus 1198891Δ1199061= 1199061[minus11988611(1199061minus 1) minus
119886121199062
1198981+ 1199061
]
1205971199062
120597119905minus 1198892Δ1199062
= 1199062[minus1198862+ intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times119886211198981(1199061(119904 119910) minus
1)
(1198981+ 1199061(119904 119910)) (119898
1+ 1)119889119904 119889119910
+119886211
1198981+ 1
minus 119886221199062minus119886231199063
1198982+ 1199062
]
1205971199063
120597119905minus 1198893Δ1199063
= 1199063[minus1198863+ intΩ
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times119886321199062(119904 119910)
1198982+ 1199062(119904 119910)
119889119904 119889119910 minus 119886331199063]
(53)
Define
119881 (119905) = 120572intΩ
[1199061minus 1minus 1log 11990611
] 119889119909 + intΩ
1199062119889119909
+ 120573intΩ
1199063119889119909
(54)
Calculating the derivative of 119881(119905) along 1198641 we get from (54)
that
1198811015840(119905) le minusint
Ω
[120572(11988611minus119886121205981
1198981
)] (1199061minus 1)2
119889119909
+ intΩ
[11988622minus12057211988612
411989811205981
minus11988623
412059821198982
minus11988621
411989811205983
] 1199062
2119889119909
+ intΩ
[12057311988633minus119886231205982
1198982
] 1199062
3119889119909
minus intΩ
[(1198862minus119886211
1198981+ 1
)1199062+ 11988631199063] 119889119909
+119886211205983
1198981
intΩ
int
119905
minusinfin
1198701(119909 119910 119905 minus 119904)
times (1199061(119904 119910) minus
1)2
119889119904 119889119910 119889119909
+11988632120573
412059841198982
∬Ω
int
119905
minusinfin
1198702(119909 119910 119905 minus 119904)
times 1199062
2(119904 119910) 119889119904 119889119910 119889119909
(55)
Define119864 (119905) = 119881 (119905)
+119886211205983
1198981
∬Ω
int
+infin
0
int
119905
119905minus119903
1198701(119909 119910 119903)
times (1199061(119904 119910) minus 119906
lowast
1)2
119889119904 119889119903 119889119910 119889119909
+11988632120573
412059841198982
∬Ω
int
+infin
0
int
119905
119905minus119903
1198702(119909 119910 119903)
times (1199062(119904 119910) minus 119906
lowast
2)2
119889119904 119889119903 119889119910 119889119909
(56)
It is easy to see that
1198641015840(119905) le minus120572119889
11intΩ
1003816100381610038161003816nabla119906110038161003816100381610038162
11990621
119889119909
minus intΩ
[120572(11988611minus119886121205981
1198981
) minus119886211205983
1198981
] (1199061minus 1)2
119889119909
+ intΩ
[11988622minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632
411989821205984
] 1199062
2119889119909
+ intΩ
[12057311988633minus119886231205982
1198982
minus119886321205731205984
1198982
] 1199062
3119889119909
minus intΩ
[(1198862minus119886211
1198981+ 1
)1199062minus 11988631205731199063] 119889119909
(57)
Assume that1198665 1198861(11988621minus 1198862) lt 119898
1119886211988611
1198666 119886221198982
2gt1198862311988632
11988633
+211988612119886211198982
2
1198861111989821
(58)
Let
1198971= 12057211988611minus120572119886121205981
1198981
minus119886211205983
1198981
1198972= 11988622minus12057211988612
411989811205981
minus11988623
411989821205982
minus11988621
411989811205983
minus11988632120573
411989821205984
1198973= 11988633120573 minus
119886231205982
1198982
minus120573119886321205984
1198982
(59)
Choose
120572 =11988621
11988612
120573 =11988623
11988632
1205981= 1205983=119886111198981
411988612
1205982= 1205984=119898211988633
411988632
(60)
Then we get 119897119894gt 0 (119894 = 1 2 3) Therefore we have
lim119905rarrinfin
1199061(119905 119909) =
1 (61)
uniformly onΩ
10 Abstract and Applied Analysis
tx
u1
02
46
8
01
23
40
1
2
3
4
(a)
02
46
8
01
23
40
1
2
3
4
t
x
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 2 Global stability of the positive equilibrium 119864lowast with an initial condition (1206011 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos 119909 27 + 05 sin119909)
Next we consider the asymptotic behavior of 1199062(119905 119909) and
1199063(119905 119909) For any 119879 gt 0 integrating (57) over [0 119879] yields
119864 (119879) + 1205721198891
10038171003817100381710038171003817100381710038171003817
10038161003816100381610038161003816100381610038161003816
nabla1199061
1199061
10038161003816100381610038161003816100381610038161003816
10038171003817100381710038171003817100381710038171003817
2
2
+ 1198971
100381710038171003817100381710038161003816100381610038161199061 minus 1
100381610038161003816100381610038171003817100381710038172
2
+ 1198972
100381710038171003817100381710038161003816100381610038161199062100381610038161003816100381610038171003817100381710038172
2+ 1198973
100381710038171003817100381710038161003816100381610038161199063100381610038161003816100381610038171003817100381710038172
2le 119864 (0)
(62)
where |119906119894|2
2= int119879
0intΩ1199062
119894119889119909 119889119905 It implies that |119906
119894|2le 119862119894(119894 =
2 3) for the constant 119862119894which is independent of 119879 Now we
consider the boundedness of |nabla1199062|2and |nabla119906
3|2 From the
Greenrsquos identity we obtain
1198892int
119879
0
intΩ
1003816100381610038161003816nabla1199062 (119905 119909)10038161003816100381610038162
119889119909 119889119905
= minus1198892int
119879
0
intΩ
1199062(119905 119909) Δ119906
2(119905 119909) 119889119909 119889119905
= minusint
119879
0
intΩ
1199062
1205971199062
120597119905119889119909 119889119905 minus 119886
2int
119879
0
intΩ
1199062
2119889119909 119889119905
minus 11988622int
119879
0
intΩ
1199063
2119889119909 119889119905 minus 119886
23int
119879
0
intΩ
1199062
21199063
1198982+ 1199063
119889119909 119889119905
+ 11988621int
119879
0
∬Ω
int
119905
minusinfin
1199061(119904 119910) 119906
2
2(119905 119909)
1198981+ 1199061(119904 119910)
times 1198701(119909 119910 119905 minus 119904) 119889119904 119889119910 119889119909 119889119905
(63)
Note that
int
119879
0
intΩ
1199062(119905 119909)
1205971199062(119905 119909)
120597119905119889119909 119889119905
= int
119879
0
intΩ
1
2
1205971199062
2
120597119905119889119905 119889119909
=1
2intΩ
1199062
2(119879 119909) 119889119909 minus
1
2intΩ
1199062
2(0 119909) 119889119909 le 119872
2
2|Ω|
int
119879
0
intΩ
1199062
21199063119889119909 119889119905 le 119872
3int
119879
0
intΩ
1199062
2119889119909 119889119905
Abstract and Applied Analysis 11
02
46
8
01
23
40
1
2
3
4
tx
u1
(a)
02
46
8
01
23
40
1
2
3
4
tx
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 3 Global stability of the positive equilibrium 1198642with an initial condition (120601
1 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos 119909 27 + 05 sin119909)
int
119879
0
intΩ
1199063
2119889119909 119889119905 le 119872
2int
119879
0
intΩ
1199062
2119889119909 119889119905
int
119879
0
∬Ω
int
119905
minusinfin
1199061(119904 119910) 119906
2
2(119905 119909)
1198981+ 1199061(119904 119910)
1198701(119909 119910 119905 minus 119904) 119889119904 119889119910 119889119909 119889119905
le int
119879
0
intΩ
1199062
2(119905 119909) 119889119909 119889119905
(64)
Thus we get |nabla1199062|2le 1198624 In a similar way we have
|nabla1199063|2le 1198625 Here 119862
4and 119862
5are independent of 119879
It is easy to see that 1199062(119905 119909) 119906
3(119905 119909) isin 119871
2((0infin)
11988212(Ω)) These imply that
lim119905rarrinfin
1003817100381710038171003817119906119894 (sdot 119905)100381710038171003817100381711988212(Ω) = 0 119894 = 2 3 (65)
From the Sobolev compact embedding theorem we know
lim119905rarrinfin
1003817100381710038171003817119906119894 (sdot 119905)1003817100381710038171003817119862(Ω) = 0 119894 = 2 3 (66)
In the end we show that the trivial solution 1198640is an
unstable equilibrium Similarly to the local stability to 119864lowast wecan get the characteristic equation of 119864
0as
(120582 + 1205831198941198631minus 1198861) (120582 + 120583
1198941198632+ 1198862) (120582 + 120583
1198941198633+ 1198863) = 0 (67)
If 119894 = 1 then 1205831= 0 It is easy to see that this equation admits
a positive solution 120582 = 1198861 According to Theorem 51 in [23]
we have the following result
Theorem 11 The trivial equilibrium 1198640is an unstable equilib-
rium of system (1)ndash(3)
4 Numerical Illustrations
In this section we performnumerical simulations to illustratethe theoretical results given in Section 3
12 Abstract and Applied Analysis
u1
02
46
8
01
23
40
1
2
3
4
tx
(a)
02
46
8
01
23
40
1
2
3
4
tx
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 4 Global stability of the positive equilibrium 1198641with an initial condition (120601
1 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos119909 27 + 05 sin119909)
In the following we always take Ω = [0 120587] 119870119894(119909 119910 119905) =
119866119894(119909 119910 119905)119896
119894(119905) where 119896
119894(119905) = (1120591
lowast)119890minus119905120591lowast
(119894 = 1 2) and
1198661(119909 119910 119905) =
1
120587+2
120587
infin
sum
119899=1
119890minus11988921198992
119905 cos 119899119909 cos 119899119910
1198662(119909 119910 119905) =
1
120587+2
120587
infin
sum
119899=1
119890minus11988931198992
119905 cos 119899119909 cos 119899119910
(68)
However it is difficult for us to simulate our results directlybecause of the nonlocal term Similar to [25] the equationsin (1) can be rewritten as follows
1205971199061
120597119905minus 1198891Δ1199061= 1199061(1198861minus 119886111199061minus119886121199062
1198981+ 1199061
)
1205971199062
120597119905minus 1198892Δ1199062= 1199062(minus1198862+ 11988621V1minus 119886221199062minus119886231199063
1198982+ 1199063
)
1205971199063
120597119905minus 1198893Δ1199063= 1199063(minus1198863+ 11988632V2minus 119886331199063)
120597V1
120597119905minus 1198892ΔV1=1
120591lowast(
1199061
1198981+ 1199061
minus V1)
120597V2
120597119905minus 1198893ΔV2=1
120591lowast(
1199062
1198982+ 1199062
minus V2)
(69)
where
V119894= int
120587
0
int
119905
minusinfin
119866119894(119909 119910 119905 minus 119904)
1
120591lowast119890minus(119905minus119904)120591
lowast 119906119894(119904 119910)
119898119894+ 119906119894(119904 119910)
119889119910 119889119904
119894 = 1 2
(70)
Each component is considered with homogeneous Neumannboundary conditions and the initial condition of V
119894is
V119894(0 119909) = int
120587
0
int
0
minusinfin
119866119894(119909 119910 minus119904)
1
120591lowast119890119904120591lowast 119906119894(119904 119910)
119898119894+ 119906119894(119904 119910)
119889119910 119889119904
119894 = 1 2
(71)
Abstract and Applied Analysis 13
In the following examples we fix some coefficients andassume that 119889
1= 1198892= 1198893= 1 119886
1= 3 119886
12= 1 119898
1= 1198982=
1 1198862= 1 119886
22= 12 119886
23= 1 119886
33= 12 and 120591lowast = 1 The
asymptotic behaviors of system (1)ndash(3) are shownby choosingdifferent coefficients 119886
11 11988621 1198863 and 119886
32
Example 12 Let 11988611= 539 119886
21= 8113 119886
3= 113 and
11988632= 14Then it is easy to see that the system admits a unique
positive equilibrium 119864lowast(12 112 112) By Theorem 8 we
see that the positive solution (1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of
system (1)ndash(3) converges to 119864lowast as 119905 rarr infin See Figure 2
Example 13 Let 11988611
= 6 11988621
= 12 1198863= 14 and
11988632= 1 Clearly 119867
2does not hold Hence the positive
steady state is not feasible System (1) admits two semi-trivial steady state 119864
2(04732 02372 0) and 119864
1(12 0 0)
According to Theorem 9 we know that the positive solution(1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of system (1)ndash(3) converges to 119864
2
as 119905 rarr infin See Figure 3
Example 14 Let 11988611= 6 119886
21= 2 119886
3= 14 and 119886
32= 1
Clearly1198671does not hold Hence 119864lowast and 119864
2are not feasible
System (1) has a unique semi-trivial steady state 1198641(12 0 0)
According to Theorem 10 we know that the positive solution(1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of system (1)ndash(3) converges to 119864
1
as 119905 rarr infin See Figure 4
5 Discussion
In this paper we incorporate nonlocal delay into a three-species food chain model with Michaelis-Menten functionalresponse to represent a delay due to the gestation of the preda-tor The conditions under which the spatial homogeneousequilibria are asymptotically stable are given by using theLyapunov functional
We now summarize the ecological meanings of ourtheoretical results Firstly the positive equilibrium 119864
lowast ofsystem (1)ndash(3) exists under the high birth rate of the prey(1198861) and low death rates (119886
2and 119886
3) of predator and top
predator 119864lowast is globally stable if the intraspecific competition11988611is neither too big nor too small and the maximum harvest
rates 11988612 11988623
are small enough Secondly the semi-trivialequilibrium 119864
2of system (1)ndash(3) exists if the birth rate of the
prey (1198861) is high death rate of predator 119886
2is low and the death
rate (1198863) exceeds the conversion rate from predator to top
predator (11988632) 1198642is globally stable if the maximum harvest
rates 11988612 11988623are small and the intra-specific competition 119886
11
is neither too big nor too small Thirdly system (1)ndash(3) hasonly one semi-trivial equilibrium 119864
1when the death rate (119886
2)
exceeds the conversion rate from prey to predator (11988621) 1198641
is globally stable if intra-specific competitions (11988611 11988622 and
11988633) are strong Finally 119864
0is unstable and the non-stability of
trivial equilibrium tells us that not all of the populations goto extinction Furthermore our main results imply that thenonlocal delay is harmless for stabilities of all non-negativesteady states of system (1)ndash(3)
There are still many interesting and challenging problemswith respect to system (1)ndash(3) for example the permanence
and stability of periodic solution or almost periodic solutionThese problems are clearly worthy for further investigations
Acknowledgments
This work is partially supported by PRC Grants NSFC11102076 and 11071209 andNSF of theHigher Education Insti-tutions of Jiangsu Province (12KJD110002 and 12KJD110008)
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] H F Huo andW T Li ldquoPeriodic solution of a delayed predator-prey system with Michaelis-Menten type functional responserdquoJournal of Computational and AppliedMathematics vol 166 no2 pp 453ndash463 2004
[3] Z Lin and M Pedersen ldquoStability in a diffusive food-chainmodel with Michaelis-Menten functional responserdquo NonlinearAnalysis Theory Methods amp Applications vol 57 no 3 pp 421ndash433 2004
[4] B X Dai N Zhang and J Z Zou ldquoPermanence for theMichaelis-Menten type discrete three-species ratio-dependentfood chain model with delayrdquo Journal of Mathematical Analysisand Applications vol 324 no 1 pp 728ndash738 2006
[5] N F Britton ldquoSpatial structures and periodic travelling wavesin an integro-differential reaction-diffusion population modelrdquoSIAM Journal on Applied Mathematics vol 50 no 6 pp 1663ndash1688 1990
[6] S A Gourley and N F Britton ldquoInstability of travelling wavesolutions of a population model with nonlocal effectsrdquo IMAJournal of Applied Mathematics vol 51 no 3 pp 299ndash310 1993
[7] X Zhang and R Xu ldquoTraveling waves of a diffusive predator-prey model with nonlocal delay and stage structurerdquo Journal ofMathematical Analysis and Applications vol 373 no 2 pp 475ndash484 2011
[8] Z C Wang W T Li and S G Ruan ldquoExistence and stability oftraveling wave fronts in reaction advection diffusion equationswith nonlocal delayrdquo Journal of Differential Equations vol 238no 1 pp 153ndash200 2007
[9] S A Gourley and S G Ruan ldquoConvergence and travellingfronts in functional differential equations with nonlocal termsa competition modelrdquo SIAM Journal on Mathematical Analysisvol 35 no 3 pp 806ndash822 2003
[10] X H Lai and T X Yao ldquoExponential stability of impul-sive delayed reaction-diffusion cellular neural networks viaPoincare integral inequalityrdquoAbstract and Applied Analysis vol2013 Article ID 131836 10 pages 2013
[11] R Xu and Z E Ma ldquoGlobal stability of a reaction-diffusionpredator-prey model with a nonlocal delayrdquo Mathematical andComputer Modelling vol 50 no 1-2 pp 194ndash206 2009
[12] Z Li F D Chen and M X He ldquoAsymptotic behavior of thereaction-diffusion model of plankton allelopathy with nonlocaldelaysrdquo Nonlinear Analysis Real World Applications vol 12 no3 pp 1748ndash1758 2011
[13] Y Kuang and H L Smith ldquoGlobal stability in diffusive delayLotka-Volterra systemsrdquoDifferential and Integral Equations vol4 no 1 pp 117ndash128 1991
14 Abstract and Applied Analysis
[14] Z G Lin M Pedersen and L Zhang ldquoA predator-prey systemwith stage-structure for predator and nonlocal delayrdquoNonlinearAnalysis Theory Methods amp Applications vol 72 no 3-4 pp2019ndash2030 2010
[15] R Xu ldquoA reaction-diffusion predator-prey model with stagestructure and nonlocal delayrdquo Applied Mathematics and Com-putation vol 175 no 2 pp 984ndash1006 2006
[16] S S Chen and J P Shi ldquoStability and Hopf bifurcation in adiffusive logistic population model with nonlocal delay effectrdquoJournal of Differential Equations vol 253 no 12 pp 3440ndash34702012
[17] H I Freedman Deterministic Mathematical Models in Popula-tion Ecology vol 57 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1980
[18] P A Abrams and L R Ginzburg ldquoThe nature of predation preydepedent ratio dependent or neitherrdquo Trends in Ecology andEvolution vol 15 no 8 pp 337ndash341 2000
[19] D L DeAngelis R A Goldstein and R V OrsquoNeill ldquoA modelfor trophic iterationrdquo Ecology vol 56 no 4 pp 881ndash892 1975
[20] C V Pao ldquoDynamics of nonlinear parabolic systems with timedelaysrdquo Journal of Mathematical Analysis and Applications vol198 no 3 pp 751ndash779 1996
[21] F Xu Y Hou and Z G Lin ldquoTime delay parabolic system ina predator-prey model with stage structurerdquo Acta MathematicaSinica vol 48 no 6 pp 1121ndash1130 2005 (Chinese)
[22] P Y H Pang and M X Wang ldquoStrategy and stationary patternin a three-species predator-prey modelrdquo Journal of DifferentialEquations vol 200 no 2 pp 245ndash273 2004
[23] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981
[24] K J Brown P C Dunne and R A Gardner ldquoA semilinearparabolic system arising in the theory of superconductivityrdquoJournal of Differential Equations vol 40 no 2 pp 232ndash252 1981
[25] S A Gourley and J W H So ldquoDynamics of a food-limitedpopulation model incorporating nonlocal delays on a finitedomainrdquo Journal of Mathematical Biology vol 44 no 1 pp 49ndash78 2002
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
tx
u1
02
46
8
01
23
40
1
2
3
4
(a)
02
46
8
01
23
40
1
2
3
4
t
x
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 2 Global stability of the positive equilibrium 119864lowast with an initial condition (1206011 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos 119909 27 + 05 sin119909)
Next we consider the asymptotic behavior of 1199062(119905 119909) and
1199063(119905 119909) For any 119879 gt 0 integrating (57) over [0 119879] yields
119864 (119879) + 1205721198891
10038171003817100381710038171003817100381710038171003817
10038161003816100381610038161003816100381610038161003816
nabla1199061
1199061
10038161003816100381610038161003816100381610038161003816
10038171003817100381710038171003817100381710038171003817
2
2
+ 1198971
100381710038171003817100381710038161003816100381610038161199061 minus 1
100381610038161003816100381610038171003817100381710038172
2
+ 1198972
100381710038171003817100381710038161003816100381610038161199062100381610038161003816100381610038171003817100381710038172
2+ 1198973
100381710038171003817100381710038161003816100381610038161199063100381610038161003816100381610038171003817100381710038172
2le 119864 (0)
(62)
where |119906119894|2
2= int119879
0intΩ1199062
119894119889119909 119889119905 It implies that |119906
119894|2le 119862119894(119894 =
2 3) for the constant 119862119894which is independent of 119879 Now we
consider the boundedness of |nabla1199062|2and |nabla119906
3|2 From the
Greenrsquos identity we obtain
1198892int
119879
0
intΩ
1003816100381610038161003816nabla1199062 (119905 119909)10038161003816100381610038162
119889119909 119889119905
= minus1198892int
119879
0
intΩ
1199062(119905 119909) Δ119906
2(119905 119909) 119889119909 119889119905
= minusint
119879
0
intΩ
1199062
1205971199062
120597119905119889119909 119889119905 minus 119886
2int
119879
0
intΩ
1199062
2119889119909 119889119905
minus 11988622int
119879
0
intΩ
1199063
2119889119909 119889119905 minus 119886
23int
119879
0
intΩ
1199062
21199063
1198982+ 1199063
119889119909 119889119905
+ 11988621int
119879
0
∬Ω
int
119905
minusinfin
1199061(119904 119910) 119906
2
2(119905 119909)
1198981+ 1199061(119904 119910)
times 1198701(119909 119910 119905 minus 119904) 119889119904 119889119910 119889119909 119889119905
(63)
Note that
int
119879
0
intΩ
1199062(119905 119909)
1205971199062(119905 119909)
120597119905119889119909 119889119905
= int
119879
0
intΩ
1
2
1205971199062
2
120597119905119889119905 119889119909
=1
2intΩ
1199062
2(119879 119909) 119889119909 minus
1
2intΩ
1199062
2(0 119909) 119889119909 le 119872
2
2|Ω|
int
119879
0
intΩ
1199062
21199063119889119909 119889119905 le 119872
3int
119879
0
intΩ
1199062
2119889119909 119889119905
Abstract and Applied Analysis 11
02
46
8
01
23
40
1
2
3
4
tx
u1
(a)
02
46
8
01
23
40
1
2
3
4
tx
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 3 Global stability of the positive equilibrium 1198642with an initial condition (120601
1 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos 119909 27 + 05 sin119909)
int
119879
0
intΩ
1199063
2119889119909 119889119905 le 119872
2int
119879
0
intΩ
1199062
2119889119909 119889119905
int
119879
0
∬Ω
int
119905
minusinfin
1199061(119904 119910) 119906
2
2(119905 119909)
1198981+ 1199061(119904 119910)
1198701(119909 119910 119905 minus 119904) 119889119904 119889119910 119889119909 119889119905
le int
119879
0
intΩ
1199062
2(119905 119909) 119889119909 119889119905
(64)
Thus we get |nabla1199062|2le 1198624 In a similar way we have
|nabla1199063|2le 1198625 Here 119862
4and 119862
5are independent of 119879
It is easy to see that 1199062(119905 119909) 119906
3(119905 119909) isin 119871
2((0infin)
11988212(Ω)) These imply that
lim119905rarrinfin
1003817100381710038171003817119906119894 (sdot 119905)100381710038171003817100381711988212(Ω) = 0 119894 = 2 3 (65)
From the Sobolev compact embedding theorem we know
lim119905rarrinfin
1003817100381710038171003817119906119894 (sdot 119905)1003817100381710038171003817119862(Ω) = 0 119894 = 2 3 (66)
In the end we show that the trivial solution 1198640is an
unstable equilibrium Similarly to the local stability to 119864lowast wecan get the characteristic equation of 119864
0as
(120582 + 1205831198941198631minus 1198861) (120582 + 120583
1198941198632+ 1198862) (120582 + 120583
1198941198633+ 1198863) = 0 (67)
If 119894 = 1 then 1205831= 0 It is easy to see that this equation admits
a positive solution 120582 = 1198861 According to Theorem 51 in [23]
we have the following result
Theorem 11 The trivial equilibrium 1198640is an unstable equilib-
rium of system (1)ndash(3)
4 Numerical Illustrations
In this section we performnumerical simulations to illustratethe theoretical results given in Section 3
12 Abstract and Applied Analysis
u1
02
46
8
01
23
40
1
2
3
4
tx
(a)
02
46
8
01
23
40
1
2
3
4
tx
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 4 Global stability of the positive equilibrium 1198641with an initial condition (120601
1 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos119909 27 + 05 sin119909)
In the following we always take Ω = [0 120587] 119870119894(119909 119910 119905) =
119866119894(119909 119910 119905)119896
119894(119905) where 119896
119894(119905) = (1120591
lowast)119890minus119905120591lowast
(119894 = 1 2) and
1198661(119909 119910 119905) =
1
120587+2
120587
infin
sum
119899=1
119890minus11988921198992
119905 cos 119899119909 cos 119899119910
1198662(119909 119910 119905) =
1
120587+2
120587
infin
sum
119899=1
119890minus11988931198992
119905 cos 119899119909 cos 119899119910
(68)
However it is difficult for us to simulate our results directlybecause of the nonlocal term Similar to [25] the equationsin (1) can be rewritten as follows
1205971199061
120597119905minus 1198891Δ1199061= 1199061(1198861minus 119886111199061minus119886121199062
1198981+ 1199061
)
1205971199062
120597119905minus 1198892Δ1199062= 1199062(minus1198862+ 11988621V1minus 119886221199062minus119886231199063
1198982+ 1199063
)
1205971199063
120597119905minus 1198893Δ1199063= 1199063(minus1198863+ 11988632V2minus 119886331199063)
120597V1
120597119905minus 1198892ΔV1=1
120591lowast(
1199061
1198981+ 1199061
minus V1)
120597V2
120597119905minus 1198893ΔV2=1
120591lowast(
1199062
1198982+ 1199062
minus V2)
(69)
where
V119894= int
120587
0
int
119905
minusinfin
119866119894(119909 119910 119905 minus 119904)
1
120591lowast119890minus(119905minus119904)120591
lowast 119906119894(119904 119910)
119898119894+ 119906119894(119904 119910)
119889119910 119889119904
119894 = 1 2
(70)
Each component is considered with homogeneous Neumannboundary conditions and the initial condition of V
119894is
V119894(0 119909) = int
120587
0
int
0
minusinfin
119866119894(119909 119910 minus119904)
1
120591lowast119890119904120591lowast 119906119894(119904 119910)
119898119894+ 119906119894(119904 119910)
119889119910 119889119904
119894 = 1 2
(71)
Abstract and Applied Analysis 13
In the following examples we fix some coefficients andassume that 119889
1= 1198892= 1198893= 1 119886
1= 3 119886
12= 1 119898
1= 1198982=
1 1198862= 1 119886
22= 12 119886
23= 1 119886
33= 12 and 120591lowast = 1 The
asymptotic behaviors of system (1)ndash(3) are shownby choosingdifferent coefficients 119886
11 11988621 1198863 and 119886
32
Example 12 Let 11988611= 539 119886
21= 8113 119886
3= 113 and
11988632= 14Then it is easy to see that the system admits a unique
positive equilibrium 119864lowast(12 112 112) By Theorem 8 we
see that the positive solution (1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of
system (1)ndash(3) converges to 119864lowast as 119905 rarr infin See Figure 2
Example 13 Let 11988611
= 6 11988621
= 12 1198863= 14 and
11988632= 1 Clearly 119867
2does not hold Hence the positive
steady state is not feasible System (1) admits two semi-trivial steady state 119864
2(04732 02372 0) and 119864
1(12 0 0)
According to Theorem 9 we know that the positive solution(1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of system (1)ndash(3) converges to 119864
2
as 119905 rarr infin See Figure 3
Example 14 Let 11988611= 6 119886
21= 2 119886
3= 14 and 119886
32= 1
Clearly1198671does not hold Hence 119864lowast and 119864
2are not feasible
System (1) has a unique semi-trivial steady state 1198641(12 0 0)
According to Theorem 10 we know that the positive solution(1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of system (1)ndash(3) converges to 119864
1
as 119905 rarr infin See Figure 4
5 Discussion
In this paper we incorporate nonlocal delay into a three-species food chain model with Michaelis-Menten functionalresponse to represent a delay due to the gestation of the preda-tor The conditions under which the spatial homogeneousequilibria are asymptotically stable are given by using theLyapunov functional
We now summarize the ecological meanings of ourtheoretical results Firstly the positive equilibrium 119864
lowast ofsystem (1)ndash(3) exists under the high birth rate of the prey(1198861) and low death rates (119886
2and 119886
3) of predator and top
predator 119864lowast is globally stable if the intraspecific competition11988611is neither too big nor too small and the maximum harvest
rates 11988612 11988623
are small enough Secondly the semi-trivialequilibrium 119864
2of system (1)ndash(3) exists if the birth rate of the
prey (1198861) is high death rate of predator 119886
2is low and the death
rate (1198863) exceeds the conversion rate from predator to top
predator (11988632) 1198642is globally stable if the maximum harvest
rates 11988612 11988623are small and the intra-specific competition 119886
11
is neither too big nor too small Thirdly system (1)ndash(3) hasonly one semi-trivial equilibrium 119864
1when the death rate (119886
2)
exceeds the conversion rate from prey to predator (11988621) 1198641
is globally stable if intra-specific competitions (11988611 11988622 and
11988633) are strong Finally 119864
0is unstable and the non-stability of
trivial equilibrium tells us that not all of the populations goto extinction Furthermore our main results imply that thenonlocal delay is harmless for stabilities of all non-negativesteady states of system (1)ndash(3)
There are still many interesting and challenging problemswith respect to system (1)ndash(3) for example the permanence
and stability of periodic solution or almost periodic solutionThese problems are clearly worthy for further investigations
Acknowledgments
This work is partially supported by PRC Grants NSFC11102076 and 11071209 andNSF of theHigher Education Insti-tutions of Jiangsu Province (12KJD110002 and 12KJD110008)
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] H F Huo andW T Li ldquoPeriodic solution of a delayed predator-prey system with Michaelis-Menten type functional responserdquoJournal of Computational and AppliedMathematics vol 166 no2 pp 453ndash463 2004
[3] Z Lin and M Pedersen ldquoStability in a diffusive food-chainmodel with Michaelis-Menten functional responserdquo NonlinearAnalysis Theory Methods amp Applications vol 57 no 3 pp 421ndash433 2004
[4] B X Dai N Zhang and J Z Zou ldquoPermanence for theMichaelis-Menten type discrete three-species ratio-dependentfood chain model with delayrdquo Journal of Mathematical Analysisand Applications vol 324 no 1 pp 728ndash738 2006
[5] N F Britton ldquoSpatial structures and periodic travelling wavesin an integro-differential reaction-diffusion population modelrdquoSIAM Journal on Applied Mathematics vol 50 no 6 pp 1663ndash1688 1990
[6] S A Gourley and N F Britton ldquoInstability of travelling wavesolutions of a population model with nonlocal effectsrdquo IMAJournal of Applied Mathematics vol 51 no 3 pp 299ndash310 1993
[7] X Zhang and R Xu ldquoTraveling waves of a diffusive predator-prey model with nonlocal delay and stage structurerdquo Journal ofMathematical Analysis and Applications vol 373 no 2 pp 475ndash484 2011
[8] Z C Wang W T Li and S G Ruan ldquoExistence and stability oftraveling wave fronts in reaction advection diffusion equationswith nonlocal delayrdquo Journal of Differential Equations vol 238no 1 pp 153ndash200 2007
[9] S A Gourley and S G Ruan ldquoConvergence and travellingfronts in functional differential equations with nonlocal termsa competition modelrdquo SIAM Journal on Mathematical Analysisvol 35 no 3 pp 806ndash822 2003
[10] X H Lai and T X Yao ldquoExponential stability of impul-sive delayed reaction-diffusion cellular neural networks viaPoincare integral inequalityrdquoAbstract and Applied Analysis vol2013 Article ID 131836 10 pages 2013
[11] R Xu and Z E Ma ldquoGlobal stability of a reaction-diffusionpredator-prey model with a nonlocal delayrdquo Mathematical andComputer Modelling vol 50 no 1-2 pp 194ndash206 2009
[12] Z Li F D Chen and M X He ldquoAsymptotic behavior of thereaction-diffusion model of plankton allelopathy with nonlocaldelaysrdquo Nonlinear Analysis Real World Applications vol 12 no3 pp 1748ndash1758 2011
[13] Y Kuang and H L Smith ldquoGlobal stability in diffusive delayLotka-Volterra systemsrdquoDifferential and Integral Equations vol4 no 1 pp 117ndash128 1991
14 Abstract and Applied Analysis
[14] Z G Lin M Pedersen and L Zhang ldquoA predator-prey systemwith stage-structure for predator and nonlocal delayrdquoNonlinearAnalysis Theory Methods amp Applications vol 72 no 3-4 pp2019ndash2030 2010
[15] R Xu ldquoA reaction-diffusion predator-prey model with stagestructure and nonlocal delayrdquo Applied Mathematics and Com-putation vol 175 no 2 pp 984ndash1006 2006
[16] S S Chen and J P Shi ldquoStability and Hopf bifurcation in adiffusive logistic population model with nonlocal delay effectrdquoJournal of Differential Equations vol 253 no 12 pp 3440ndash34702012
[17] H I Freedman Deterministic Mathematical Models in Popula-tion Ecology vol 57 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1980
[18] P A Abrams and L R Ginzburg ldquoThe nature of predation preydepedent ratio dependent or neitherrdquo Trends in Ecology andEvolution vol 15 no 8 pp 337ndash341 2000
[19] D L DeAngelis R A Goldstein and R V OrsquoNeill ldquoA modelfor trophic iterationrdquo Ecology vol 56 no 4 pp 881ndash892 1975
[20] C V Pao ldquoDynamics of nonlinear parabolic systems with timedelaysrdquo Journal of Mathematical Analysis and Applications vol198 no 3 pp 751ndash779 1996
[21] F Xu Y Hou and Z G Lin ldquoTime delay parabolic system ina predator-prey model with stage structurerdquo Acta MathematicaSinica vol 48 no 6 pp 1121ndash1130 2005 (Chinese)
[22] P Y H Pang and M X Wang ldquoStrategy and stationary patternin a three-species predator-prey modelrdquo Journal of DifferentialEquations vol 200 no 2 pp 245ndash273 2004
[23] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981
[24] K J Brown P C Dunne and R A Gardner ldquoA semilinearparabolic system arising in the theory of superconductivityrdquoJournal of Differential Equations vol 40 no 2 pp 232ndash252 1981
[25] S A Gourley and J W H So ldquoDynamics of a food-limitedpopulation model incorporating nonlocal delays on a finitedomainrdquo Journal of Mathematical Biology vol 44 no 1 pp 49ndash78 2002
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
02
46
8
01
23
40
1
2
3
4
tx
u1
(a)
02
46
8
01
23
40
1
2
3
4
tx
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 3 Global stability of the positive equilibrium 1198642with an initial condition (120601
1 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos 119909 27 + 05 sin119909)
int
119879
0
intΩ
1199063
2119889119909 119889119905 le 119872
2int
119879
0
intΩ
1199062
2119889119909 119889119905
int
119879
0
∬Ω
int
119905
minusinfin
1199061(119904 119910) 119906
2
2(119905 119909)
1198981+ 1199061(119904 119910)
1198701(119909 119910 119905 minus 119904) 119889119904 119889119910 119889119909 119889119905
le int
119879
0
intΩ
1199062
2(119905 119909) 119889119909 119889119905
(64)
Thus we get |nabla1199062|2le 1198624 In a similar way we have
|nabla1199063|2le 1198625 Here 119862
4and 119862
5are independent of 119879
It is easy to see that 1199062(119905 119909) 119906
3(119905 119909) isin 119871
2((0infin)
11988212(Ω)) These imply that
lim119905rarrinfin
1003817100381710038171003817119906119894 (sdot 119905)100381710038171003817100381711988212(Ω) = 0 119894 = 2 3 (65)
From the Sobolev compact embedding theorem we know
lim119905rarrinfin
1003817100381710038171003817119906119894 (sdot 119905)1003817100381710038171003817119862(Ω) = 0 119894 = 2 3 (66)
In the end we show that the trivial solution 1198640is an
unstable equilibrium Similarly to the local stability to 119864lowast wecan get the characteristic equation of 119864
0as
(120582 + 1205831198941198631minus 1198861) (120582 + 120583
1198941198632+ 1198862) (120582 + 120583
1198941198633+ 1198863) = 0 (67)
If 119894 = 1 then 1205831= 0 It is easy to see that this equation admits
a positive solution 120582 = 1198861 According to Theorem 51 in [23]
we have the following result
Theorem 11 The trivial equilibrium 1198640is an unstable equilib-
rium of system (1)ndash(3)
4 Numerical Illustrations
In this section we performnumerical simulations to illustratethe theoretical results given in Section 3
12 Abstract and Applied Analysis
u1
02
46
8
01
23
40
1
2
3
4
tx
(a)
02
46
8
01
23
40
1
2
3
4
tx
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 4 Global stability of the positive equilibrium 1198641with an initial condition (120601
1 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos119909 27 + 05 sin119909)
In the following we always take Ω = [0 120587] 119870119894(119909 119910 119905) =
119866119894(119909 119910 119905)119896
119894(119905) where 119896
119894(119905) = (1120591
lowast)119890minus119905120591lowast
(119894 = 1 2) and
1198661(119909 119910 119905) =
1
120587+2
120587
infin
sum
119899=1
119890minus11988921198992
119905 cos 119899119909 cos 119899119910
1198662(119909 119910 119905) =
1
120587+2
120587
infin
sum
119899=1
119890minus11988931198992
119905 cos 119899119909 cos 119899119910
(68)
However it is difficult for us to simulate our results directlybecause of the nonlocal term Similar to [25] the equationsin (1) can be rewritten as follows
1205971199061
120597119905minus 1198891Δ1199061= 1199061(1198861minus 119886111199061minus119886121199062
1198981+ 1199061
)
1205971199062
120597119905minus 1198892Δ1199062= 1199062(minus1198862+ 11988621V1minus 119886221199062minus119886231199063
1198982+ 1199063
)
1205971199063
120597119905minus 1198893Δ1199063= 1199063(minus1198863+ 11988632V2minus 119886331199063)
120597V1
120597119905minus 1198892ΔV1=1
120591lowast(
1199061
1198981+ 1199061
minus V1)
120597V2
120597119905minus 1198893ΔV2=1
120591lowast(
1199062
1198982+ 1199062
minus V2)
(69)
where
V119894= int
120587
0
int
119905
minusinfin
119866119894(119909 119910 119905 minus 119904)
1
120591lowast119890minus(119905minus119904)120591
lowast 119906119894(119904 119910)
119898119894+ 119906119894(119904 119910)
119889119910 119889119904
119894 = 1 2
(70)
Each component is considered with homogeneous Neumannboundary conditions and the initial condition of V
119894is
V119894(0 119909) = int
120587
0
int
0
minusinfin
119866119894(119909 119910 minus119904)
1
120591lowast119890119904120591lowast 119906119894(119904 119910)
119898119894+ 119906119894(119904 119910)
119889119910 119889119904
119894 = 1 2
(71)
Abstract and Applied Analysis 13
In the following examples we fix some coefficients andassume that 119889
1= 1198892= 1198893= 1 119886
1= 3 119886
12= 1 119898
1= 1198982=
1 1198862= 1 119886
22= 12 119886
23= 1 119886
33= 12 and 120591lowast = 1 The
asymptotic behaviors of system (1)ndash(3) are shownby choosingdifferent coefficients 119886
11 11988621 1198863 and 119886
32
Example 12 Let 11988611= 539 119886
21= 8113 119886
3= 113 and
11988632= 14Then it is easy to see that the system admits a unique
positive equilibrium 119864lowast(12 112 112) By Theorem 8 we
see that the positive solution (1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of
system (1)ndash(3) converges to 119864lowast as 119905 rarr infin See Figure 2
Example 13 Let 11988611
= 6 11988621
= 12 1198863= 14 and
11988632= 1 Clearly 119867
2does not hold Hence the positive
steady state is not feasible System (1) admits two semi-trivial steady state 119864
2(04732 02372 0) and 119864
1(12 0 0)
According to Theorem 9 we know that the positive solution(1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of system (1)ndash(3) converges to 119864
2
as 119905 rarr infin See Figure 3
Example 14 Let 11988611= 6 119886
21= 2 119886
3= 14 and 119886
32= 1
Clearly1198671does not hold Hence 119864lowast and 119864
2are not feasible
System (1) has a unique semi-trivial steady state 1198641(12 0 0)
According to Theorem 10 we know that the positive solution(1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of system (1)ndash(3) converges to 119864
1
as 119905 rarr infin See Figure 4
5 Discussion
In this paper we incorporate nonlocal delay into a three-species food chain model with Michaelis-Menten functionalresponse to represent a delay due to the gestation of the preda-tor The conditions under which the spatial homogeneousequilibria are asymptotically stable are given by using theLyapunov functional
We now summarize the ecological meanings of ourtheoretical results Firstly the positive equilibrium 119864
lowast ofsystem (1)ndash(3) exists under the high birth rate of the prey(1198861) and low death rates (119886
2and 119886
3) of predator and top
predator 119864lowast is globally stable if the intraspecific competition11988611is neither too big nor too small and the maximum harvest
rates 11988612 11988623
are small enough Secondly the semi-trivialequilibrium 119864
2of system (1)ndash(3) exists if the birth rate of the
prey (1198861) is high death rate of predator 119886
2is low and the death
rate (1198863) exceeds the conversion rate from predator to top
predator (11988632) 1198642is globally stable if the maximum harvest
rates 11988612 11988623are small and the intra-specific competition 119886
11
is neither too big nor too small Thirdly system (1)ndash(3) hasonly one semi-trivial equilibrium 119864
1when the death rate (119886
2)
exceeds the conversion rate from prey to predator (11988621) 1198641
is globally stable if intra-specific competitions (11988611 11988622 and
11988633) are strong Finally 119864
0is unstable and the non-stability of
trivial equilibrium tells us that not all of the populations goto extinction Furthermore our main results imply that thenonlocal delay is harmless for stabilities of all non-negativesteady states of system (1)ndash(3)
There are still many interesting and challenging problemswith respect to system (1)ndash(3) for example the permanence
and stability of periodic solution or almost periodic solutionThese problems are clearly worthy for further investigations
Acknowledgments
This work is partially supported by PRC Grants NSFC11102076 and 11071209 andNSF of theHigher Education Insti-tutions of Jiangsu Province (12KJD110002 and 12KJD110008)
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] H F Huo andW T Li ldquoPeriodic solution of a delayed predator-prey system with Michaelis-Menten type functional responserdquoJournal of Computational and AppliedMathematics vol 166 no2 pp 453ndash463 2004
[3] Z Lin and M Pedersen ldquoStability in a diffusive food-chainmodel with Michaelis-Menten functional responserdquo NonlinearAnalysis Theory Methods amp Applications vol 57 no 3 pp 421ndash433 2004
[4] B X Dai N Zhang and J Z Zou ldquoPermanence for theMichaelis-Menten type discrete three-species ratio-dependentfood chain model with delayrdquo Journal of Mathematical Analysisand Applications vol 324 no 1 pp 728ndash738 2006
[5] N F Britton ldquoSpatial structures and periodic travelling wavesin an integro-differential reaction-diffusion population modelrdquoSIAM Journal on Applied Mathematics vol 50 no 6 pp 1663ndash1688 1990
[6] S A Gourley and N F Britton ldquoInstability of travelling wavesolutions of a population model with nonlocal effectsrdquo IMAJournal of Applied Mathematics vol 51 no 3 pp 299ndash310 1993
[7] X Zhang and R Xu ldquoTraveling waves of a diffusive predator-prey model with nonlocal delay and stage structurerdquo Journal ofMathematical Analysis and Applications vol 373 no 2 pp 475ndash484 2011
[8] Z C Wang W T Li and S G Ruan ldquoExistence and stability oftraveling wave fronts in reaction advection diffusion equationswith nonlocal delayrdquo Journal of Differential Equations vol 238no 1 pp 153ndash200 2007
[9] S A Gourley and S G Ruan ldquoConvergence and travellingfronts in functional differential equations with nonlocal termsa competition modelrdquo SIAM Journal on Mathematical Analysisvol 35 no 3 pp 806ndash822 2003
[10] X H Lai and T X Yao ldquoExponential stability of impul-sive delayed reaction-diffusion cellular neural networks viaPoincare integral inequalityrdquoAbstract and Applied Analysis vol2013 Article ID 131836 10 pages 2013
[11] R Xu and Z E Ma ldquoGlobal stability of a reaction-diffusionpredator-prey model with a nonlocal delayrdquo Mathematical andComputer Modelling vol 50 no 1-2 pp 194ndash206 2009
[12] Z Li F D Chen and M X He ldquoAsymptotic behavior of thereaction-diffusion model of plankton allelopathy with nonlocaldelaysrdquo Nonlinear Analysis Real World Applications vol 12 no3 pp 1748ndash1758 2011
[13] Y Kuang and H L Smith ldquoGlobal stability in diffusive delayLotka-Volterra systemsrdquoDifferential and Integral Equations vol4 no 1 pp 117ndash128 1991
14 Abstract and Applied Analysis
[14] Z G Lin M Pedersen and L Zhang ldquoA predator-prey systemwith stage-structure for predator and nonlocal delayrdquoNonlinearAnalysis Theory Methods amp Applications vol 72 no 3-4 pp2019ndash2030 2010
[15] R Xu ldquoA reaction-diffusion predator-prey model with stagestructure and nonlocal delayrdquo Applied Mathematics and Com-putation vol 175 no 2 pp 984ndash1006 2006
[16] S S Chen and J P Shi ldquoStability and Hopf bifurcation in adiffusive logistic population model with nonlocal delay effectrdquoJournal of Differential Equations vol 253 no 12 pp 3440ndash34702012
[17] H I Freedman Deterministic Mathematical Models in Popula-tion Ecology vol 57 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1980
[18] P A Abrams and L R Ginzburg ldquoThe nature of predation preydepedent ratio dependent or neitherrdquo Trends in Ecology andEvolution vol 15 no 8 pp 337ndash341 2000
[19] D L DeAngelis R A Goldstein and R V OrsquoNeill ldquoA modelfor trophic iterationrdquo Ecology vol 56 no 4 pp 881ndash892 1975
[20] C V Pao ldquoDynamics of nonlinear parabolic systems with timedelaysrdquo Journal of Mathematical Analysis and Applications vol198 no 3 pp 751ndash779 1996
[21] F Xu Y Hou and Z G Lin ldquoTime delay parabolic system ina predator-prey model with stage structurerdquo Acta MathematicaSinica vol 48 no 6 pp 1121ndash1130 2005 (Chinese)
[22] P Y H Pang and M X Wang ldquoStrategy and stationary patternin a three-species predator-prey modelrdquo Journal of DifferentialEquations vol 200 no 2 pp 245ndash273 2004
[23] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981
[24] K J Brown P C Dunne and R A Gardner ldquoA semilinearparabolic system arising in the theory of superconductivityrdquoJournal of Differential Equations vol 40 no 2 pp 232ndash252 1981
[25] S A Gourley and J W H So ldquoDynamics of a food-limitedpopulation model incorporating nonlocal delays on a finitedomainrdquo Journal of Mathematical Biology vol 44 no 1 pp 49ndash78 2002
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
u1
02
46
8
01
23
40
1
2
3
4
tx
(a)
02
46
8
01
23
40
1
2
3
4
tx
u2
(b)
02
46
8
01
23
40
1
2
3
4
tx
u3
(c)
Figure 4 Global stability of the positive equilibrium 1198641with an initial condition (120601
1 1206012 1206013) = (27 + 05 sin119909 27 + 05 cos119909 27 + 05 sin119909)
In the following we always take Ω = [0 120587] 119870119894(119909 119910 119905) =
119866119894(119909 119910 119905)119896
119894(119905) where 119896
119894(119905) = (1120591
lowast)119890minus119905120591lowast
(119894 = 1 2) and
1198661(119909 119910 119905) =
1
120587+2
120587
infin
sum
119899=1
119890minus11988921198992
119905 cos 119899119909 cos 119899119910
1198662(119909 119910 119905) =
1
120587+2
120587
infin
sum
119899=1
119890minus11988931198992
119905 cos 119899119909 cos 119899119910
(68)
However it is difficult for us to simulate our results directlybecause of the nonlocal term Similar to [25] the equationsin (1) can be rewritten as follows
1205971199061
120597119905minus 1198891Δ1199061= 1199061(1198861minus 119886111199061minus119886121199062
1198981+ 1199061
)
1205971199062
120597119905minus 1198892Δ1199062= 1199062(minus1198862+ 11988621V1minus 119886221199062minus119886231199063
1198982+ 1199063
)
1205971199063
120597119905minus 1198893Δ1199063= 1199063(minus1198863+ 11988632V2minus 119886331199063)
120597V1
120597119905minus 1198892ΔV1=1
120591lowast(
1199061
1198981+ 1199061
minus V1)
120597V2
120597119905minus 1198893ΔV2=1
120591lowast(
1199062
1198982+ 1199062
minus V2)
(69)
where
V119894= int
120587
0
int
119905
minusinfin
119866119894(119909 119910 119905 minus 119904)
1
120591lowast119890minus(119905minus119904)120591
lowast 119906119894(119904 119910)
119898119894+ 119906119894(119904 119910)
119889119910 119889119904
119894 = 1 2
(70)
Each component is considered with homogeneous Neumannboundary conditions and the initial condition of V
119894is
V119894(0 119909) = int
120587
0
int
0
minusinfin
119866119894(119909 119910 minus119904)
1
120591lowast119890119904120591lowast 119906119894(119904 119910)
119898119894+ 119906119894(119904 119910)
119889119910 119889119904
119894 = 1 2
(71)
Abstract and Applied Analysis 13
In the following examples we fix some coefficients andassume that 119889
1= 1198892= 1198893= 1 119886
1= 3 119886
12= 1 119898
1= 1198982=
1 1198862= 1 119886
22= 12 119886
23= 1 119886
33= 12 and 120591lowast = 1 The
asymptotic behaviors of system (1)ndash(3) are shownby choosingdifferent coefficients 119886
11 11988621 1198863 and 119886
32
Example 12 Let 11988611= 539 119886
21= 8113 119886
3= 113 and
11988632= 14Then it is easy to see that the system admits a unique
positive equilibrium 119864lowast(12 112 112) By Theorem 8 we
see that the positive solution (1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of
system (1)ndash(3) converges to 119864lowast as 119905 rarr infin See Figure 2
Example 13 Let 11988611
= 6 11988621
= 12 1198863= 14 and
11988632= 1 Clearly 119867
2does not hold Hence the positive
steady state is not feasible System (1) admits two semi-trivial steady state 119864
2(04732 02372 0) and 119864
1(12 0 0)
According to Theorem 9 we know that the positive solution(1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of system (1)ndash(3) converges to 119864
2
as 119905 rarr infin See Figure 3
Example 14 Let 11988611= 6 119886
21= 2 119886
3= 14 and 119886
32= 1
Clearly1198671does not hold Hence 119864lowast and 119864
2are not feasible
System (1) has a unique semi-trivial steady state 1198641(12 0 0)
According to Theorem 10 we know that the positive solution(1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of system (1)ndash(3) converges to 119864
1
as 119905 rarr infin See Figure 4
5 Discussion
In this paper we incorporate nonlocal delay into a three-species food chain model with Michaelis-Menten functionalresponse to represent a delay due to the gestation of the preda-tor The conditions under which the spatial homogeneousequilibria are asymptotically stable are given by using theLyapunov functional
We now summarize the ecological meanings of ourtheoretical results Firstly the positive equilibrium 119864
lowast ofsystem (1)ndash(3) exists under the high birth rate of the prey(1198861) and low death rates (119886
2and 119886
3) of predator and top
predator 119864lowast is globally stable if the intraspecific competition11988611is neither too big nor too small and the maximum harvest
rates 11988612 11988623
are small enough Secondly the semi-trivialequilibrium 119864
2of system (1)ndash(3) exists if the birth rate of the
prey (1198861) is high death rate of predator 119886
2is low and the death
rate (1198863) exceeds the conversion rate from predator to top
predator (11988632) 1198642is globally stable if the maximum harvest
rates 11988612 11988623are small and the intra-specific competition 119886
11
is neither too big nor too small Thirdly system (1)ndash(3) hasonly one semi-trivial equilibrium 119864
1when the death rate (119886
2)
exceeds the conversion rate from prey to predator (11988621) 1198641
is globally stable if intra-specific competitions (11988611 11988622 and
11988633) are strong Finally 119864
0is unstable and the non-stability of
trivial equilibrium tells us that not all of the populations goto extinction Furthermore our main results imply that thenonlocal delay is harmless for stabilities of all non-negativesteady states of system (1)ndash(3)
There are still many interesting and challenging problemswith respect to system (1)ndash(3) for example the permanence
and stability of periodic solution or almost periodic solutionThese problems are clearly worthy for further investigations
Acknowledgments
This work is partially supported by PRC Grants NSFC11102076 and 11071209 andNSF of theHigher Education Insti-tutions of Jiangsu Province (12KJD110002 and 12KJD110008)
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] H F Huo andW T Li ldquoPeriodic solution of a delayed predator-prey system with Michaelis-Menten type functional responserdquoJournal of Computational and AppliedMathematics vol 166 no2 pp 453ndash463 2004
[3] Z Lin and M Pedersen ldquoStability in a diffusive food-chainmodel with Michaelis-Menten functional responserdquo NonlinearAnalysis Theory Methods amp Applications vol 57 no 3 pp 421ndash433 2004
[4] B X Dai N Zhang and J Z Zou ldquoPermanence for theMichaelis-Menten type discrete three-species ratio-dependentfood chain model with delayrdquo Journal of Mathematical Analysisand Applications vol 324 no 1 pp 728ndash738 2006
[5] N F Britton ldquoSpatial structures and periodic travelling wavesin an integro-differential reaction-diffusion population modelrdquoSIAM Journal on Applied Mathematics vol 50 no 6 pp 1663ndash1688 1990
[6] S A Gourley and N F Britton ldquoInstability of travelling wavesolutions of a population model with nonlocal effectsrdquo IMAJournal of Applied Mathematics vol 51 no 3 pp 299ndash310 1993
[7] X Zhang and R Xu ldquoTraveling waves of a diffusive predator-prey model with nonlocal delay and stage structurerdquo Journal ofMathematical Analysis and Applications vol 373 no 2 pp 475ndash484 2011
[8] Z C Wang W T Li and S G Ruan ldquoExistence and stability oftraveling wave fronts in reaction advection diffusion equationswith nonlocal delayrdquo Journal of Differential Equations vol 238no 1 pp 153ndash200 2007
[9] S A Gourley and S G Ruan ldquoConvergence and travellingfronts in functional differential equations with nonlocal termsa competition modelrdquo SIAM Journal on Mathematical Analysisvol 35 no 3 pp 806ndash822 2003
[10] X H Lai and T X Yao ldquoExponential stability of impul-sive delayed reaction-diffusion cellular neural networks viaPoincare integral inequalityrdquoAbstract and Applied Analysis vol2013 Article ID 131836 10 pages 2013
[11] R Xu and Z E Ma ldquoGlobal stability of a reaction-diffusionpredator-prey model with a nonlocal delayrdquo Mathematical andComputer Modelling vol 50 no 1-2 pp 194ndash206 2009
[12] Z Li F D Chen and M X He ldquoAsymptotic behavior of thereaction-diffusion model of plankton allelopathy with nonlocaldelaysrdquo Nonlinear Analysis Real World Applications vol 12 no3 pp 1748ndash1758 2011
[13] Y Kuang and H L Smith ldquoGlobal stability in diffusive delayLotka-Volterra systemsrdquoDifferential and Integral Equations vol4 no 1 pp 117ndash128 1991
14 Abstract and Applied Analysis
[14] Z G Lin M Pedersen and L Zhang ldquoA predator-prey systemwith stage-structure for predator and nonlocal delayrdquoNonlinearAnalysis Theory Methods amp Applications vol 72 no 3-4 pp2019ndash2030 2010
[15] R Xu ldquoA reaction-diffusion predator-prey model with stagestructure and nonlocal delayrdquo Applied Mathematics and Com-putation vol 175 no 2 pp 984ndash1006 2006
[16] S S Chen and J P Shi ldquoStability and Hopf bifurcation in adiffusive logistic population model with nonlocal delay effectrdquoJournal of Differential Equations vol 253 no 12 pp 3440ndash34702012
[17] H I Freedman Deterministic Mathematical Models in Popula-tion Ecology vol 57 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1980
[18] P A Abrams and L R Ginzburg ldquoThe nature of predation preydepedent ratio dependent or neitherrdquo Trends in Ecology andEvolution vol 15 no 8 pp 337ndash341 2000
[19] D L DeAngelis R A Goldstein and R V OrsquoNeill ldquoA modelfor trophic iterationrdquo Ecology vol 56 no 4 pp 881ndash892 1975
[20] C V Pao ldquoDynamics of nonlinear parabolic systems with timedelaysrdquo Journal of Mathematical Analysis and Applications vol198 no 3 pp 751ndash779 1996
[21] F Xu Y Hou and Z G Lin ldquoTime delay parabolic system ina predator-prey model with stage structurerdquo Acta MathematicaSinica vol 48 no 6 pp 1121ndash1130 2005 (Chinese)
[22] P Y H Pang and M X Wang ldquoStrategy and stationary patternin a three-species predator-prey modelrdquo Journal of DifferentialEquations vol 200 no 2 pp 245ndash273 2004
[23] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981
[24] K J Brown P C Dunne and R A Gardner ldquoA semilinearparabolic system arising in the theory of superconductivityrdquoJournal of Differential Equations vol 40 no 2 pp 232ndash252 1981
[25] S A Gourley and J W H So ldquoDynamics of a food-limitedpopulation model incorporating nonlocal delays on a finitedomainrdquo Journal of Mathematical Biology vol 44 no 1 pp 49ndash78 2002
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
In the following examples we fix some coefficients andassume that 119889
1= 1198892= 1198893= 1 119886
1= 3 119886
12= 1 119898
1= 1198982=
1 1198862= 1 119886
22= 12 119886
23= 1 119886
33= 12 and 120591lowast = 1 The
asymptotic behaviors of system (1)ndash(3) are shownby choosingdifferent coefficients 119886
11 11988621 1198863 and 119886
32
Example 12 Let 11988611= 539 119886
21= 8113 119886
3= 113 and
11988632= 14Then it is easy to see that the system admits a unique
positive equilibrium 119864lowast(12 112 112) By Theorem 8 we
see that the positive solution (1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of
system (1)ndash(3) converges to 119864lowast as 119905 rarr infin See Figure 2
Example 13 Let 11988611
= 6 11988621
= 12 1198863= 14 and
11988632= 1 Clearly 119867
2does not hold Hence the positive
steady state is not feasible System (1) admits two semi-trivial steady state 119864
2(04732 02372 0) and 119864
1(12 0 0)
According to Theorem 9 we know that the positive solution(1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of system (1)ndash(3) converges to 119864
2
as 119905 rarr infin See Figure 3
Example 14 Let 11988611= 6 119886
21= 2 119886
3= 14 and 119886
32= 1
Clearly1198671does not hold Hence 119864lowast and 119864
2are not feasible
System (1) has a unique semi-trivial steady state 1198641(12 0 0)
According to Theorem 10 we know that the positive solution(1199061(119905 119909) 119906
2(119905 119909) 119906
3(119905 119909)) of system (1)ndash(3) converges to 119864
1
as 119905 rarr infin See Figure 4
5 Discussion
In this paper we incorporate nonlocal delay into a three-species food chain model with Michaelis-Menten functionalresponse to represent a delay due to the gestation of the preda-tor The conditions under which the spatial homogeneousequilibria are asymptotically stable are given by using theLyapunov functional
We now summarize the ecological meanings of ourtheoretical results Firstly the positive equilibrium 119864
lowast ofsystem (1)ndash(3) exists under the high birth rate of the prey(1198861) and low death rates (119886
2and 119886
3) of predator and top
predator 119864lowast is globally stable if the intraspecific competition11988611is neither too big nor too small and the maximum harvest
rates 11988612 11988623
are small enough Secondly the semi-trivialequilibrium 119864
2of system (1)ndash(3) exists if the birth rate of the
prey (1198861) is high death rate of predator 119886
2is low and the death
rate (1198863) exceeds the conversion rate from predator to top
predator (11988632) 1198642is globally stable if the maximum harvest
rates 11988612 11988623are small and the intra-specific competition 119886
11
is neither too big nor too small Thirdly system (1)ndash(3) hasonly one semi-trivial equilibrium 119864
1when the death rate (119886
2)
exceeds the conversion rate from prey to predator (11988621) 1198641
is globally stable if intra-specific competitions (11988611 11988622 and
11988633) are strong Finally 119864
0is unstable and the non-stability of
trivial equilibrium tells us that not all of the populations goto extinction Furthermore our main results imply that thenonlocal delay is harmless for stabilities of all non-negativesteady states of system (1)ndash(3)
There are still many interesting and challenging problemswith respect to system (1)ndash(3) for example the permanence
and stability of periodic solution or almost periodic solutionThese problems are clearly worthy for further investigations
Acknowledgments
This work is partially supported by PRC Grants NSFC11102076 and 11071209 andNSF of theHigher Education Insti-tutions of Jiangsu Province (12KJD110002 and 12KJD110008)
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] H F Huo andW T Li ldquoPeriodic solution of a delayed predator-prey system with Michaelis-Menten type functional responserdquoJournal of Computational and AppliedMathematics vol 166 no2 pp 453ndash463 2004
[3] Z Lin and M Pedersen ldquoStability in a diffusive food-chainmodel with Michaelis-Menten functional responserdquo NonlinearAnalysis Theory Methods amp Applications vol 57 no 3 pp 421ndash433 2004
[4] B X Dai N Zhang and J Z Zou ldquoPermanence for theMichaelis-Menten type discrete three-species ratio-dependentfood chain model with delayrdquo Journal of Mathematical Analysisand Applications vol 324 no 1 pp 728ndash738 2006
[5] N F Britton ldquoSpatial structures and periodic travelling wavesin an integro-differential reaction-diffusion population modelrdquoSIAM Journal on Applied Mathematics vol 50 no 6 pp 1663ndash1688 1990
[6] S A Gourley and N F Britton ldquoInstability of travelling wavesolutions of a population model with nonlocal effectsrdquo IMAJournal of Applied Mathematics vol 51 no 3 pp 299ndash310 1993
[7] X Zhang and R Xu ldquoTraveling waves of a diffusive predator-prey model with nonlocal delay and stage structurerdquo Journal ofMathematical Analysis and Applications vol 373 no 2 pp 475ndash484 2011
[8] Z C Wang W T Li and S G Ruan ldquoExistence and stability oftraveling wave fronts in reaction advection diffusion equationswith nonlocal delayrdquo Journal of Differential Equations vol 238no 1 pp 153ndash200 2007
[9] S A Gourley and S G Ruan ldquoConvergence and travellingfronts in functional differential equations with nonlocal termsa competition modelrdquo SIAM Journal on Mathematical Analysisvol 35 no 3 pp 806ndash822 2003
[10] X H Lai and T X Yao ldquoExponential stability of impul-sive delayed reaction-diffusion cellular neural networks viaPoincare integral inequalityrdquoAbstract and Applied Analysis vol2013 Article ID 131836 10 pages 2013
[11] R Xu and Z E Ma ldquoGlobal stability of a reaction-diffusionpredator-prey model with a nonlocal delayrdquo Mathematical andComputer Modelling vol 50 no 1-2 pp 194ndash206 2009
[12] Z Li F D Chen and M X He ldquoAsymptotic behavior of thereaction-diffusion model of plankton allelopathy with nonlocaldelaysrdquo Nonlinear Analysis Real World Applications vol 12 no3 pp 1748ndash1758 2011
[13] Y Kuang and H L Smith ldquoGlobal stability in diffusive delayLotka-Volterra systemsrdquoDifferential and Integral Equations vol4 no 1 pp 117ndash128 1991
14 Abstract and Applied Analysis
[14] Z G Lin M Pedersen and L Zhang ldquoA predator-prey systemwith stage-structure for predator and nonlocal delayrdquoNonlinearAnalysis Theory Methods amp Applications vol 72 no 3-4 pp2019ndash2030 2010
[15] R Xu ldquoA reaction-diffusion predator-prey model with stagestructure and nonlocal delayrdquo Applied Mathematics and Com-putation vol 175 no 2 pp 984ndash1006 2006
[16] S S Chen and J P Shi ldquoStability and Hopf bifurcation in adiffusive logistic population model with nonlocal delay effectrdquoJournal of Differential Equations vol 253 no 12 pp 3440ndash34702012
[17] H I Freedman Deterministic Mathematical Models in Popula-tion Ecology vol 57 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1980
[18] P A Abrams and L R Ginzburg ldquoThe nature of predation preydepedent ratio dependent or neitherrdquo Trends in Ecology andEvolution vol 15 no 8 pp 337ndash341 2000
[19] D L DeAngelis R A Goldstein and R V OrsquoNeill ldquoA modelfor trophic iterationrdquo Ecology vol 56 no 4 pp 881ndash892 1975
[20] C V Pao ldquoDynamics of nonlinear parabolic systems with timedelaysrdquo Journal of Mathematical Analysis and Applications vol198 no 3 pp 751ndash779 1996
[21] F Xu Y Hou and Z G Lin ldquoTime delay parabolic system ina predator-prey model with stage structurerdquo Acta MathematicaSinica vol 48 no 6 pp 1121ndash1130 2005 (Chinese)
[22] P Y H Pang and M X Wang ldquoStrategy and stationary patternin a three-species predator-prey modelrdquo Journal of DifferentialEquations vol 200 no 2 pp 245ndash273 2004
[23] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981
[24] K J Brown P C Dunne and R A Gardner ldquoA semilinearparabolic system arising in the theory of superconductivityrdquoJournal of Differential Equations vol 40 no 2 pp 232ndash252 1981
[25] S A Gourley and J W H So ldquoDynamics of a food-limitedpopulation model incorporating nonlocal delays on a finitedomainrdquo Journal of Mathematical Biology vol 44 no 1 pp 49ndash78 2002
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
[14] Z G Lin M Pedersen and L Zhang ldquoA predator-prey systemwith stage-structure for predator and nonlocal delayrdquoNonlinearAnalysis Theory Methods amp Applications vol 72 no 3-4 pp2019ndash2030 2010
[15] R Xu ldquoA reaction-diffusion predator-prey model with stagestructure and nonlocal delayrdquo Applied Mathematics and Com-putation vol 175 no 2 pp 984ndash1006 2006
[16] S S Chen and J P Shi ldquoStability and Hopf bifurcation in adiffusive logistic population model with nonlocal delay effectrdquoJournal of Differential Equations vol 253 no 12 pp 3440ndash34702012
[17] H I Freedman Deterministic Mathematical Models in Popula-tion Ecology vol 57 of Monographs and Textbooks in Pure andAppliedMathematicsMarcelDekker NewYorkNYUSA 1980
[18] P A Abrams and L R Ginzburg ldquoThe nature of predation preydepedent ratio dependent or neitherrdquo Trends in Ecology andEvolution vol 15 no 8 pp 337ndash341 2000
[19] D L DeAngelis R A Goldstein and R V OrsquoNeill ldquoA modelfor trophic iterationrdquo Ecology vol 56 no 4 pp 881ndash892 1975
[20] C V Pao ldquoDynamics of nonlinear parabolic systems with timedelaysrdquo Journal of Mathematical Analysis and Applications vol198 no 3 pp 751ndash779 1996
[21] F Xu Y Hou and Z G Lin ldquoTime delay parabolic system ina predator-prey model with stage structurerdquo Acta MathematicaSinica vol 48 no 6 pp 1121ndash1130 2005 (Chinese)
[22] P Y H Pang and M X Wang ldquoStrategy and stationary patternin a three-species predator-prey modelrdquo Journal of DifferentialEquations vol 200 no 2 pp 245ndash273 2004
[23] D Henry Geometric Theory of Semilinear Parabolic Equationsvol 840 of Lecture Notes in Mathematics Springer New YorkNY USA 1981
[24] K J Brown P C Dunne and R A Gardner ldquoA semilinearparabolic system arising in the theory of superconductivityrdquoJournal of Differential Equations vol 40 no 2 pp 232ndash252 1981
[25] S A Gourley and J W H So ldquoDynamics of a food-limitedpopulation model incorporating nonlocal delays on a finitedomainrdquo Journal of Mathematical Biology vol 44 no 1 pp 49ndash78 2002
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
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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
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Stochastic AnalysisInternational Journal of
