Research Article Dynamic Behaviors of a Leslie-Gower...

Research ArticleDynamic Behaviors of a Leslie-Gower Ecoepidemiological Model

Aihua Kang12 Yakui Xue13 and Jianping Fu1

1College of Mechatronic Engineering North University of China Taiyuan Shanxi 030051 China2Shuozhou Advanced Normal College Shuozhou Shanxi 036000 China3Department of Mathematics North University of China Taiyuan Shanxi 030051 China

Correspondence should be addressed to Aihua Kang kangaihua1982163com

Received 11 August 2015 Accepted 2 November 2015

Academic Editor Yi Wang

Copyright copy 2015 Aihua Kang 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

A Leslie-Gower ecoepidemic model with disease in the predators is constructed and analyzed The total population is subdividedinto three subclasses namely susceptible predator infected predator and prey population The positivity boundness of solutionsand the existence of the equilibria are studied and the sufficient conditions of local asymptotic stability of the equilibria are obtainedby the Routh-Hurwitz criterionWe analyze the global stability of the interior equilibria by using Lyapunov functions It is observedthat a Hopf bifurcation may occur around the interior equilibrium At last numeric simulations are performed in support of thefeasibility of the main result

1 Introduction

Since the pioneering work of Anderson and May [1] manyresearchers have paid great attention to the modeling andanalysis of ecoepidemiological systems recently Venturino[2] Haque et al [3] Xiao and Chen [4 5] Tewa et al[6] Rahman and Chakravarty [7] and so forth discussedthe dynamics of prey-predator system with disease in preypopulation Haque et al [8] analyzed the dynamical behaviorof predator-prey system with disease in predator populationHsieh and Hsiao [9] proposed and discussed the dynamics ofa predator-preymodel with disease in both prey and predatorpopulations The boundness and stability of the equilibriaare studied There are mainly two types functional responseHolling-type functional response and Leslie-Gower func-tional responseMost scholars discussed theHopf bifurcationand the Bogdanov-Takens bifurcation near the boundaryequilibrium

The Leslie-Gower functional response is first proposedby Leslie [10] which introduced the following predator-preymodel where the ldquocarrying capacityrdquo of the predatorrsquos envi-ronment is proportional to the number of prey populations

The first and second Leslie-Gower predator-prey models areas follows

119889119867

119889119905= (1199031minus 1198861119875)119867

119889119875

119889119905= (1199032minus 1198862

119875

119867)119875

(1)

119889119867

119889119905= (1199031minus 1198861119875 minus 1198871119867)119867

119889119875

119889119905= (1199032minus 1198862

119875

119867)119875

(2)

where 119867 and 119875 are the density of prey species and thepredator species at time 119905 respectively Because of thecomplex mathematical expressions involved in the analy-sis Korobeinikov [11] introduced a Lyapunov function forboth models (1) and (2) to prove their global stabilitiesAfter the work of Korobeinikov many scholars have doneworks on Leslie-type predator prey ecosystem The modifiedLeslie-Gower and Holling-type II predator-prey model isgeneralized in the context of ecoepidemiology with diseasespreading only among the prey species [12] Hopf bifurcation

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2015 Article ID 169242 7 pageshttpdxdoiorg1011552015169242

2 Discrete Dynamics in Nature and Society

is studied for a modified Leslie-Gower predator-prey sys-tem with harvesting [13] Aziz-Alaoui [14] studied dynamicbehaviors of three Leslie-Gower-type species food chainsystems Chen et al [15] incorporated a prey refuge to system(1) and showed that the refuge has no influence on thepersistent property of the system A predator-prey Leslie-Gowermodel with disease in prey has been developed whereit is observed that a Hopf bifurcation may occur around theinterior equilibrium taking refuge parameter as bifurcationparameter [16] Some similar kinds of models have appearedin the recent literature the main new distinctive feature is theinclusion of an infectious disease in prey population But thedisease also can spread in predator because of food parasitemating and so on

In the present research we formulate a predator-preyLeslie-Gower model with disease in predator The totalpopulation have been divided into three classes namelysusceptible predator infected predator and prey populationThe construction and model assumptions are discussed inSection 2 In Section 3 positivity and boundedness of thesolutions of the model are discussed Section 4 deals withtheir existence and stability analysis of the equilibriumpoints In Section 5 a detailed study of the Hopf bifurcationaround the interior equilibrium is carried out Numericalillustrations are performed finally in order to validate theapplicability of the model under consideration

2 The Mathematical Model

We construct the following model

119889119883

119889119905= 1199031119883(1 minus

119883

119896) minus 119886119883119884

1

1198891198841

119889119905= 11990321198841[1 minus

ℎ (1198841+ 1198842)

119883] minus 120573119884

11198842minus 11988911198841

1198891198842

119889119905= 12057311988411198842minus 11988921198842

(3)

with initial conditions119883 (0) ge 0

1198841(0) ge 0

1198842(0) ge 0

(4)

where119883(119905)1198841(119905) and119884

2(119905) are the density of prey susceptible

predator and infected predator populations respectively attime 119905 The prey population grows according to a logisticfashion with carrying capacity 119896 and an intrinsic birth rateconstant 119903

1 1199032is the intrinsic growth rate of susceptible

predator populations 120573 is the transmission coefficient fromsusceptible predator to infected predator ℎ is the maximumvalue of the per capita reduction rate of119883 due to119884 = (119884

1+1198842)

The second equation of system (3) contains the so-calledLeslie-Gower term namely ℎ(119884

1+ 1198842)119883 119889

1is the natural

death rate of susceptible predator 1198892is death rate of infected

predator including natural death rate and disease relateddeath rate in the absence of predator The model parameters1199031 1199032 119896 119886 ℎ 120573 119889

1 and 119889

2are all positive constants

3 Some Preliminary Results

Theorem 1 Every solution of system (3) with initial conditions(4) exists in the interval [0 +infin) and 119883(119905) ge 0 119884

1(119905) ge 0

1198842(119905) ge 0 for all 119905 gt 0

Proof Since the right-hand side of system (3) is completelycontinuous and locally Lipschitzian on 119862 the solution(119883(119905) 119884

1(119905) 1198842(119905)) of (3) with initial conditions (4) exists and

is unique on [0 120577) where 0 lt 120577 le +infin [17] From system (3)with initial conditions (4) we have

119883 (119905) ge 119883 (0) int

+infin

0

1199031[119883 (119904) (1 minus

119883 (119904)

119896)

minus 119886119883 (119904) 1198841(119904)] 119889119904 ge 0

1198841(119905) ge 119884

1(0) int

+infin

0

11990321198841(119904) [1 minus

ℎ (1198841(119904) + 119884

2(119904))

119883 (119904)]

minus 1205731198841(119904) 1198842(119904) minus 119889

11198841(119904) 119889119904 ge 0

1198842(119905) ge 119884

2(0) int

+infin

0

[1205731198841(119904) 1198842(119904) minus 119889

21198842(119904)] 119889119904 ge 0

(5)

which completes the proof

Theorem 2 All solutions of system (3) initiating 1198773 are

ultimately bounded

Proof We consider first 119883(119905) le 119896 forall119905 gt 0

119889119883

119889119905= 1199031119883(1 minus

119883

119896) minus 119886119883119884

1le 1199031119883(1 minus

119883

119896) (6)

We get 119883 = 1(119862119890minus1199031119905+ 1119896) If 119905 rarr infin 119883 rarr 119896

1198891198841

119889119905+

1198891198842

119889119905= 11990321198841[1 minus

ℎ (1198841+ 1198842)

119883] minus 11988911198841minus 11988921198842

le 11990321198841[1 minus

ℎ (1198841+ 1198842)

119883]

le 11990321198841[1 minus

ℎ (1198841+ 1198842)

119896]

le 1199032(1198841+ 1198842) minus

1199032ℎ

119896(1198841+ 1198842)2

(7)

So again by solving the above linear differential inequalitywe have

0 lt 1198841+ 1198842lt

1

119890minus1199032119905 + ℎ119896 (8)

If 119905 rarr infin 1198841+ 1198842

rarr ℎ119896 The proof is completed

Therefore the feasible region Γ defined by

Γ

= (119883 (119905) 1198841(119905) 1198842(119905)) isin 119877

3

+ 119883 le 119896 119884

1+ 1198842le

ℎ

119896

(9)

Discrete Dynamics in Nature and Society 3

with119883(0) ge 01198841(0) ge 0 and119884

2(0) ge 0 is positively invariant

of model (3)

4 Stability Analysis

41 Existence of Equilibrium Points All equilibrium points ofsystem (3) are as follows

(1) trivial equilibrium 1198640= (0 0 0)

(2) axial equilibrium 1198641= (119896 0 0)

(3) if 1199032

gt 1198891 the planar equilibrium 119864

2= (119896119903

2ℎ1199031

(11990311199032ℎ+119886119896(119903

2minus1198891)) 1198961199031(1199032minus1198891)(11990311199032ℎ+119886119896(119903

2minus1198891)) 0)

exists(4) if 120573119903

1gt 1198861198892 and 119896(119903

2minus 1198891)(1205731199031minus 1198861198892) minus 1199032ℎ11990311198892

gt

0 the interior equilibrium 119864lowast

= (119883lowast 1198841lowast

1198842lowast

) existswhere 119883

lowast= 119896(120573119903

1minus 1198861198892)1205731199031 1198841lowast

= 1198892120573 and 119884

2lowast=

(119896(1199032minus 1198891)(1205731199031minus 1198861198892) minus 1199032ℎ11990311198892)(1199032ℎ1205731199031+ 120573(119896120573119903

1minus

1198961198861198892))

42 Local Stability Let 119864 = (119883 1198841 1198842) be an equilibrium

point of model (3) the Jacobian matrix of system (3) at theequilibrium point is

(

1199031minus

21199031

119896119883 minus 119886119884

1minus119886119883 0

ℎ11990321198841(1198841+ 1198842)2

1198831199032minus

21199032ℎ1198841

119883minus1199032ℎ1198841

119883minus 1205731198841

0 1205731198842

1205731198841minus 1198892

) (10)

Through judging the positive or negative of the eigenvalueswhich is the characteristic equation we can know localasymptotic stability of all equilibrium points Through calcu-lation we have the following results

(I) Eigenvalues of the characteristic equation of 1198641are

1205821= minus1199031 1205822= 1199032minus 1198891 and 120582

3= minus1198892 It is clear that

if 1199032lt 1198891 1205822lt 0 the equilibrium point 119864

1is locally

asymptotically stable(II) The variational matrix of system (3) at 119864

2= (1198832 11988412

0) is given by

(

minus1199031

1198961198832

minus1198861198832

0

ℎ11990321198842

12

11988322

minus1199032ℎ11988412

1198832

minus1199032ℎ11988412

1198832

minus 12057311988412

0 0 12057311988412

minus 1198892

) (11)

With regard to the equilibrium point 1198642 its charac-

teristic equation is

1205833+ 11986011205832+ 1198612120583 + 119860

3= 0 (12)

where 1198601= minus(119886

11+ 11988622

+ 11988633) 1198602= 1198861111988612

minus 1198861211988621

+

1198861111988633

+ 1198862211988633 1198603= minus11988633(1198861111988612

minus 1198861211988621) and 119886

11=

minus1198961199032ℎ1199032

1(11990311199032ℎ+119886119896(119903

2minus1198891)) 11988612

= minus1198861198961199032ℎ1199031(11990311199032ℎ+

119886119896(1199032minus 1198891)) 11988621

= (1199032minus 1198891)21199032ℎ 11988622

= minus(1199032minus 1198891)

(11990311199032ℎ+119886119896(119903

2minus1198891)) and 119886

33= 120573119896119903

1(1199032minus1198891)(11990311199032ℎ+

119886119896(1199032minus 1198891)) minus 119889

2

If 119896(1199032

minus 1198891)(1205731199031

minus 1198861198892) minus 1199032ℎ11990311198892

lt 0 11988633

lt 0Obviously 119860

1gt 0 119860

3gt 0 and 119860

11198602minus 1198603gt 0

By the Routh-Hurwitz rule the equilibrium point 1198642

is locally asymptotically stable in the region Γ(III) The variational matrix of system (3) at 119864

lowast=

(119883lowast 1198841lowast

1198842lowast

) is given by

(

(

minus1199031

119896119883lowast

minus119886119883lowast

0

ℎ11990321198841lowast

(1198841lowast

+ 1198842lowast

)

1198832lowast

1199032ℎ (1198842lowast

minus 1198841lowast

)

119883lowast

minus1199032ℎ1198841lowast

119883lowast

minus 1205731198841lowast

0 1205731198842lowast

0

)

)

(13)

With regard to the equilibrium point 119864lowast its charac-


1205833+ 11986111205832+ 1198612120583 + 1198613= 0 (14)

where 1198611= minus(11988711

+ 11988722) 1198612= 1198871111988722

minus 1198871211988721

minus 1198872311988732

1198613

= 119887111198872311988732

and 11988711

= minus(1205731199031

minus 1198861198892)120573 119887

12=

minus119886119896(1205731199031minus1198861198892)1205731199031 11988721

= 1205731199032

1ℎ1199032(1+1199032minus1198891)119896(120573119903

1minus

1198861198892)(1199032ℎ1199031

+ 119896(1205731199031

minus 1198861198892)) 11988722

= 1199032ℎ1199031(11989611990321205731199031

+

11989611988911198861198892+1198961198861198892

2minus11989611990321198861198892minus11989611988911205731199031minus211990311199032ℎ1198892minus11988921198961199031120573)

119896(1205731199031minus 1198861198892)(1199032ℎ1199031+ 119896(120573119903

1minus 1198861198892)) 11988723

= minus(11990311199032ℎ1198892+

1198892119896(1205731199031

minus 1198861198892))119896(120573119903

1minus 1198861198892) and 119887

32= (119896(119903

2minus

1198891)(1205731199031minus 1198861198892) minus 1199032ℎ11990311198892)(1199032ℎ1199031+ (119896120573119903

1minus 119896119886119889

2))

If 119896(1199032minus 1198891)(1205731199031minus 1198861198892) minus 1199032ℎ11990311198892

gt 0 11988732

gt 0 and1198613gt 0

If 1198842lowast

minus 1198841lowast

lt 0 in other words 11989611990321205731199031+ 11989611988911198861198892+

1198961198861198892

2gt 11989611990321198861198892+ 11989611988911205731199031+ 211990311199032ℎ1198892+ 11988921198961199031120573 11988722

lt 0and 119861

1gt 0

11986111198612minus1198613= minus11988711(1198871111988722

minus1198871211988721)+11988722(1198871211988721

+1198872311988732) gt

0

By the Routh-Hurwitz rule the equilibrium point 119864lowast

is locally asymptotically stable in the region Γ

So we come to the following results

Theorem 3 If 119896(1199032minus1198891)(1205731199031minus 1198861198892) minus 1199032ℎ11990311198892lt 0 the planar

equilibrium point 1198642is locally asymptotically stable

Theorem 4 If 11989611990321205731199031+ 11989611988911198861198892+ 119896119886119889

2

2lt 11989611990321198861198892+ 11989611988911205731199031+

211990311199032ℎ1198892+ 11988921198961199031120573 the interior equilibrium point 119864

lowastis locally

asymptotically stable

43 Globally Asymptotically Stable Defining the Lyapunovfunction we can judge the global asymptotically stability ofthe interior equilibrium point

Theorem 5 The equilibrium point 119864lowastis locally asymptotically

stable meaning that it is globally asymptotically stable in Σ =

(119878 119868 119884) 119878 gt 0 119868 gt 0 119884 gt 0

Proof Construct the Lyapunov function

119881 (119883 1198841 1198842) = 1198811(119883 1198841 1198842) + 1198812(119883 1198841 1198842)

+ 1198813(119883 1198841 1198842)

(15)


where1198811(119883 1198841 1198842) = 119883minus119883

lowastminus119883lowastln(119883119883

lowast)1198812(119883 1198841 1198842) =

1198841minus 1198841lowast

minus 1198841lowastln(11988411198841lowast

) and 1198813(119883 1198841 1198842) = 119884


minus

1198842lowastln(11988421198842lowast

)

Since the solutions of the system are bounded andultimately enter the set Γ we restrict the study for this setThe time derivative of 119881

1along the solutions of system (3) is

1198891198811

119889119905=

119883 minus 119883lowast

119883[1199031minus

119883

119896minus 1198861198841]119883

= minus1

119896(119883 minus 119883

lowast)2

minus 119886 (119883 minus 119883lowast) (1198841minus 1198841lowast

)

(16)

Similarly

1198891198812

119889119905=


1198841

[1199032minus

1199032ℎ (1198841+ 1198842)

119883minus 1205731198842minus 1198891]1198841

= minus(120573 +1199032ℎ

119883) (1198841minus 1198841lowast

) (1198842minus 1198842lowast

)

minus1199032ℎ

119883(1198841minus 1198841lowast

)2

+1199032ℎ (1198841lowast

+ 1198842lowast

)

119883119883lowast

(119883 minus 119883lowast) (1198841minus 1198841lowast

)

1198891198813

119889119905= 120573 (119884


) (1198842minus 1198842lowast

)

(17)

The above equation can be written as

119889119881

119889119905= minus (119883 minus 119883

lowast 1198841minus 1198841lowast


)

sdot (

1

119896minus119892 (119883 119884

1 1198842) 0

119892 (119883 1198841 1198842)

1199032ℎ

119883

1199032ℎ

2119883

01199032ℎ

21198830

)(




)

(18)

where 119892(119883 1198841 1198842) = (12)[119886 + 119903

2ℎ(1198841lowast

+ 1198842lowast

)119883119883lowast]

This matrix is positive definite if all upper-left submatri-ces are positive

Through calculating all upper-left submatrices

1198721=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

119896minus119892 (119883 119884

1 1198842)

119892 (119883 1198841 1198842)

1199032ℎ

119883

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=1199032ℎ

119896119883+ 1198922(119883 1198841 1198842) gt 0

1198722=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

119896minus119892 (119883 119884

1 1198842) 0

119892 (119883 1198841 1198842)

1199032ℎ

119883

1199032ℎ

2119883

01199032ℎ

21198830

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=1

119896(1199032ℎ

2119883)

2

gt 0

(19)

it is obvious that 119889119881119889119905 lt 0 So the interior equilibrium point119864lowast


1198842lowast

) is globally asymptotically stable

5 Hopf Bifurcation

Theorem 6 The dynamical system undergoes Hopf bifurca-tion around the interior equilibrium points 119864

lowastwhenever the

critical parameter value 120573 = 120573119867is included in the domain

119863 = 120573119867

isin 119877+

1198611(120573) 1198612(120573) minus 119861

3(120573)

1003816100381610038161003816120573=120573119867=0

with 1198612

gt 01198611(120573) 1198612(120573) minus 119861

3(120573)

119889120573

100381610038161003816100381610038161003816100381610038161003816120573=120573119867

= 0

(20)

Proof The characteristic equation of system (3) at 119864lowast

=


1198842lowast

) 119879 is given by

1205823+ 11986111205822+ 1198612120582 + 1198613= 0 (21)

The conditions 1198611(120573)1198612(120573) minus 119861

3(120573)|120573=120573119867

= 0 give

11988722

(1198871211988721

+ 1198872311988732) minus 11988711

(1198871111988722

minus 1198871211988721)1003816100381610038161003816120573=120573119867

= 0 (22)

From (21) we should have

(1205822+ 1198612) (120582 + 119861

1) = 0 (23)

which has three roots 1205821

= +119894radic1198612 1205822

= minus119894radic1198612 and 120582

3=

minus1198611

Differentiating the characteristic (21) with regard to 120573 wehave

119889120582

119889120573= minus

12058221+ 1205822+ 3

31205822 + 21198611120582 + 1198613

100381610038161003816100381610038161003816100381610038161003816120582=119894radic1198612

=3minus 11986121+ 1198942radic1198612

2 (1198612minus 1198941198611radic1198612)

=3minus (11986121+ 11986112)

2 (11986121+ 1198612)

+ 119894radic1198612(11986113+ 11986122minus 119861111198612)

21198612(11986121+ 1198612)

= minus119889 (1198611(120573) 1198612(120573) minus 119861

3(120573)) 119889120573

2 (11986121+ 1198612)

+ 119894 [radic11986122

21198612

minus1198611radic1198612(119889 (1198611(120573) 1198612(120573) minus 119861

3(120573)) 119889120573)

21198612(11986121+ 1198612)

]

(24)

Hence119889

119889120582(Re (120582 (120573)))

1003816100381610038161003816120573=120573119867

= minus119889 (1198611(120573) 1198612(120573) minus 119861

3(120573)) 119889120573

2 (11986121+ 1198612)

100381610038161003816100381610038161003816100381610038161003816120573=120573119867

= 0

(25)

and 1198611(120573119867) lt 0

We can easily establish the condition of the theorem(119889119889120582)(Re(120582(120573))) = 0 which completes the proof


minus5

0

5

10

15

20

25Th

e pro

port

ion

of p

opul

atio

ns

100 200 300 400 5000Time t

Y1

Y2

X

(a)

minus5

0

5

10

15

20

25

The p

ropo

rtio

n of

pop

ulat

ions

100 200 300 400 5000

Y1

Y2

X

Time t

(b)

0

5

10

15

20

25

The p

ropo

rtio

n of

pop

ulat

ions

100 200 300 400 5000

Y1

Y2

X

Time t

(c)

Figure 1 The local stability around all equilibriums of system (3)

6 Number Simulations

For the purpose of making qualitative analysis of the presentstudy numerical simulations have been carried out by mak-ing use of MATLAB

The parametric values were given as follows 1199031= 05 119896 =

2 119886 = 03 1199032= 0002 ℎ = 5 120573 = 0003 119889

1= 003 119889

2= 004

and 1199032minus 1198891

= minus0028 lt 0 The eigenvalues of the Jacobianmatrix at 119864

1are 1205821

= minus05 1205821

= minus0028 and 1205823

= minus004which satisfied the condition of the local asymptotic stabilityof axial equilibrium 119864

1= (2 0 0) (see Figure 1(a))

The parametric values were given as follows 1199031

= 02119896 = 20 119886 = 008 119903

2= 3 ℎ = 06 120573 = 0023 119889

1= 0003

1198892= 008 and 119896(119903

2minus 1198891)(1205731199031minus 1198861198892) minus 11990311199032ℎ1198892= minus01367 lt

0 which satisfied the condition of the local asymptoticstability of planar equilibrium 119864

2 So the planar equilibrium

1198642

= (13966 23254 0) is locally asymptotically stable (seeFigure 1(b))


20 119886 = 003 1199032= 3 ℎ = 05 120573 = 0023 119889

1= 003 119889

2= 004

and 1205731199031minus1198861198892= 00036 gt 0 119896(119903

2minus1198891)(1205731199031minus1198861198892)minus1199032ℎ11990311198892=

02018 gt 0 which satisfied the conditions for existence ofinterior equilibrium solution 119864

lowast 11989611990321205731199031+ 11989611988911198861198892+ 119896119886119889

2

2minus

11989611990321198861198892

minus 11989611988911205731199031

minus 211990311199032ℎ1198892

minus 11988921198961199031120573 = minus01626 lt 0

the conditions for the local asymptotic stability of interiorequilibrium solution 119864

lowastare well satisfied Hence the positive


X

200 400 600 800 1000 1200 140000

1

2

3

4

5

6Th

e pro

port

ion

of p

rey

popu

latio

ns

Time t

(a)

Y1

200 400 600 800 1000 1200 140000

1

2

3

4

5

6

The p

ropo

rtio

n of

susc

eptib

le p

reda

tor p

opul

atio

ns

Time t

(b)

Y2

200 400 600 800 1000 1200 140000

1

2

3

4

5

6

The p

ropo

rtio

n of

infe

cted

pre

dato

r pop

ulat

ions

Time t

(c)

Figure 2 Hopf bifurcation around the interior equilibrium of system (3)

interior equilibrium 119864lowastis locally asymptotically stable in the

neighborhood of 119864lowast(see Figure 1(c)) Since the condition

for the global asymptotic stability of 119864lowastholds good (see

Theorem 5) the unique interior equilibrium solution 119864lowast=

(149542 17391 224433) is a global attractor (see Figure 3)We have performed a bifurcation analysis of the model

and obtained a critical value 120573119867 When the transmission

coefficient from susceptible predator to infected predator 120573

passes through 120573119867 the system undergoes a Hopf bifurcation

around the stationary state of coexistence (see Figure 2)

7 Conclusions

In this paper we have proposed and analyzed a Leslie-Gowerecoepidemiological model that divided the total population

into three different populations namely prey (119883) susceptiblepredator (119884

1) and infected predator (119884

2) The conditions

for existence and stability of the all equilibria of the systemhave been given The bifurcation situations have also beenobserved around the interior equilibrium point

The system has four equilibriums 1198640 1198641 1198642 and 119864

lowast We

have obtained epidemiological threshold quantities for ourmodel 1198771

0= 11990321198891 11987720

= 119896(1199032

minus 1198891)(1205731199031

minus 1198861198892)1199032ℎ11990311198892

and 1198773

0= (119896119903

21205731199031

+ 11989611988911198861198892

+ 1198961198861198892

2)(11989611990321198861198892

+ 11989611988911205731199031

+

211990311199032ℎ1198892+ 11988921198961199031120573) 1198640is unstable for all times If 1198771

0lt 1 the

axial equilibrium1198641is locally asymptotically stable If1198772

0lt 1

the planar equilibrium 1198642is locally asymptotically stable It

is observed that the infected predator does not survive andcould make the system free from disease If 119877

2

0gt 1 the

planer equilibrium 1198642is unstable which is the conditions of


010

2030

05

10150

2

4

6

8

10

X

Y1

Y2

minus10

Figure 3 The global stability around the equilibriums 119864lowastof system

(3)

the existence of the the interior equilibrium 119864lowast If 11987730lt 1 the

interior equilibrium119864lowastis locally asymptotically stable which

means the global asymptotic stabilityWe analyze theHopf bifurcation around119864

lowast whichmeans

that the susceptible predator coexists with the prey and theinfected predator showing oscillatory balance behavior

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors would like to thank the reviewers for theircareful reading and constructive suggestions to the originalpaper that significantly contributed to improve the qualityof the paper And the authors gratefully acknowledge MrZhigang Chen for modifying the English grammatical errorsin the revised paper This work is supported by the NationalSciences Foundation of China (10471040) and the NationalSciences Foundation of Shanxi Province (2009011005-1)

References

[1] R M Anderson and R M May ldquoRegulation and stability ofhost-parasite population interactions I Regulatory processesrdquoJournal of Animal Ecology vol 47 no 1 pp 219ndash247 1978

[2] E Venturino ldquoEpidemics in predator-prey models diseases inthe preyrdquo in Mathematical Population Dynamics Analysis ofHeterogeneity vol 1 ofTheory of Epidemics pp 381ndash393 WuerzPublishing Winnipeg Canada 1995

[3] M Haque J Zhen and E Venturino ldquoRich dynamics of Lotka-Volterra type predatorprey model system with viral disease inPrey speciesrdquoMathematicalMethods in the Applied Sciences vol32 pp 875ndash898 2009

[4] Y Xiao and L Chen ldquoAnalysis of a three species eco-epidemiological modelrdquo Journal of Mathematical Analysis andApplications vol 258 no 2 pp 733ndash754 2001

[5] Y Xiao and L Chen ldquoModelling and analysis of a predator-preymodel with disease in the preyrdquo Mathematical Biosciences vol171 no 1 pp 59ndash82 2001

[6] J J Tewa V Y Djeumen and S Bowong ldquoPredator-prey modelwith Holling response function of type II and SIS infectiousdiseaserdquo Applied Mathematical Modelling vol 37 no 7 pp4825ndash4841 2013

[7] M S Rahman and S Chakravarty ldquoA predator-preymodel withdisease in preyrdquo Nonlinear Analysis Modelling and Control vol18 no 2 pp 191ndash209 2013

[8] M Haque S Sarwardi S Preston and E Venturino ldquoEffectof delay in a Lotka-Volterra type predator-prey model witha transmission disease in the predator speciesrdquo MathematicalBiosciences vol 234 no 1 pp 47ndash57 2011

[9] Y-HHsieh andC-KHsiao ldquoPredator-preymodel with diseaseinfection in both populationsrdquo Mathematical Medicine andBiology vol 25 no 3 pp 247ndash266 2008

[10] P H Leslie ldquoSome further notes on the use of matrices inpopulationmathematicsrdquoBiometrika vol 35 pp 213ndash245 1948

[11] A Korobeinikov ldquoA lyapunov function for leslie-gowerpredator-prey modelsrdquo Applied Mathematics Letters vol 14 no6 pp 697ndash699 2001

[12] S Sarwardi M Haque and E Venturino ldquoA Leslie-GowerHolling-type II ecoepidemic modelrdquo Journal of Applied Math-ematics and Computing vol 35 no 1-2 pp 263ndash280 2011

[13] S Sharma and G P Samanta ldquoA Leslie-Gower predator-preymodel with disease in prey incorporating a prey refugerdquo ChaosSolitons amp Fractals vol 70 pp 69ndash84 2015

[14] M A Aziz-Alaoui ldquoStudy of a Leslie-Gower-type tritrophicpopulation modelrdquo Chaos Solitons and Fractals vol 14 no 8pp 1275ndash1293 2002

[15] F Chen L Chen and X Xie ldquoOn a Leslie-Gower predator-preymodel incorporating a prey refugerdquo Nonlinear Analysis RealWorld Applications vol 10 no 5 pp 2905ndash2908 2009

[16] W Liu andC Fu ldquoHopf bifurcation of amodified lesliemdashGowerpredatormdashprey systemrdquoCognitive Computation vol 5 no 1 pp40ndash47 2013

[17] J K HaleTheory of Functional Differential Equations SpringerNew York NY USA 1977

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


is studied for a modified Leslie-Gower predator-prey sys-tem with harvesting [13] Aziz-Alaoui [14] studied dynamicbehaviors of three Leslie-Gower-type species food chainsystems Chen et al [15] incorporated a prey refuge to system(1) and showed that the refuge has no influence on thepersistent property of the system A predator-prey Leslie-Gowermodel with disease in prey has been developed whereit is observed that a Hopf bifurcation may occur around theinterior equilibrium taking refuge parameter as bifurcationparameter [16] Some similar kinds of models have appearedin the recent literature the main new distinctive feature is theinclusion of an infectious disease in prey population But thedisease also can spread in predator because of food parasitemating and so on

In the present research we formulate a predator-preyLeslie-Gower model with disease in predator The totalpopulation have been divided into three classes namelysusceptible predator infected predator and prey populationThe construction and model assumptions are discussed inSection 2 In Section 3 positivity and boundedness of thesolutions of the model are discussed Section 4 deals withtheir existence and stability analysis of the equilibriumpoints In Section 5 a detailed study of the Hopf bifurcationaround the interior equilibrium is carried out Numericalillustrations are performed finally in order to validate theapplicability of the model under consideration

2 The Mathematical Model

We construct the following model

119889119883

119889119905= 1199031119883(1 minus

119883

119896) minus 119886119883119884

1

1198891198841

119889119905= 11990321198841[1 minus

ℎ (1198841+ 1198842)

119883] minus 120573119884

11198842minus 11988911198841

1198891198842

119889119905= 12057311988411198842minus 11988921198842

(3)

with initial conditions119883 (0) ge 0

1198841(0) ge 0

1198842(0) ge 0

(4)

where119883(119905)1198841(119905) and119884

2(119905) are the density of prey susceptible

predator and infected predator populations respectively attime 119905 The prey population grows according to a logisticfashion with carrying capacity 119896 and an intrinsic birth rateconstant 119903

1 1199032is the intrinsic growth rate of susceptible

predator populations 120573 is the transmission coefficient fromsusceptible predator to infected predator ℎ is the maximumvalue of the per capita reduction rate of119883 due to119884 = (119884

1+1198842)

The second equation of system (3) contains the so-calledLeslie-Gower term namely ℎ(119884

1+ 1198842)119883 119889

1is the natural

death rate of susceptible predator 1198892is death rate of infected

predator including natural death rate and disease relateddeath rate in the absence of predator The model parameters1199031 1199032 119896 119886 ℎ 120573 119889

1 and 119889

2are all positive constants

3 Some Preliminary Results

Theorem 1 Every solution of system (3) with initial conditions(4) exists in the interval [0 +infin) and 119883(119905) ge 0 119884

1(119905) ge 0

1198842(119905) ge 0 for all 119905 gt 0

Proof Since the right-hand side of system (3) is completelycontinuous and locally Lipschitzian on 119862 the solution(119883(119905) 119884

1(119905) 1198842(119905)) of (3) with initial conditions (4) exists and

is unique on [0 120577) where 0 lt 120577 le +infin [17] From system (3)with initial conditions (4) we have

119883 (119905) ge 119883 (0) int

+infin

0

1199031[119883 (119904) (1 minus

119883 (119904)

119896)

minus 119886119883 (119904) 1198841(119904)] 119889119904 ge 0

1198841(119905) ge 119884

1(0) int

+infin

0

11990321198841(119904) [1 minus

ℎ (1198841(119904) + 119884

2(119904))

119883 (119904)]

minus 1205731198841(119904) 1198842(119904) minus 119889

11198841(119904) 119889119904 ge 0

1198842(119905) ge 119884

2(0) int

+infin

0

[1205731198841(119904) 1198842(119904) minus 119889

21198842(119904)] 119889119904 ge 0

(5)

which completes the proof

Theorem 2 All solutions of system (3) initiating 1198773 are

ultimately bounded

Proof We consider first 119883(119905) le 119896 forall119905 gt 0

119889119883

119889119905= 1199031119883(1 minus

119883

119896) minus 119886119883119884

1le 1199031119883(1 minus

119883

119896) (6)

We get 119883 = 1(119862119890minus1199031119905+ 1119896) If 119905 rarr infin 119883 rarr 119896

1198891198841

119889119905+

1198891198842

119889119905= 11990321198841[1 minus

ℎ (1198841+ 1198842)

119883] minus 11988911198841minus 11988921198842

le 11990321198841[1 minus

ℎ (1198841+ 1198842)

119883]

le 11990321198841[1 minus

ℎ (1198841+ 1198842)

119896]

le 1199032(1198841+ 1198842) minus

1199032ℎ

119896(1198841+ 1198842)2

(7)

So again by solving the above linear differential inequalitywe have

0 lt 1198841+ 1198842lt

1

119890minus1199032119905 + ℎ119896 (8)

If 119905 rarr infin 1198841+ 1198842

rarr ℎ119896 The proof is completed

Therefore the feasible region Γ defined by

Γ

= (119883 (119905) 1198841(119905) 1198842(119905)) isin 119877

3

+ 119883 le 119896 119884

1+ 1198842le

ℎ

119896

(9)


with119883(0) ge 01198841(0) ge 0 and119884


of model (3)





(3) if 1199032


2= (119896119903

2ℎ1199031

(11990311199032ℎ+119886119896(119903

2minus1198891)) 1198961199031(1199032minus1198891)(11990311199032ℎ+119886119896(119903

2minus1198891)) 0)

exists(4) if 120573119903

1gt 1198861198892 and 119896(119903


gt



1198842lowast


lowast= 119896(120573119903

1minus 1198861198892)1205731199031 1198841lowast

= 1198892120573 and 119884

2lowast=


1minus

1198961198861198892))



(

1199031minus

21199031

119896119883 minus 119886119884

1minus119886119883 0

ℎ11990321198841(1198841+ 1198842)2

1198831199032minus

21199032ℎ1198841

119883minus1199032ℎ1198841

119883minus 1205731198841

0 1205731198842

1205731198841minus 1198892

) (10)



1205821= minus1199031 1205822= 1199032minus 1198891 and 120582



1is locally


2= (1198832 11988412

0) is given by

(

minus1199031

1198961198832

minus1198861198832

0

ℎ11990321198842

12

11988322

minus1199032ℎ11988412

1198832

minus1199032ℎ11988412

1198832

minus 12057311988412

0 0 12057311988412

minus 1198892

) (11)



1205833+ 11986011205832+ 1198612120583 + 119860

3= 0 (12)

where 1198601= minus(119886

11+ 11988622

+ 11988633) 1198602= 1198861111988612

minus 1198861211988621

+

1198861111988633

+ 1198862211988633 1198603= minus11988633(1198861111988612

minus 1198861211988621) and 119886

11=

minus1198961199032ℎ1199032

1(11990311199032ℎ+119886119896(119903

2minus1198891)) 11988612

= minus1198861198961199032ℎ1199031(11990311199032ℎ+

119886119896(1199032minus 1198891)) 11988621

= (1199032minus 1198891)21199032ℎ 11988622

= minus(1199032minus 1198891)

(11990311199032ℎ+119886119896(119903

2minus1198891)) and 119886

33= 120573119896119903

1(1199032minus1198891)(11990311199032ℎ+

119886119896(1199032minus 1198891)) minus 119889

2

If 119896(1199032

minus 1198891)(1205731199031

minus 1198861198892) minus 1199032ℎ11990311198892

lt 0 11988633


1gt 0 119860

3gt 0 and 119860

11198602minus 1198603gt 0



lowast=


1198842lowast

) is given by

(

(

minus1199031

119896119883lowast


0

ℎ11990321198841lowast

(1198841lowast

+ 1198842lowast

)

1198832lowast

1199032ℎ (1198842lowast

minus 1198841lowast

)

119883lowast


119883lowast


0 1205731198842lowast

0

)

)

(13)



1205833+ 11986111205832+ 1198612120583 + 1198613= 0 (14)

where 1198611= minus(11988711

+ 11988722) 1198612= 1198871111988722

minus 1198871211988721

minus 1198872311988732

1198613

= 119887111198872311988732

and 11988711

= minus(1205731199031

minus 1198861198892)120573 119887

12=

minus119886119896(1205731199031minus1198861198892)1205731199031 11988721

= 1205731199032

1ℎ1199032(1+1199032minus1198891)119896(120573119903

1minus

1198861198892)(1199032ℎ1199031

+ 119896(1205731199031

minus 1198861198892)) 11988722

= 1199032ℎ1199031(11989611990321205731199031

+

11989611988911198861198892+1198961198861198892


119896(1205731199031minus 1198861198892)(1199032ℎ1199031+ 119896(120573119903

1minus 1198861198892)) 11988723

= minus(11990311199032ℎ1198892+

1198892119896(1205731199031

minus 1198861198892))119896(120573119903

1minus 1198861198892) and 119887

32= (119896(119903

2minus

1198891)(1205731199031minus 1198861198892) minus 1199032ℎ11990311198892)(1199032ℎ1199031+ (119896120573119903

1minus 119896119886119889

2))


gt 0 11988732

gt 0 and1198613gt 0

If 1198842lowast

minus 1198841lowast

lt 0 in other words 11989611990321205731199031+ 11989611988911198861198892+

1198961198861198892

2gt 11989611990321198861198892+ 11989611988911205731199031+ 211990311199032ℎ1198892+ 11988921198961199031120573 11988722

lt 0and 119861

1gt 0

11986111198612minus1198613= minus11988711(1198871111988722

minus1198871211988721)+11988722(1198871211988721

+1198872311988732) gt

0






Theorem 4 If 11989611990321205731199031+ 11989611988911198861198892+ 119896119886119889

2

2lt 11989611990321198861198892+ 11989611988911205731199031+


lowastis locally





(119878 119868 119884) 119878 gt 0 119868 gt 0 119884 gt 0


119881 (119883 1198841 1198842) = 1198811(119883 1198841 1198842) + 1198812(119883 1198841 1198842)

+ 1198813(119883 1198841 1198842)

(15)


where1198811(119883 1198841 1198842) = 119883minus119883


lowast)1198812(119883 1198841 1198842) =



) and 1198813(119883 1198841 1198842) = 119884


minus

1198842lowastln(11988421198842lowast

)



1198891198811

119889119905=


119883[1199031minus

119883

119896minus 1198861198841]119883

= minus1

119896(119883 minus 119883

lowast)2


)

(16)

Similarly

1198891198812

119889119905=


1198841

[1199032minus

1199032ℎ (1198841+ 1198842)

119883minus 1205731198842minus 1198891]1198841

= minus(120573 +1199032ℎ

119883) (1198841minus 1198841lowast

) (1198842minus 1198842lowast

)

minus1199032ℎ

119883(1198841minus 1198841lowast

)2

+1199032ℎ (1198841lowast

+ 1198842lowast

)

119883119883lowast


)

1198891198813

119889119905= 120573 (119884


) (1198842minus 1198842lowast

)

(17)


119889119881

119889119905= minus (119883 minus 119883



)

sdot (

1

119896minus119892 (119883 119884

1 1198842) 0

119892 (119883 1198841 1198842)

1199032ℎ

119883

1199032ℎ

2119883

01199032ℎ

21198830

)(




)

(18)

where 119892(119883 1198841 1198842) = (12)[119886 + 119903

2ℎ(1198841lowast

+ 1198842lowast

)119883119883lowast]



1198721=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

119896minus119892 (119883 119884

1 1198842)

119892 (119883 1198841 1198842)

1199032ℎ

119883

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=1199032ℎ

119896119883+ 1198922(119883 1198841 1198842) gt 0

1198722=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

119896minus119892 (119883 119884

1 1198842) 0

119892 (119883 1198841 1198842)

1199032ℎ

119883

1199032ℎ

2119883

01199032ℎ

21198830

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=1

119896(1199032ℎ

2119883)

2

gt 0

(19)



1198842lowast


5 Hopf Bifurcation


lowastwhenever the


119863 = 120573119867

isin 119877+

1198611(120573) 1198612(120573) minus 119861

3(120573)

1003816100381610038161003816120573=120573119867=0

with 1198612

gt 01198611(120573) 1198612(120573) minus 119861

3(120573)

119889120573

100381610038161003816100381610038161003816100381610038161003816120573=120573119867

= 0

(20)


=


1198842lowast


1205823+ 11986111205822+ 1198612120582 + 1198613= 0 (21)


3(120573)|120573=120573119867

= 0 give

11988722

(1198871211988721

+ 1198872311988732) minus 11988711

(1198871111988722

minus 1198871211988721)1003816100381610038161003816120573=120573119867

= 0 (22)


(1205822+ 1198612) (120582 + 119861

1) = 0 (23)


= +119894radic1198612 1205822

= minus119894radic1198612 and 120582

3=

minus1198611


119889120582

119889120573= minus

12058221+ 1205822+ 3

31205822 + 21198611120582 + 1198613

100381610038161003816100381610038161003816100381610038161003816120582=119894radic1198612

=3minus 11986121+ 1198942radic1198612

2 (1198612minus 1198941198611radic1198612)

=3minus (11986121+ 11986112)

2 (11986121+ 1198612)

+ 119894radic1198612(11986113+ 11986122minus 119861111198612)

21198612(11986121+ 1198612)

= minus119889 (1198611(120573) 1198612(120573) minus 119861

3(120573)) 119889120573

2 (11986121+ 1198612)

+ 119894 [radic11986122

21198612


3(120573)) 119889120573)

21198612(11986121+ 1198612)

]

(24)

Hence119889

119889120582(Re (120582 (120573)))

1003816100381610038161003816120573=120573119867

= minus119889 (1198611(120573) 1198612(120573) minus 119861

3(120573)) 119889120573

2 (11986121+ 1198612)

100381610038161003816100381610038161003816100381610038161003816120573=120573119867

= 0

(25)

and 1198611(120573119867) lt 0



minus5

0

5

10

15

20

25Th

e pro

port

ion

of p

opul

atio

ns

100 200 300 400 5000Time t

Y1

Y2

X

(a)

minus5

0

5

10

15

20

25

The p

ropo

rtio

n of

pop

ulat

ions

100 200 300 400 5000

Y1

Y2

X

Time t

(b)

0

5

10

15

20

25

The p

ropo

rtio

n of

pop

ulat

ions

100 200 300 400 5000

Y1

Y2

X

Time t

(c)





2 119886 = 03 1199032= 0002 ℎ = 5 120573 = 0003 119889

1= 003 119889

2= 004

and 1199032minus 1198891


1are 1205821

= minus05 1205821

= minus0028 and 1205823


1= (2 0 0) (see Figure 1(a))


= 02119896 = 20 119886 = 008 119903

2= 3 ℎ = 06 120573 = 0023 119889

1= 0003

1198892= 008 and 119896(119903




1198642



20 119886 = 003 1199032= 3 ℎ = 05 120573 = 0023 119889

1= 003 119889

2= 004

and 1205731199031minus1198861198892= 00036 gt 0 119896(119903



lowast 11989611990321205731199031+ 11989611988911198861198892+ 119896119886119889

2

2minus

11989611990321198861198892

minus 11989611988911205731199031

minus 211990311199032ℎ1198892

minus 11988921198961199031120573 = minus01626 lt 0




X

200 400 600 800 1000 1200 140000

1

2

3

4

5

6Th

e pro

port

ion

of p

rey

popu

latio

ns

Time t

(a)

Y1

200 400 600 800 1000 1200 140000

1

2

3

4

5

6

The p

ropo

rtio

n of

susc

eptib

le p

reda

tor p

opul

atio

ns

Time t

(b)

Y2

200 400 600 800 1000 1200 140000

1

2

3

4

5

6

The p

ropo

rtio

n of

infe

cted

pre

dato

r pop

ulat

ions

Time t

(c)











7 Conclusions




2) The conditions



lowast We


0= 11990321198891 11987720

= 119896(1199032

minus 1198891)(1205731199031

minus 1198861198892)1199032ℎ11990311198892

and 1198773

0= (119896119903

21205731199031

+ 11989611988911198861198892

+ 1198961198861198892

2)(11989611990321198861198892

+ 11989611988911205731199031

+


0lt 1 the


0lt 1



2

0gt 1 the



010

2030

05

10150

2

4

6

8

10

X

Y1

Y2

minus10


(3)




lowast whichmeans




Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










with119883(0) ge 01198841(0) ge 0 and119884


of model (3)





(3) if 1199032


2= (119896119903

2ℎ1199031

(11990311199032ℎ+119886119896(119903

2minus1198891)) 1198961199031(1199032minus1198891)(11990311199032ℎ+119886119896(119903

2minus1198891)) 0)

exists(4) if 120573119903

1gt 1198861198892 and 119896(119903


gt



1198842lowast


lowast= 119896(120573119903

1minus 1198861198892)1205731199031 1198841lowast

= 1198892120573 and 119884

2lowast=


1minus

1198961198861198892))



(

1199031minus

21199031

119896119883 minus 119886119884

1minus119886119883 0

ℎ11990321198841(1198841+ 1198842)2

1198831199032minus

21199032ℎ1198841

119883minus1199032ℎ1198841

119883minus 1205731198841

0 1205731198842

1205731198841minus 1198892

) (10)



1205821= minus1199031 1205822= 1199032minus 1198891 and 120582



1is locally


2= (1198832 11988412

0) is given by

(

minus1199031

1198961198832

minus1198861198832

0

ℎ11990321198842

12

11988322

minus1199032ℎ11988412

1198832

minus1199032ℎ11988412

1198832

minus 12057311988412

0 0 12057311988412

minus 1198892

) (11)



1205833+ 11986011205832+ 1198612120583 + 119860

3= 0 (12)

where 1198601= minus(119886

11+ 11988622

+ 11988633) 1198602= 1198861111988612

minus 1198861211988621

+

1198861111988633

+ 1198862211988633 1198603= minus11988633(1198861111988612

minus 1198861211988621) and 119886

11=

minus1198961199032ℎ1199032

1(11990311199032ℎ+119886119896(119903

2minus1198891)) 11988612

= minus1198861198961199032ℎ1199031(11990311199032ℎ+

119886119896(1199032minus 1198891)) 11988621

= (1199032minus 1198891)21199032ℎ 11988622

= minus(1199032minus 1198891)

(11990311199032ℎ+119886119896(119903

2minus1198891)) and 119886

33= 120573119896119903

1(1199032minus1198891)(11990311199032ℎ+

119886119896(1199032minus 1198891)) minus 119889

2

If 119896(1199032

minus 1198891)(1205731199031

minus 1198861198892) minus 1199032ℎ11990311198892

lt 0 11988633


1gt 0 119860

3gt 0 and 119860

11198602minus 1198603gt 0



lowast=


1198842lowast

) is given by

(

(

minus1199031

119896119883lowast


0

ℎ11990321198841lowast

(1198841lowast

+ 1198842lowast

)

1198832lowast

1199032ℎ (1198842lowast

minus 1198841lowast

)

119883lowast


119883lowast


0 1205731198842lowast

0

)

)

(13)



1205833+ 11986111205832+ 1198612120583 + 1198613= 0 (14)

where 1198611= minus(11988711

+ 11988722) 1198612= 1198871111988722

minus 1198871211988721

minus 1198872311988732

1198613

= 119887111198872311988732

and 11988711

= minus(1205731199031

minus 1198861198892)120573 119887

12=

minus119886119896(1205731199031minus1198861198892)1205731199031 11988721

= 1205731199032

1ℎ1199032(1+1199032minus1198891)119896(120573119903

1minus

1198861198892)(1199032ℎ1199031

+ 119896(1205731199031

minus 1198861198892)) 11988722

= 1199032ℎ1199031(11989611990321205731199031

+

11989611988911198861198892+1198961198861198892


119896(1205731199031minus 1198861198892)(1199032ℎ1199031+ 119896(120573119903

1minus 1198861198892)) 11988723

= minus(11990311199032ℎ1198892+

1198892119896(1205731199031

minus 1198861198892))119896(120573119903

1minus 1198861198892) and 119887

32= (119896(119903

2minus

1198891)(1205731199031minus 1198861198892) minus 1199032ℎ11990311198892)(1199032ℎ1199031+ (119896120573119903

1minus 119896119886119889

2))


gt 0 11988732

gt 0 and1198613gt 0

If 1198842lowast

minus 1198841lowast

lt 0 in other words 11989611990321205731199031+ 11989611988911198861198892+

1198961198861198892

2gt 11989611990321198861198892+ 11989611988911205731199031+ 211990311199032ℎ1198892+ 11988921198961199031120573 11988722

lt 0and 119861

1gt 0

11986111198612minus1198613= minus11988711(1198871111988722

minus1198871211988721)+11988722(1198871211988721

+1198872311988732) gt

0






Theorem 4 If 11989611990321205731199031+ 11989611988911198861198892+ 119896119886119889

2

2lt 11989611990321198861198892+ 11989611988911205731199031+


lowastis locally





(119878 119868 119884) 119878 gt 0 119868 gt 0 119884 gt 0


119881 (119883 1198841 1198842) = 1198811(119883 1198841 1198842) + 1198812(119883 1198841 1198842)

+ 1198813(119883 1198841 1198842)

(15)


where1198811(119883 1198841 1198842) = 119883minus119883


lowast)1198812(119883 1198841 1198842) =



) and 1198813(119883 1198841 1198842) = 119884


minus

1198842lowastln(11988421198842lowast

)



1198891198811

119889119905=


119883[1199031minus

119883

119896minus 1198861198841]119883

= minus1

119896(119883 minus 119883

lowast)2


)

(16)

Similarly

1198891198812

119889119905=


1198841

[1199032minus

1199032ℎ (1198841+ 1198842)

119883minus 1205731198842minus 1198891]1198841

= minus(120573 +1199032ℎ

119883) (1198841minus 1198841lowast

) (1198842minus 1198842lowast

)

minus1199032ℎ

119883(1198841minus 1198841lowast

)2

+1199032ℎ (1198841lowast

+ 1198842lowast

)

119883119883lowast


)

1198891198813

119889119905= 120573 (119884


) (1198842minus 1198842lowast

)

(17)


119889119881

119889119905= minus (119883 minus 119883



)

sdot (

1

119896minus119892 (119883 119884

1 1198842) 0

119892 (119883 1198841 1198842)

1199032ℎ

119883

1199032ℎ

2119883

01199032ℎ

21198830

)(




)

(18)

where 119892(119883 1198841 1198842) = (12)[119886 + 119903

2ℎ(1198841lowast

+ 1198842lowast

)119883119883lowast]



1198721=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

119896minus119892 (119883 119884

1 1198842)

119892 (119883 1198841 1198842)

1199032ℎ

119883

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=1199032ℎ

119896119883+ 1198922(119883 1198841 1198842) gt 0

1198722=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

119896minus119892 (119883 119884

1 1198842) 0

119892 (119883 1198841 1198842)

1199032ℎ

119883

1199032ℎ

2119883

01199032ℎ

21198830

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=1

119896(1199032ℎ

2119883)

2

gt 0

(19)



1198842lowast


5 Hopf Bifurcation


lowastwhenever the


119863 = 120573119867

isin 119877+

1198611(120573) 1198612(120573) minus 119861

3(120573)

1003816100381610038161003816120573=120573119867=0

with 1198612

gt 01198611(120573) 1198612(120573) minus 119861

3(120573)

119889120573

100381610038161003816100381610038161003816100381610038161003816120573=120573119867

= 0

(20)


=


1198842lowast


1205823+ 11986111205822+ 1198612120582 + 1198613= 0 (21)


3(120573)|120573=120573119867

= 0 give

11988722

(1198871211988721

+ 1198872311988732) minus 11988711

(1198871111988722

minus 1198871211988721)1003816100381610038161003816120573=120573119867

= 0 (22)


(1205822+ 1198612) (120582 + 119861

1) = 0 (23)


= +119894radic1198612 1205822

= minus119894radic1198612 and 120582

3=

minus1198611


119889120582

119889120573= minus

12058221+ 1205822+ 3

31205822 + 21198611120582 + 1198613

100381610038161003816100381610038161003816100381610038161003816120582=119894radic1198612

=3minus 11986121+ 1198942radic1198612

2 (1198612minus 1198941198611radic1198612)

=3minus (11986121+ 11986112)

2 (11986121+ 1198612)

+ 119894radic1198612(11986113+ 11986122minus 119861111198612)

21198612(11986121+ 1198612)

= minus119889 (1198611(120573) 1198612(120573) minus 119861

3(120573)) 119889120573

2 (11986121+ 1198612)

+ 119894 [radic11986122

21198612


3(120573)) 119889120573)

21198612(11986121+ 1198612)

]

(24)

Hence119889

119889120582(Re (120582 (120573)))

1003816100381610038161003816120573=120573119867

= minus119889 (1198611(120573) 1198612(120573) minus 119861

3(120573)) 119889120573

2 (11986121+ 1198612)

100381610038161003816100381610038161003816100381610038161003816120573=120573119867

= 0

(25)

and 1198611(120573119867) lt 0



minus5

0

5

10

15

20

25Th

e pro

port

ion

of p

opul

atio

ns

100 200 300 400 5000Time t

Y1

Y2

X

(a)

minus5

0

5

10

15

20

25

The p

ropo

rtio

n of

pop

ulat

ions

100 200 300 400 5000

Y1

Y2

X

Time t

(b)

0

5

10

15

20

25

The p

ropo

rtio

n of

pop

ulat

ions

100 200 300 400 5000

Y1

Y2

X

Time t

(c)





2 119886 = 03 1199032= 0002 ℎ = 5 120573 = 0003 119889

1= 003 119889

2= 004

and 1199032minus 1198891


1are 1205821

= minus05 1205821

= minus0028 and 1205823


1= (2 0 0) (see Figure 1(a))


= 02119896 = 20 119886 = 008 119903

2= 3 ℎ = 06 120573 = 0023 119889

1= 0003

1198892= 008 and 119896(119903




1198642



20 119886 = 003 1199032= 3 ℎ = 05 120573 = 0023 119889

1= 003 119889

2= 004

and 1205731199031minus1198861198892= 00036 gt 0 119896(119903



lowast 11989611990321205731199031+ 11989611988911198861198892+ 119896119886119889

2

2minus

11989611990321198861198892

minus 11989611988911205731199031

minus 211990311199032ℎ1198892

minus 11988921198961199031120573 = minus01626 lt 0




X

200 400 600 800 1000 1200 140000

1

2

3

4

5

6Th

e pro

port

ion

of p

rey

popu

latio

ns

Time t

(a)

Y1

200 400 600 800 1000 1200 140000

1

2

3

4

5

6

The p

ropo

rtio

n of

susc

eptib

le p

reda

tor p

opul

atio

ns

Time t

(b)

Y2

200 400 600 800 1000 1200 140000

1

2

3

4

5

6

The p

ropo

rtio

n of

infe

cted

pre

dato

r pop

ulat

ions

Time t

(c)











7 Conclusions




2) The conditions



lowast We


0= 11990321198891 11987720

= 119896(1199032

minus 1198891)(1205731199031

minus 1198861198892)1199032ℎ11990311198892

and 1198773

0= (119896119903

21205731199031

+ 11989611988911198861198892

+ 1198961198861198892

2)(11989611990321198861198892

+ 11989611988911205731199031

+


0lt 1 the


0lt 1



2

0gt 1 the



010

2030

05

10150

2

4

6

8

10

X

Y1

Y2

minus10


(3)




lowast whichmeans




Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










where1198811(119883 1198841 1198842) = 119883minus119883


lowast)1198812(119883 1198841 1198842) =



) and 1198813(119883 1198841 1198842) = 119884


minus

1198842lowastln(11988421198842lowast

)



1198891198811

119889119905=


119883[1199031minus

119883

119896minus 1198861198841]119883

= minus1

119896(119883 minus 119883

lowast)2


)

(16)

Similarly

1198891198812

119889119905=


1198841

[1199032minus

1199032ℎ (1198841+ 1198842)

119883minus 1205731198842minus 1198891]1198841

= minus(120573 +1199032ℎ

119883) (1198841minus 1198841lowast

) (1198842minus 1198842lowast

)

minus1199032ℎ

119883(1198841minus 1198841lowast

)2

+1199032ℎ (1198841lowast

+ 1198842lowast

)

119883119883lowast


)

1198891198813

119889119905= 120573 (119884


) (1198842minus 1198842lowast

)

(17)


119889119881

119889119905= minus (119883 minus 119883



)

sdot (

1

119896minus119892 (119883 119884

1 1198842) 0

119892 (119883 1198841 1198842)

1199032ℎ

119883

1199032ℎ

2119883

01199032ℎ

21198830

)(




)

(18)

where 119892(119883 1198841 1198842) = (12)[119886 + 119903

2ℎ(1198841lowast

+ 1198842lowast

)119883119883lowast]



1198721=

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

119896minus119892 (119883 119884

1 1198842)

119892 (119883 1198841 1198842)

1199032ℎ

119883

10038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=1199032ℎ

119896119883+ 1198922(119883 1198841 1198842) gt 0

1198722=

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

1

119896minus119892 (119883 119884

1 1198842) 0

119892 (119883 1198841 1198842)

1199032ℎ

119883

1199032ℎ

2119883

01199032ℎ

21198830

1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816

=1

119896(1199032ℎ

2119883)

2

gt 0

(19)



1198842lowast


5 Hopf Bifurcation


lowastwhenever the


119863 = 120573119867

isin 119877+

1198611(120573) 1198612(120573) minus 119861

3(120573)

1003816100381610038161003816120573=120573119867=0

with 1198612

gt 01198611(120573) 1198612(120573) minus 119861

3(120573)

119889120573

100381610038161003816100381610038161003816100381610038161003816120573=120573119867

= 0

(20)


=


1198842lowast


1205823+ 11986111205822+ 1198612120582 + 1198613= 0 (21)


3(120573)|120573=120573119867

= 0 give

11988722

(1198871211988721

+ 1198872311988732) minus 11988711

(1198871111988722

minus 1198871211988721)1003816100381610038161003816120573=120573119867

= 0 (22)


(1205822+ 1198612) (120582 + 119861

1) = 0 (23)


= +119894radic1198612 1205822

= minus119894radic1198612 and 120582

3=

minus1198611


119889120582

119889120573= minus

12058221+ 1205822+ 3

31205822 + 21198611120582 + 1198613

100381610038161003816100381610038161003816100381610038161003816120582=119894radic1198612

=3minus 11986121+ 1198942radic1198612

2 (1198612minus 1198941198611radic1198612)

=3minus (11986121+ 11986112)

2 (11986121+ 1198612)

+ 119894radic1198612(11986113+ 11986122minus 119861111198612)

21198612(11986121+ 1198612)

= minus119889 (1198611(120573) 1198612(120573) minus 119861

3(120573)) 119889120573

2 (11986121+ 1198612)

+ 119894 [radic11986122

21198612


3(120573)) 119889120573)

21198612(11986121+ 1198612)

]

(24)

Hence119889

119889120582(Re (120582 (120573)))

1003816100381610038161003816120573=120573119867

= minus119889 (1198611(120573) 1198612(120573) minus 119861

3(120573)) 119889120573

2 (11986121+ 1198612)

100381610038161003816100381610038161003816100381610038161003816120573=120573119867

= 0

(25)

and 1198611(120573119867) lt 0



minus5

0

5

10

15

20

25Th

e pro

port

ion

of p

opul

atio

ns

100 200 300 400 5000Time t

Y1

Y2

X

(a)

minus5

0

5

10

15

20

25

The p

ropo

rtio

n of

pop

ulat

ions

100 200 300 400 5000

Y1

Y2

X

Time t

(b)

0

5

10

15

20

25

The p

ropo

rtio

n of

pop

ulat

ions

100 200 300 400 5000

Y1

Y2

X

Time t

(c)





2 119886 = 03 1199032= 0002 ℎ = 5 120573 = 0003 119889

1= 003 119889

2= 004

and 1199032minus 1198891


1are 1205821

= minus05 1205821

= minus0028 and 1205823


1= (2 0 0) (see Figure 1(a))


= 02119896 = 20 119886 = 008 119903

2= 3 ℎ = 06 120573 = 0023 119889

1= 0003

1198892= 008 and 119896(119903




1198642



20 119886 = 003 1199032= 3 ℎ = 05 120573 = 0023 119889

1= 003 119889

2= 004

and 1205731199031minus1198861198892= 00036 gt 0 119896(119903



lowast 11989611990321205731199031+ 11989611988911198861198892+ 119896119886119889

2

2minus

11989611990321198861198892

minus 11989611988911205731199031

minus 211990311199032ℎ1198892

minus 11988921198961199031120573 = minus01626 lt 0




X

200 400 600 800 1000 1200 140000

1

2

3

4

5

6Th

e pro

port

ion

of p

rey

popu

latio

ns

Time t

(a)

Y1

200 400 600 800 1000 1200 140000

1

2

3

4

5

6

The p

ropo

rtio

n of

susc

eptib

le p

reda

tor p

opul

atio

ns

Time t

(b)

Y2

200 400 600 800 1000 1200 140000

1

2

3

4

5

6

The p

ropo

rtio

n of

infe

cted

pre

dato

r pop

ulat

ions

Time t

(c)











7 Conclusions




2) The conditions



lowast We


0= 11990321198891 11987720

= 119896(1199032

minus 1198891)(1205731199031

minus 1198861198892)1199032ℎ11990311198892

and 1198773

0= (119896119903

21205731199031

+ 11989611988911198861198892

+ 1198961198861198892

2)(11989611990321198861198892

+ 11989611988911205731199031

+


0lt 1 the


0lt 1



2

0gt 1 the



010

2030

05

10150

2

4

6

8

10

X

Y1

Y2

minus10


(3)




lowast whichmeans




Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus5

0

5

10

15

20

25Th

e pro

port

ion

of p

opul

atio

ns

100 200 300 400 5000Time t

Y1

Y2

X

(a)

minus5

0

5

10

15

20

25

The p

ropo

rtio

n of

pop

ulat

ions

100 200 300 400 5000

Y1

Y2

X

Time t

(b)

0

5

10

15

20

25

The p

ropo

rtio

n of

pop

ulat

ions

100 200 300 400 5000

Y1

Y2

X

Time t

(c)





2 119886 = 03 1199032= 0002 ℎ = 5 120573 = 0003 119889

1= 003 119889

2= 004

and 1199032minus 1198891


1are 1205821

= minus05 1205821

= minus0028 and 1205823


1= (2 0 0) (see Figure 1(a))


= 02119896 = 20 119886 = 008 119903

2= 3 ℎ = 06 120573 = 0023 119889

1= 0003

1198892= 008 and 119896(119903




1198642



20 119886 = 003 1199032= 3 ℎ = 05 120573 = 0023 119889

1= 003 119889

2= 004

and 1205731199031minus1198861198892= 00036 gt 0 119896(119903



lowast 11989611990321205731199031+ 11989611988911198861198892+ 119896119886119889

2

2minus

11989611990321198861198892

minus 11989611988911205731199031

minus 211990311199032ℎ1198892

minus 11988921198961199031120573 = minus01626 lt 0




X

200 400 600 800 1000 1200 140000

1

2

3

4

5

6Th

e pro

port

ion

of p

rey

popu

latio

ns

Time t

(a)

Y1

200 400 600 800 1000 1200 140000

1

2

3

4

5

6

The p

ropo

rtio

n of

susc

eptib

le p

reda

tor p

opul

atio

ns

Time t

(b)

Y2

200 400 600 800 1000 1200 140000

1

2

3

4

5

6

The p

ropo

rtio

n of

infe

cted

pre

dato

r pop

ulat

ions

Time t

(c)











7 Conclusions




2) The conditions



lowast We


0= 11990321198891 11987720

= 119896(1199032

minus 1198891)(1205731199031

minus 1198861198892)1199032ℎ11990311198892

and 1198773

0= (119896119903

21205731199031

+ 11989611988911198861198892

+ 1198961198861198892

2)(11989611990321198861198892

+ 11989611988911205731199031

+


0lt 1 the


0lt 1



2

0gt 1 the



010

2030

05

10150

2

4

6

8

10

X

Y1

Y2

minus10


(3)




lowast whichmeans




Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










X

200 400 600 800 1000 1200 140000

1

2

3

4

5

6Th

e pro

port

ion

of p

rey

popu

latio

ns

Time t

(a)

Y1

200 400 600 800 1000 1200 140000

1

2

3

4

5

6

The p

ropo

rtio

n of

susc

eptib

le p

reda

tor p

opul

atio

ns

Time t

(b)

Y2

200 400 600 800 1000 1200 140000

1

2

3

4

5

6

The p

ropo

rtio

n of

infe

cted

pre

dato

r pop

ulat

ions

Time t

(c)











7 Conclusions




2) The conditions



lowast We


0= 11990321198891 11987720

= 119896(1199032

minus 1198891)(1205731199031

minus 1198861198892)1199032ℎ11990311198892

and 1198773

0= (119896119903

21205731199031

+ 11989611988911198861198892

+ 1198961198861198892

2)(11989611990321198861198892

+ 11989611988911205731199031

+


0lt 1 the


0lt 1



2

0gt 1 the



010

2030

05

10150

2

4

6

8

10

X

Y1

Y2

minus10


(3)




lowast whichmeans




Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










010

2030

05

10150

2

4

6

8

10

X

Y1

Y2

minus10


(3)




lowast whichmeans




Acknowledgments


References

























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Research Article Dynamic Behaviors of a Leslie-Gower...

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