# Dynamic Behaviors of a Leslie-Gower Ecoepidemiological Model

Authors: Aihua Kang; Yakui Xue; Jianping Fu
Journal: Discrete Dynamics in Nature and Society
 (2015)

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## Abstract

A Leslie-Gower ecoepidemic model with disease in the predators is constructed and analyzed. The total population is subdivided into three subclasses, namely, susceptible predator, infected predator, and prey population. The positivity, boundness of solutions, and the existence of the equilibria are studied, and the sufficient conditions of local asymptotic stability of the equilibria are obtained by the Routh-Hurwitz criterion. We analyze the global stability of the interior equilibria by using Lyapunov functions. It is observed that a Hopf bifurcation may occur around the interior equilibrium. At last, numeric simulations are performed in support of the feasibility of the main result.


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## Body

## 1. Introduction

Since the pioneering work of Anderson and May [1], many researchers have paid great attention to the modeling and analysis 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 discussed the dynamics of prey-predator system with disease in prey population. Haque et al. [8] analyzed the dynamical behavior of predator-prey system with disease in predator population. Hsieh and Hsiao [9] proposed and discussed the dynamics of a predator-prey model with disease in both prey and predator populations. The boundness and stability of the equilibria are studied. There are mainly two types functional response: Holling-type functional response and Leslie-Gower functional response. Most scholars discussed the Hopf bifurcation and the Bogdanov-Takens bifurcation near the boundary equilibrium.The Leslie-Gower functional response is first proposed by Leslie [10], which introduced the following predator-prey model where the “carrying capacity” of the predator’s environment is proportional to the number of prey populations. The first and second Leslie-Gower predator-prey models are as follows: (1)dHdt=r1-a1PH,dPdt=r2-a2PHP,(2)dHdt=r1-a1P-b1HH,dPdt=r2-a2PHP,where H and P are the density of prey species and the predator species at time t, respectively. Because of the complex mathematical expressions involved in the analysis, Korobeinikov [11] introduced a Lyapunov function for both models (1) and (2) to prove their global stabilities. After the work of Korobeinikov, many scholars have done works on Leslie-type predator prey ecosystem. The modified Leslie-Gower and Holling-type II predator-prey model is generalized in the context of ecoepidemiology, with disease spreading only among the prey species [12]. Hopf bifurcation is studied for a modified Leslie-Gower predator-prey system with harvesting [13]. Aziz-Alaoui [14] studied dynamic behaviors of three Leslie-Gower-type species food chain systems. Chen et al. [15] incorporated a prey refuge to system (1) and showed that the refuge has no influence on the persistent property of the system. A predator-prey Leslie-Gower model with disease in prey has been developed, where it is observed that a Hopf bifurcation may occur around the interior equilibrium taking refuge parameter as bifurcation parameter [16]. Some similar kinds of models have appeared in the recent literature; the main new distinctive feature is the inclusion of an infectious disease in prey population. But the disease also can spread in predator because of food, parasite, mating, and so on.In the present research, we formulate a predator-prey Leslie-Gower model with disease in predator. The total population have been divided into three classes, namely, susceptible predator, infected predator, and prey population. The construction and model assumptions are discussed in Section2. In Section 3, positivity and boundedness of the solutions of the model are discussed. Section 4 deals with their existence and stability analysis of the equilibrium points. In Section 5, a detailed study of the Hopf bifurcation around the interior equilibrium is carried out. Numerical illustrations are performed finally in order to validate the applicability of the model under consideration.


## 2. The Mathematical Model

We construct the following model:(3)dXdt=r1X1-Xk-aXY1,dY1dt=r2Y11-hY1+Y2X-βY1Y2-d1Y1,dY2dt=βY1Y2-d2Y2with initial conditions (4)X0≥0,Y10≥0,Y20≥0,where X(t), Y1(t), and Y2(t) are the density of prey, susceptible predator, and infected predator populations, respectively, at time t. The prey population grows according to a logistic fashion with carrying capacity k and an intrinsic birth rate constant r1. r2 is the intrinsic growth rate of susceptible predator populations. β is the transmission coefficient from susceptible predator to infected predator. h is the maximum value of the per capita reduction rate of X due to Y=(Y1+Y2). The second equation of system (3) contains the so-called Leslie-Gower term, namely, h(Y1+Y2)/X. d1 is the natural death rate of susceptible predator. d2 is death rate of infected predator including natural death rate and disease related death rate in the absence of predator. The model parameters r1; r2; k; a; h; β; d1; and d2 are all positive constants.


## 3. Some Preliminary Results

Theorem 1.
Every solution of system (3) with initial conditions (4) exists in the interval [0,+∞) and X(t)≥0, Y1(t)≥0, Y2(t)≥0 for all t>0.Proof.
Since the right-hand side of system (3) is completely continuous and locally Lipschitzian on C, the solution (X(t),Y1(t),Y2(t)) of (3) with initial conditions (4) exists and is unique on [0,ζ), where 0<ζ≤+∞ [17]. From system (3) with initial conditions (4), we have (5)Xt≥X0∫0+∞r1Xs1-Xsk-aXsY1sds≥0,Y1t≥Y10∫0+∞r2Y1s1-hY1s+Y2sXs-βY1sY2s-d1Y1sds≥0,Y2t≥Y20∫0+∞βY1sY2s-d2Y2sds≥0which completes the proof.Theorem 2.
All solutions of system (3) initiating R3 are ultimately bounded.Proof.
We consider firstX(t)≤k, ∀t>0:(6)dXdt=r1X1-Xk-aXY1≤r1X1-Xk.We get X=1/Ce-r1t+1/k. If t→∞, X→k:(7)dY1dt+dY2dt=r2Y11-hY1+Y2X-d1Y1-d2Y2≤r2Y11-hY1+Y2X≤r2Y11-hY1+Y2k≤r2Y1+Y2-r2hkY1+Y22.
So again by solving the above linear differential inequality, we have(8)0<Y1+Y2<1e-r2t+h/k.If t→∞, Y1+Y2→h/k. The proof is completed.Therefore, the feasible regionΓ defined by(9)Γ=Xt,Y1t,Y2t∈R+3:X≤k,Y1+Y2≤hkwith X(0)≥0, Y1(0)≥0, and Y2(0)≥0 is positively invariant of model (3).


## 4. Stability Analysis

### 4.1. Existence of Equilibrium Points

All equilibrium points of system (3) are as follows:(1)
trivial equilibrium:E0=(0,0,0);(2)
axial equilibrium:E1=(k,0,0);(3)
ifr2>d1, the planar equilibrium E2=(kr2hr1/r1r2h+ak(r2-d1),kr1(r2-d1)/r1r2h+ak(r2-d1),0) exists;(4)
ifβr1>ad2, and k(r2-d1)(βr1-ad2)-r2hr1d2>0, the interior equilibrium E∗=(X∗,Y1∗,Y2∗) exists, where X∗=k(βr1-ad2)/βr1, Y1∗=d2/β, and Y2∗=k(r2-d1)(βr1-ad2)-r2hr1d2/r2hβr1+β(kβr1-kad2).


### 4.2. Local Stability

LetE-=(X-,Y-1,Y-2) be an equilibrium point of model (3); the Jacobian matrix of system (3) at the equilibrium point is (10)
r
1
-
2
r
1
k
X
-
-
a
Y
-
1
-
a
X
-
0
h
r
2
Y
-
1
Y
-
1
+
Y
-
2
2
X
-
r
2
-
2
r
2
h
Y
-
1
X
-
-
r
2
h
Y
-
1
X
-
-
β
Y
-
1
0
β
Y
-
2
β
Y
-
1
-
d
2
.Through judging the positive or negative of the eigenvalues which is the characteristic equation, we can know local asymptotic stability of all equilibrium points. Through calculation, we have the following results.(I)
Eigenvalues of the characteristic equation ofE1 are λ1=-r1, λ2=r2-d1, and λ3=-d2. It is clear that if r2<d1, λ2<0, the equilibrium point E1 is locally asymptotically stable.(II)
The variational matrix of system (3) at E2=(X2,Y12,0) is given by(11)
-
r
1
k
X
2
-
a
X
2
0
h
r
2
Y
12
2
X
2
2
-
r
2
h
Y
12
X
2
-
r
2
h
Y
12
X
2
-
β
Y
12
0
0
β
Y
12
-
d
2
.With regard to the equilibrium pointE2, its characteristic equation is(12)μ3+A1μ2+B2μ+A3=0,whereA1=-(a11+a22+a33), A2=a11a12-a12a21+a11a33+a22a33,  A3=-a33(a11a12-a12a21)  and  a11=-kr2hr12/(r1r2h+ak(r2-d1)), a12=-akr2hr1/r1r2h+ak(r2-d1), a21=(r2-d1)2/r2h, a22=-r2-d1/(r1r2h+ak(r2-d1)), and a33=βkr1(r2-d1)/(r1r2h+ak(r2-d1))-d2.Ifk(r2-d1)(βr1-ad2)-r2hr1d2<0, a33<0. Obviously, A1>0, A3>0, and A1A2-A3>0.By the Routh-Hurwitz rule, the equilibrium pointE2 is locally asymptotically stable in the region Γ.(III)
The variational matrix of system (3) at E∗=(X∗,Y1∗,Y2∗) is given by(13)-r1kX∗-aX∗0hr2Y1∗Y1∗+Y2∗X∗2r2hY2∗-Y1∗X∗-r2hY1∗X∗-βY1∗0βY2∗0.With regard to the equilibrium pointE∗, its characteristic equation is(14)μ3+B1μ2+B2μ+B3=0,whereB1=-(b11+b22), B2=b11b22-b12b21-b23b32, B3=b11b23b32 and b11=-(βr1-ad2)/β, b12=-ak(βr1-ad2)/βr1,  b21=βr12hr2(1+r2-d1)/k(βr1-ad2)(r2hr1+k(βr1-ad2)), b22=r2hr1(kr2βr1 + kd1ad2+kad22-kr2ad2-kd1βr1-2r1r2hd2-d2kr1β)/k(βr1-ad2)(r2hr1+k(βr1-ad2)),  b23=-r1r2hd2+d2k(βr1-ad2)/k(βr1-ad2), and  b32=(kr2-d1βr1-ad2-r2hr1d2)/(r2hr1+(kβr1-kad2)).Ifk(r2-d1)(βr1-ad2)-r2hr1d2>0, b32>0, and B3>0.IfY2∗-Y1∗<0, in other words kr2βr1+kd1ad2+kad22>kr2ad2+kd1βr1+2r1r2hd2+d2kr1β, b22<0, and B1>0.B1B2-B3=-b11(b11b22-b12b21)+b22(b12b21+b23b32)>0.By the Routh-Hurwitz rule, the equilibrium pointE∗ is locally asymptotically stable in the region Γ.So, we come to the following results.Theorem 3.
Ifk(r2-d1)(βr1-ad2)-r2hr1d2<0, the planar equilibrium point E2 is locally asymptotically stable.Theorem 4.
Ifkr2βr1+kd1ad2+kad22<kr2ad2+kd1βr1+2r1r2hd2+d2kr1β, the interior equilibrium point E∗ is locally asymptotically stable.


### 4.3. Globally Asymptotically Stable

Defining the Lyapunov function, we can judge the global asymptotically stability of the interior equilibrium point.Theorem 5.
The equilibrium pointE∗ is locally asymptotically stable, meaning that it is globally asymptotically stable in Σ={(S,I,Y):S>0,I>0,Y>0}.Proof.
Construct the Lyapunov function(15)VX,Y1,Y2=V1X,Y1,Y2+V2X,Y1,Y2+V3X,Y1,Y2,where V1(X,Y1,Y2)=X-X∗-X∗ln⁡X/X∗, V2(X,Y1,Y2)=Y1-Y1∗-Y1∗ln⁡Y1/Y1∗, and V3(X,Y1,Y2)=Y2-Y2∗-Y2∗ln⁡Y2/Y2∗
Since the solutions of the system are bounded and ultimately enter the setΓ, we restrict the study for this set. The time derivative of V1 along the solutions of system (3) is(16)dV1dt=X-X∗Xr1-Xk-aY1X=-1kX-X∗2-aX-X∗Y1-Y1∗.
Similarly,(17)dV2dt=Y1-Y1∗Y1r2-r2hY1+Y2X-βY2-d1Y1=-β+r2hXY1-Y1∗Y2-Y2∗-r2hXY1-Y1∗2+r2hY1∗+Y2∗XX∗X-X∗Y1-Y1∗,dV3dt=βY1-Y1∗Y2-Y2∗.
The above equation can be written as(18)
d
V
d
t
=
-
X
-
X
∗
,
Y
1
-
Y
1
∗
,
Y
2
-
Y
2
∗
1
k
-
g
X
,
Y
1
,
Y
2
0
g
X
,
Y
1
,
Y
2
r
2
h
X
r
2
h
2
X
0
r
2
h
2
X
0
X
-
X
∗
Y
1
-
Y
1
∗
Y
2
-
Y
2
∗
,where g(X,Y1,Y2)=1/2[a+r2h(Y1∗+Y2∗)/XX∗].
This matrix is positive definite if all upper-left submatrices are positive.
Through calculating all upper-left submatrices:(19)M1=1k-gX,Y1,Y2gX,Y1,Y2r2hX=r2hkX+g2X,Y1,Y2>0,M2=1k-gX,Y1,Y20gX,Y1,Y2r2hXr2h2X0r2h2X0=1kr2h2X2>0, it is obvious that dV/dt<0. So the interior equilibrium point E∗=(X∗,Y1∗,Y2∗) is globally asymptotically stable.


## 4.1. Existence of Equilibrium Points

All equilibrium points of system (3) are as follows:(1)
trivial equilibrium:E0=(0,0,0);(2)
axial equilibrium:E1=(k,0,0);(3)
ifr2>d1, the planar equilibrium E2=(kr2hr1/r1r2h+ak(r2-d1),kr1(r2-d1)/r1r2h+ak(r2-d1),0) exists;(4)
ifβr1>ad2, and k(r2-d1)(βr1-ad2)-r2hr1d2>0, the interior equilibrium E∗=(X∗,Y1∗,Y2∗) exists, where X∗=k(βr1-ad2)/βr1, Y1∗=d2/β, and Y2∗=k(r2-d1)(βr1-ad2)-r2hr1d2/r2hβr1+β(kβr1-kad2).


## 4.2. Local Stability

LetE-=(X-,Y-1,Y-2) be an equilibrium point of model (3); the Jacobian matrix of system (3) at the equilibrium point is (10)
r
1
-
2
r
1
k
X
-
-
a
Y
-
1
-
a
X
-
0
h
r
2
Y
-
1
Y
-
1
+
Y
-
2
2
X
-
r
2
-
2
r
2
h
Y
-
1
X
-
-
r
2
h
Y
-
1
X
-
-
β
Y
-
1
0
β
Y
-
2
β
Y
-
1
-
d
2
.Through judging the positive or negative of the eigenvalues which is the characteristic equation, we can know local asymptotic stability of all equilibrium points. Through calculation, we have the following results.(I)
Eigenvalues of the characteristic equation ofE1 are λ1=-r1, λ2=r2-d1, and λ3=-d2. It is clear that if r2<d1, λ2<0, the equilibrium point E1 is locally asymptotically stable.(II)
The variational matrix of system (3) at E2=(X2,Y12,0) is given by(11)
-
r
1
k
X
2
-
a
X
2
0
h
r
2
Y
12
2
X
2
2
-
r
2
h
Y
12
X
2
-
r
2
h
Y
12
X
2
-
β
Y
12
0
0
β
Y
12
-
d
2
.With regard to the equilibrium pointE2, its characteristic equation is(12)μ3+A1μ2+B2μ+A3=0,whereA1=-(a11+a22+a33), A2=a11a12-a12a21+a11a33+a22a33,  A3=-a33(a11a12-a12a21)  and  a11=-kr2hr12/(r1r2h+ak(r2-d1)), a12=-akr2hr1/r1r2h+ak(r2-d1), a21=(r2-d1)2/r2h, a22=-r2-d1/(r1r2h+ak(r2-d1)), and a33=βkr1(r2-d1)/(r1r2h+ak(r2-d1))-d2.Ifk(r2-d1)(βr1-ad2)-r2hr1d2<0, a33<0. Obviously, A1>0, A3>0, and A1A2-A3>0.By the Routh-Hurwitz rule, the equilibrium pointE2 is locally asymptotically stable in the region Γ.(III)
The variational matrix of system (3) at E∗=(X∗,Y1∗,Y2∗) is given by(13)-r1kX∗-aX∗0hr2Y1∗Y1∗+Y2∗X∗2r2hY2∗-Y1∗X∗-r2hY1∗X∗-βY1∗0βY2∗0.With regard to the equilibrium pointE∗, its characteristic equation is(14)μ3+B1μ2+B2μ+B3=0,whereB1=-(b11+b22), B2=b11b22-b12b21-b23b32, B3=b11b23b32 and b11=-(βr1-ad2)/β, b12=-ak(βr1-ad2)/βr1,  b21=βr12hr2(1+r2-d1)/k(βr1-ad2)(r2hr1+k(βr1-ad2)), b22=r2hr1(kr2βr1 + kd1ad2+kad22-kr2ad2-kd1βr1-2r1r2hd2-d2kr1β)/k(βr1-ad2)(r2hr1+k(βr1-ad2)),  b23=-r1r2hd2+d2k(βr1-ad2)/k(βr1-ad2), and  b32=(kr2-d1βr1-ad2-r2hr1d2)/(r2hr1+(kβr1-kad2)).Ifk(r2-d1)(βr1-ad2)-r2hr1d2>0, b32>0, and B3>0.IfY2∗-Y1∗<0, in other words kr2βr1+kd1ad2+kad22>kr2ad2+kd1βr1+2r1r2hd2+d2kr1β, b22<0, and B1>0.B1B2-B3=-b11(b11b22-b12b21)+b22(b12b21+b23b32)>0.By the Routh-Hurwitz rule, the equilibrium pointE∗ is locally asymptotically stable in the region Γ.So, we come to the following results.Theorem 3.
Ifk(r2-d1)(βr1-ad2)-r2hr1d2<0, the planar equilibrium point E2 is locally asymptotically stable.Theorem 4.
Ifkr2βr1+kd1ad2+kad22<kr2ad2+kd1βr1+2r1r2hd2+d2kr1β, the interior equilibrium point E∗ is locally asymptotically stable.


## 4.3. Globally Asymptotically Stable

Defining the Lyapunov function, we can judge the global asymptotically stability of the interior equilibrium point.Theorem 5.
The equilibrium pointE∗ is locally asymptotically stable, meaning that it is globally asymptotically stable in Σ={(S,I,Y):S>0,I>0,Y>0}.Proof.
Construct the Lyapunov function(15)VX,Y1,Y2=V1X,Y1,Y2+V2X,Y1,Y2+V3X,Y1,Y2,where V1(X,Y1,Y2)=X-X∗-X∗ln⁡X/X∗, V2(X,Y1,Y2)=Y1-Y1∗-Y1∗ln⁡Y1/Y1∗, and V3(X,Y1,Y2)=Y2-Y2∗-Y2∗ln⁡Y2/Y2∗
Since the solutions of the system are bounded and ultimately enter the setΓ, we restrict the study for this set. The time derivative of V1 along the solutions of system (3) is(16)dV1dt=X-X∗Xr1-Xk-aY1X=-1kX-X∗2-aX-X∗Y1-Y1∗.
Similarly,(17)dV2dt=Y1-Y1∗Y1r2-r2hY1+Y2X-βY2-d1Y1=-β+r2hXY1-Y1∗Y2-Y2∗-r2hXY1-Y1∗2+r2hY1∗+Y2∗XX∗X-X∗Y1-Y1∗,dV3dt=βY1-Y1∗Y2-Y2∗.
The above equation can be written as(18)
d
V
d
t
=
-
X
-
X
∗
,
Y
1
-
Y
1
∗
,
Y
2
-
Y
2
∗
1
k
-
g
X
,
Y
1
,
Y
2
0
g
X
,
Y
1
,
Y
2
r
2
h
X
r
2
h
2
X
0
r
2
h
2
X
0
X
-
X
∗
Y
1
-
Y
1
∗
Y
2
-
Y
2
∗
,where g(X,Y1,Y2)=1/2[a+r2h(Y1∗+Y2∗)/XX∗].
This matrix is positive definite if all upper-left submatrices are positive.
Through calculating all upper-left submatrices:(19)M1=1k-gX,Y1,Y2gX,Y1,Y2r2hX=r2hkX+g2X,Y1,Y2>0,M2=1k-gX,Y1,Y20gX,Y1,Y2r2hXr2h2X0r2h2X0=1kr2h2X2>0, it is obvious that dV/dt<0. So the interior equilibrium point E∗=(X∗,Y1∗,Y2∗) is globally asymptotically stable.


## 5. Hopf Bifurcation

Theorem 6.
The dynamical system undergoes Hopf bifurcation around the interior equilibrium pointsE∗ whenever the critical parameter value β=βH is included in the domain:(20)D=βH∈R+:B1βB2β-B3ββ=βH=0,withB2>0,B1βB2β-B3βdββ=βH≠0.Proof.
The characteristic equation of system (3) at E∗=(X∗,Y1∗,Y2∗)  T is given by(21)λ3+B1λ2+B2λ+B3=0.
The conditionsB1(β)B2(β)-B3(β)|β=βH=0 give(22)b22b12b21+b23b32-b11b11b22-b12b21β=βH=0.
From (21) we should have(23)λ2+B2λ+B1=0which has three roots λ1=+iB2, λ2=-iB2, and λ3=-B1.Differentiating the characteristic (21) with regard to β, we have(24)dλdβ=-λ2B˙1+λB˙2+B˙33λ2+2B1λ+B3λ=iB2=B˙3-B2B˙1+iB˙2B22B2-iB1B2=B˙3-B2B˙1+B1B˙22B12+B2+iB2B1B˙3+B2B˙2-B1B˙1B22B2B12+B2=-dB1βB2β-B3β/dβ2B12+B2+iB2B˙22B2-B1B2dB1βB2β-B3β/dβ2B2B12+B2.Hence,(25)ddλReλββ=βH=-dB1βB2β-B3β/dβ2B12+B2β=βH≠0and B1(βH)<0.We can easily establish the condition of the theoremd/dλ(Re(λ(β)))≠0, which completes the proof.


## 6. Number Simulations

For the purpose of making qualitative analysis of the present study, numerical simulations have been carried out by making use of MATLAB.The parametric values were given as follows:r1=0.5, k=2, a=0.3, r2=0.002, h=5, β=0.003, d1=0.03, d2=0.04, and r2-d1=-0.028<0. The eigenvalues of the Jacobian matrix at E1 are λ1=-0.5, λ1=-0.028, and λ3=-0.04, which satisfied the condition of the local asymptotic stability of axial equilibrium E1=(2,0,0) (see Figure 1(a)).Figure 1
The local stability around all equilibriums of system (3).
(a)
(b)
(c)The parametric values were given as follows:r1=0.2, k=20, a=0.08, r2=3, h=0.6, β=0.023, d1=0.003, d2=0.08, and k(r2-d1)(βr1-ad2)-r1r2hd2=-0.1367<0, which satisfied the condition of the local asymptotic stability of planar equilibrium E2. So the planar equilibrium E2=(1.3966,2.3254,0) is locally asymptotically stable (see Figure 1(b)).The parametric values were given as follows:r1=0.2, k=20, a=0.03, r2=3, h=0.5, β=0.023, d1=0.03, d2=0.04, and βr1-ad2=0.0036>0,  k(r2-d1)(βr1-ad2)-r2hr1d2=0.2018>0, which satisfied the conditions for existence of interior equilibrium solution E∗. kr2βr1+kd1ad2+kad22-kr2ad2-kd1βr1-2r1r2hd2-d2kr1β=-0.1626<0; the conditions for the local asymptotic stability of interior equilibrium solution E∗ are well satisfied. Hence, the positive interior equilibrium E∗ is locally asymptotically stable in the neighborhood of E∗ (see Figure 1(c)). Since the condition for the global asymptotic stability of E∗ holds good (see Theorem 5), the unique interior equilibrium solution E∗ = (14.9542,1.7391,22.4433) is a global attractor (see Figure 3).We have performed a bifurcation analysis of the model and obtained a critical valueβH. When the transmission coefficient from susceptible predator to infected predator β passes through βH, the system undergoes a Hopf bifurcation around the stationary state of coexistence (see Figure 2).Figure 2
Hopf bifurcation around the interior equilibrium of system (3).
(a)
(b)
(c)Figure 3
The global stability around the equilibriumsE∗ of system (3).


## 7. Conclusions

In this paper we have proposed and analyzed a Leslie-Gower ecoepidemiological model that divided the total population into three different populations, namely, prey (X), susceptible predator (Y1), and infected predator (Y2). The conditions for existence and stability of the all equilibria of the system have been given. The bifurcation situations have also been observed around the interior equilibrium point.The system has four equilibriumsE0, E1, E2, and E∗. We have obtained epidemiological threshold quantities for our model: R01=r2/d1, R02=k(r2-d1)(βr1-ad2)/r2hr1d2, and R03=kr2βr1+kd1ad2+kad22/kr2ad2+kd1βr1+2r1r2hd2+d2kr1β. E0 is unstable for all times. If R01<1, the axial equilibrium E1 is locally asymptotically stable. If R02<1, the planar equilibrium E2 is locally asymptotically stable. It is observed that the infected predator does not survive and could make the system free from disease. If R02>1, the planer equilibrium E2 is unstable, which is the conditions of the existence of the the interior equilibrium E∗. If R03<1, the interior equilibrium E∗ is locally asymptotically stable, which means the global asymptotic stability.We analyze the Hopf bifurcation aroundE∗, which means that the susceptible predator coexists with the prey and the infected predator showing oscillatory balance behavior.



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*Source: 169242-2015-11-23.xml*