A delayed predator-prey model with disease in the prey is investigated. The conditions for the local stability and the existence of Hopf bifurcation at the positive equilibrium of the system are derived. The effect of the two different time delays on the dynamical behavior has been given. Numerical simulations are performed to illustrate the theoretical analysis. Finally, the main conclusions are drawn.

1. Introduction

During the past decades, epidemiological models have received considerate attention since the seminal SIR model of Kermack and McKendrich [1]. Great attention has been paid to the dynamics properties of the predator-prey models which have significant biological background. Numerous excellent and interesting results have been reported. For example, Bhattacharyya and Mukhopadhyay [2] studied the spatial dynamics of nonlinear prey-predator models with prey migration and predator switching, Bhattacharyya and Mukhopadhyay [3] analyzed the local and global dynamical behavior of an ecoepidemiological model, Kar and Ghorai [4] made a detailed discussion on the local stability, global stability, influence of harvesting and bifurcation of a delayed predator-prey model with harvesting, Chakraborty et al. [5] focused on the bifurcation and control of a bio-economic model of a delayed prey-predator model. For more related research, one can see [6–19].

In 2005, Song et al. [20] investigated the stability and Hopf bifurcation of a delayed ecoepidemiological model as follows:
(1.1)S˙(t)=rS(1-S+IK)-βSI,I˙(t)=βSI-cI-pIY(t-τ-1),Y˙(t)=-dY+kpYI(t-τ-2),
where S(t), I(t), Y(t) represent the susceptible prey, infected prey and predator population, respectively. K(K>0) can be interpreted as the prey carrying capacity with an intrinsic birth rate constant r(r>0). β(β>0) is called the transmission coefficient. The predator has a death rate constant d(d>0) and the predation coefficient p(p>0). The death rate of infected prey is positive constant c. The coefficient in conversing prey into predator is k(0<k≤1). τ-1 and τ-2 are the time required for mature of predator and the time required for the gestation of predator, respectively. The more detail biological meaning of the coefficients of system (1.1), one can see [20].

For the sake of simplicity, Song et al. [20] rescales time t→βkt, then system (1.1) can be transformed into the following form:
(1.2)s˙(t)=as[1-(s+i)]-si,i˙(t)=-b2i+si-liy(t-τ1),y˙(t)=-b1y+klyi(t-τ2),
where s=S/K, i=I/K, y=Y/K, a=r/Kβ, b2=c/Kβ, b1=d/Kβ, l=p/β, τ1=βKτ-1, τ2=βKτ-2.

We would like to point out that although Song et al. [20] investigated the local stability and Hopf bifurcation of system (1.2) under the assumption τ1+τ2=τ and obtained some good results, but they did not discuss what the different time delay τ1 and τ2 have effect on the stability and Hopf bifurcation behavior of system (1.2). Thus it is important for us to deal with the effect of time delay on the dynamics of system (1.2). There are some work which deal with this topic [21–24]. In this paper, we will further investigate the stability and bifurcation of model (1.2) as a complementarity. We will show that the two different time delay τ1 and τ2 have different effect on the stability and Hopf bifurcation behavior of system (1.2).

The remainder of the paper is organized as follows. In Section 2, we investigate the stability of the positive equilibrium and the occurrence of local Hopf bifurcations. In Section 3, numerical simulations are carried out to illustrate the validity of the main results. Some main conclusions are drawn in Section 4.

2. Stability and Local Hopf Bifurcations

In this section, we will study the stability of the positive equilibrium and the existence of local Hopf bifurcations.

If the following condition:
(2.1)(H1)akl>b1(k+l),akl(1-b2)>b1(1+a),
holds, then system (1.2) has a unique equilibrium point E0(s*,i*,y*), where
(2.2)s*=akl-b1(k+l)akl,i*=b1kl,y*=akl(1-b2)-b1(1+a)akl2.

Let s-(t)=s(t)-s*, i-(t)=i(t)-i*, y-(t)=y(t)-y* and still denote s-(t), i-(t), y-(t) by s(t), i(t), y(t), respectively, then (1.2) reads as
(2.3)s˙(t)=-as*s-s*(a+1)i,i˙(t)=i*s-li*y(t-τ1),y˙(t)=-(b1-kli*)y+kly*i(t-τ2).
The characteristic equation of (2.3) is given by
(2.4)det(a-2as*-(a+1)i*-λ-(a+1)s*0i*s*-b2-ly*-λ-li*e-λτ10kly*e-λτ2-b1+kli*-λ)=0.
That is
(2.5)λ3+θ2λ2+θ1λ+θ0+(γ1λ+γ0)e-λ(τ1+τ2)=0,
where
(2.6)θ0=m1m2m3+m5,θ1=-(m1m2+m1m3+m2m3+m6),θ2=-(m1+m2+m3),γ0=m1m4,γ1=m4,
where
(2.7)m1=a-2as*-(a+1)i*,m2=b2+ly*-s*,m3=b1-kli*,m4=kl2i*y*,m5=-(a+1)s*i*(b1-kli*),m6=-i*(a+1)s*.
The following lemma is important for us to analyze the distribution of roots of the transcendental equation (2.5).

Lemma 2.1 (see [<xref ref-type="bibr" rid="B15">13</xref>]).

For the transcendental equation
(2.8)P(λ,e-λτ1,…,e-λτm)=λn+p1(0)λn-1+⋯+pn-1(0)λ+pn(0)+[p1(1)λn-1+⋯+pn-1(1)λ+pn(1)]e-λτ1+⋯+[p1(m)λn-1+⋯+pn-1(m)λ+pn(m)]e-λτm=0,
as (τ1,τ2,τ3,…,τm) vary, the sum of orders of the zeros of P(λ,e-λτ1,…,e-λτm) in the open right half plane can change, and only a zero appears on or crosses the imaginary axis.

In the sequel, we consider four cases.

Case 1.

τ1=τ2=0, (2.5) becomes
(2.9)λ3+θ2λ2+(θ1+γ1)λ+θ0+γ0=0.
All roots of (2.9) have a negative real part if the following condition holds:
(2.10)(H2)θ0+γ0>0,θ2(θ1+γ1)>θ0+γ0.
Then the equilibrium point E0(s*,i*,y*) is locally asymptotically stable when the conditions (H1) and (H2) are satisfied.

Case 2.

τ1=0, τ2>0, (2.5) becomes
(2.11)λ3+θ2λ2+θ1λ+θ0+(γ1λ+γ0)e-λτ2=0.
For ω>0, iω be a root of (2.11), then it follows that
(2.12)γ1ωsinωτ2+γ0cosωτ2=θ2ω2-θ0,γ1ωcosωτ2-γ0sinωτ2=ω3-θ1ω
which is equivalent to
(2.13)ω6+(θ22-2θ0θ2-2θ1)ω4+(θ12-γ12)ω2-γ02=0.
Let z=ω2, then (2.13) takes the form
(2.14)z3+r1z2+r2z+r3=0,
where r1=θ22-2θ0θ2-2θ1, r2=θ12-γ12, r3=-γ02. Denote
(2.15)h(z)=z3+r1z2+r2z+r3.
Let M=(q/2)2+(r/3)3, where r=r2-(1/3)r12, q=(2/27)r13-(1/3)r1r2+r3. There are three cases for the solutions of (2.15).

If M>0, (2.15) has a real root and a pair of conjugate complex roots. The real root is positive and is given by
(2.16)μ1=-q2+M3+-q2-M3-13r1.

If M=0, (2.15) has three real roots, of which two are equal. In particular, if r1>0, there exists only one positive root, μ1=2-q/23-r1/3; If r1<0, there exists only one positive root, μ1=2-q/23-r1/3 for -q/23>-r1/3, and there exist three positive roots for r1/6<-q/23<-r1/3, μ1=2-q/23-r1/3, μ2=μ3=--q/23-r1/3.

If M<0, there are three distinct real roots, μ1=2(|r|/3)cos(φ/3)-r1/3, μ2=2(|r|/3)cos(φ/3+2π/3)-r1/3, μ3=2(|r|/3)cos(φ/3+4π/3)-r1/3, where cosφ=-q/2(|r|/3)3. Furthermore, if r1>0, there exists only one positive root. Otherwise, if r1<0, there may exist either one or three positive real roots. If there is only one positive real root, it is equal to max(μ1,μ2,μ3).

Obviously, the number of positive real roots of (2.15) depends on the sign of r1. If r1≥0, (2.15) has only one positive real root. Otherwise, there may exist three positive roots.

Without loss of generality, we assume that (2.14) has three positive roots, defined by z1, z2, z3, respectively. Then (2.13) has three positive roots,
(2.17)ω1=z1,ω2=z2,ω3=z3.
By (2.12), we have
(2.18)cosωkτ2=(θ2ωk2-θ0)γ0+(ωk3-θ1ωk)γ1ωkγ02+γ12ωk2.
Thus, if we denote
(2.19)τ2k(j)=1ωk{arccos[(θ2ωk2-θ0)γ0+(ωk3-θ1ωk)γ1ωkγ02+γ12ωk2]+2jπ},
where k=1,2,3; j=0,1,…, then ±iωk are a pair of purely imaginary roots of (2.11) with τ2k(j). Define
(2.20)τ20=τk0(0)=mink∈{1,2,3}{τ2k(0)},ω0=ωk0.

Based on above analysis, we have the following result.

Lemma 2.2.

If (H1) and (H2) hold, then all roots of (1.2) have a negative real part when τ2∈[0,τ20) and (1.2) admits a pair of purely imaginary roots ±ωki when τ2=τ2k(j)(k=1,2,3; j=0,1,2,…).

Let λ(τ2)=α(τ2)+iω(τ2) be a root of (2.11) near τ2=τ2k(j), and α(τ2k(j))=0, and ω(τ2k(j))=ω0. Due to functional differential equation theory, for every τ2k(j),k=1,2,3; j=0,1,2,…, there exists ε>0 such that λ(τ2) is continuously differentiable in τ2 for |τ2-τ2k(j)|<ε. Substituting λ(τ2) into the left hand side of (2.11) and taking derivative with respect to τ2, we have
(2.21)(dλdτ2)-1=-(3λ2+2θ2λ+θ1)eλτ2λ(γ1λ+γ0)+γ1λ(γ1λ+γ0)-τ2λ.
We can easily obtain
(2.22)[d(Re(τ2))dτ2]τ2=τ2k(j)-1=Re{-(3λ2+2θ2λ+θ1)eλτ2λ(γ1λ+γ0)}+Re{γ1λ(γ1λ+γ0)}=1Λ{γ1ωk2[(θ1-3ωk2)cosωkτ2k(j)-2θ2ωksinωkτ2k(j)]+(-γ0ωk)[2θ2ωkcosωkτ2k(j)+(θ1-3ωk2)sinωkτ2k(j)]+γ12ωk2}=1Λ{(θ1-3ωk2)ωk[γ0sinωkτ2k(j)+γ1ωkcosωkτ2k(j)]+2θ2ωk2[-γ0cosωkτ2k(j)-γ1ωksinωkτ2k(j)]+γ12ωk2}=1Λ[(3ωk6+(2θ22-4θ1)ωk4+(θ12-2θ0θ2+γ12)ωk2)]=1Λ(3ωk6+2r1ωk4+r2ωk2)=1Λ[zk(3zk2+2r1zk+r2)]=zkΛh′(zk),
where Λ=(γ1ωk2)2+(γ0ωk)2>0. Thus, we have
(2.23)sign{d(Reλ(τ2))dτ2}τ2=τ2k(j)=sign{d(Reλ(τ2))dτ2}τ2=τ2k(j)-1=sign{zkΛh′(zk)}≠0.
Since Λ, zk>0, we can conclude that the sign of [d(Reλ(τ2))/dτ2]τ2=τ2k(j) is determined by that of h′(zk).

The analysis above leads to the following result.

Theorem 2.3.

Suppose that zk=ωk2 and h′(zk)≠0, where h(z) is defined by (2.15). Then
(2.24)[d(Reλ(τ2))dτ]τ2=τ2k(j)≠0
and the sign of [d(Reλ(τ2))/dτ2]τ2=τ2k(j) is consistent with that of h′(zk).

In the sequel, we assume that
(2.25)(H3)h′(zk)≠0.
According to above analysis and the results of Kuang [25] and Hale [26], we have the following.

Theorem 2.4.

For τ1=0, if (H1) and (H2) hold, then the positive equilibrium E0(s*,i*,y*) of system (1.2) is asymptotically stable for τ2∈[0,τ20). In addition to the conditions (H1) and (H2), we further assume that (H3) holds, then system (1.2) undergoes a Hopf bifurcation at the positive equilibrium E0(s*,i*,y*) when τ2=τ2k(j), k=1,2,3; j=0,1,2,….

Case 3.

τ1>0, τ2=0, (2.5) takes the form
(2.26)λ3+θ2λ2+θ1λ+θ0+(γ1λ+γ0)e-λτ1=0.
For ω*>0, iω* be a root of (2.26), then it follows that
(2.27)γ1ω*sinω*τ1+γ0cosω*τ1=θ2ω*2-θ0,γ1ω*cosω*τ1-γ0sinω*τ1=ω*3-θ1ω*
which is equivalent to
(2.28)ω*6+(θ22-2θ0θ2-2θ1)ω*4+(θ12-γ12)ω*2-γ02=0.
Let z*=ω*2, then (2.13) takes the form
(2.29)z*3+p2z*2+p1z*+p0=0,
where p0=-γ02, p1=θ12-γ12, p2=θ22-2θ0θ2-2θ1. Denote
(2.30)h*(z*)=z*3+r1z*2+r2z*+r3.
Let M=(q/2)2+(r/3)3, where r=r2-(1/3)r12, q=(2/27)r13-(1/3)r1r2+r3. For (2.15), Similar analysis on the solutions of system (2.30) as that in Case 2. Here we omit it.

Without loss of generality, we assume that (2.30) has three positive roots, defined by z*1, z*2, z*3, respectively. Then (2.29) has three positive roots
(2.31)ω*1=z*1,ω*2=z*2,ω*3=z*3.
By (2.27), we have
(2.32)cosω*kτ1=(θ2ω*k2-θ0)γ0+(ω*k3-θ1ω*k)γ1ω*kγ02+γ12ω*k2.
Thus, if we denote
(2.33)τ1k(j)=1ω*k{arccos[(θ2ω*k2-θ0)γ0+(ω*k3-θ1ω*k)γ1ω*kγ02+γ12ω*k2]+2jπ},
where k=1,2,3; j=0,1,…, then ±iω*k are a pair of purely imaginary roots of (2.26) with τ1k(j). Define
(2.34)τ10=τk0(0)=mink∈{1,2,3}{τ1k(0)},ω*0=ω*k0.

The above analysis leads to the following result.

Lemma 2.5.

If (H1) and (H2) hold, then all roots of (1.2) have a negative real part when τ1∈[0,τ10) and (1.2) admits a pair of purely imaginary roots ±ωki when τ1=τ1k(j)(k=1,2,3; j=0,1,2,…).

Let λ(τ1)=α(τ1)+iω(τ1) be a root of (2.26) near τ1=τ1k(j), and α(τ1k(j))=0, and ω(τ1k(j))=ω*0. Due to functional differential equation theory, for every τ1k(j), k=1,2,3; j=0,1,2,…, there exists ε*>0 such that λ(τ1) is continuously differentiable in τ1 for |τ1-τ1k(j)|<ε. Substituting λ(τ1) into the left hand side of (2.26) and taking derivative with respect to τ1, we have
(2.35)(dλdτ1)-1=-(3λ2+2θ2λ+θ1)eλτ2λ(γ1λ+γ0)+γ1λ(γ1λ+γ0)-τλ.
We can easily obtain
(2.36)[d(Reλ(τ1))dτ1]τ1=τ1k(j)-1=Re{-(3λ2+2θ2λ+θ1)eλτ1λ(γ1λ+γ0)}+Re{γ1λ(γ1λ+γ0)}=1Λ*{γ1ω*k2[(θ1-3ω*k2)cosω*kτ1k(j)-2θ2ω*ksinω*kτ1k(j)]+(-γ0ω*k)[2θ2ω*kcosω*kτ1k(j)+(θ1-3ω*k2)sinω*kτ1k(j)]+γ12ω*k2}=1Λ*{(θ1-3ω*k2)ω*k[γ0sinω*kτ1k(j)+γ1ω*kcosω*kτ1k(j)]+2θ2ω*k2[-γ0cosω*kτ1k(j)-γ1ω*ksinω*kτ1k(j)]+γ12ω*k2}=1Λ*[(3ω*k6+(2θ22-4θ1)ω*k4+(θ12-2θ0θ2+γ12)ω*k2)]=1Λ*(3ω*k6+2r1ω*k4+r2ω*k2)=1Λ[z*k(3z*k2+2r1z*k+r2)]=z*kΛh′(z*k),
where Λ*=(γ1ω*k2)2+(γ0ω*k)2>0. Thus, we have
(2.37)sign{d(Reλ(τ1))dτ1}τ1=τ1k(j)=sign{d(Reλ(τ1))dτ1}τ1=τ1k(j)-1=sign{z*kΛh*′(z*k)}≠0.
Since Λ*,z*k>0, we can conclude that the sign of [d(Reλ(τ1))/dτ1]τ=τ1k(j) is determined by that of h*′(z*k).

From the analysis above, we obtain the following result.

Theorem 2.6.

Suppose that z*k=ω*k2 and h*′(z*k)≠0, where h*(z*) is defined by (2.30). Then
(2.38)[d(Reλ(τ1))dτ1]τ1=τ1k(j)≠0
and the sign of [d(Reλ(τ1))/dτ1]τ=τ1k(j) is consistent with that of h*′(z*k).

In the sequel, we assume that
(2.39)(H4)h*′(z*k)≠0.
Based on above analysis and in view of Kuang [25] and Hale [26], we get the following result.

Theorem 2.7.

For τ2=0, if (H1) and (H2) hold, then the positive equilibrium E0(s*,i*,y*) of system (1.2) is asymptotically stable for τ1∈[0,τ10). In addition to the condition (H1) and (H2), one further assumes that (H4) holds, then system (1.2) undergoes a Hopf bifurcation at the positive equilibrium E0(s*,i*,y*) when τ1=τ1k(j), k=1,2,3; j=0,1,2,….

Case 4.

τ1>0, τ2>0. We consider (2.5) with τ2 in its stable interval. Regarding τ1 as a parameter. Without loss of generality, we consider system (1.2) under the assumptions (H1) and (H2). Let iω(ω>0) be a root of (2.5), then we can obtain
(2.40)k1ω6+k1ω4+k2ω2+k3=0,
where
(2.41)k1=3γ02,k2=γ1(2γ0θ2-γ1θ1)+3γ1(γ0θ2-θ1γ1)+(2θ2γ1-3γ0)(θ2γ1-γ0),k3=(2γ0θ2-γ1θ1)(γ0θ2-θ1γ1)-3γ0γ1θ1+γ0θ1(θ2γ1-γ0)+(2θ2γ1-3γ0)(θ1γ0-θ1γ1),k4=γ0θ1(θ1γ0-θ1γ1).
Denote
(2.42)H(ω)=3γ02ω4+k1ω3+k2ω2+k3ω+k4.
Assume that
(2.43)(H5)θ1γ0<θ1γ1.
It is easy to check that H(0)<0 if (H5) holds and limω→+∞H(ω)=+∞. We can obtain that (2.42) has finite positive roots ω1,ω2,…,ωn. For every fixed ωi, i=1,2,3,…,k, there exists a sequence {τ1ij∣j=1,2,3,…}, such that (2.42) holds. Let
(2.44)τ10=min{τ1ij∣i=1,2,…,k;j=1,2,…}.
When τ1=τ10, (2.5) has a pair of purely imaginary roots ±iω~* for τ2∈[0,τ20).

In the following, we assume that
(2.45)(H6)[d(Reλ)dτ1]λ=iω~*≠0.
Thus, by the general Hopf bifurcation theorem for FDEs in Hale [26], we have the following result on the stability and Hopf bifurcation in system (1.2).

Theorem 2.8.

For system (1.2), suppose (H1), (H2), (H3), (H5), and (H6) are satisfied, and τ2∈[0,τ20), then the positive equilibrium E0(s*,i*,y*) is asymptotically stable when τ1∈[0,τ10), and system (1.2) undergoes a Hopf bifurcation at the positive equilibrium E0(s*,i*,y*) when τ1=τ10.

Case 5.

τ1>0, τ2>0. We consider (2.5) with τ1 in its stable interval. Regarding τ2 as a parameter. Without loss of generality, we consider system (1.2) under the assumptions (H1) and (H2). Let iω*(ω*>0) be a root of (2.5), then we can obtain
(2.46)k1ω*6+k1ω*4+k2ω*2+k3=0,
where k1, k2, k3, and k4 are defined by (2.41). Denote
(2.47)H*(ω*)=3γ02ω*4+k1ω*3+k2ω*2+k3ω*+k4.
Obviously, H(0)<0 if (H5) holds and limω→+∞H*(ω*)=+∞. We can obtain that (2.47) has finite positive roots ω1*,ω2*,…,ωn*. For every fixed ωi*, i=1,2,3,…,k, there exists a sequence {τ2ij∣j=1,2,3,…}, such that (2.47) holds. Let
(2.48)τ20=min{τ2ij∣i=1,2,…,k;j=1,2,…}.
When τ2=τ20, (2.5) has a pair of purely imaginary roots ±iω-* for τ1∈[0,τ10).

In the following, we assume that
(2.49)(H7)[d(Reλ)dτ2]λ=iω-*≠0.
In view of the general Hopf bifurcation theorem for FDEs in Hale [26], we have the following result on the stability and Hopf bifurcation in system (1.2).

Theorem 2.9.

For system (1.2), assume that (H1), (H2), (H3), (H4), and (H7) are satisfied and τ1∈[0,τ10), then the positive equilibrium E0(s*,i*,y*) is asymptotically stable when τ2∈[0,τ20), and system (1.2) undergoes a Hopf bifurcation at the positive equilibrium E0(s*,i*,y*) when τ2=τ20.

3. Computer Simulations

In this section, we present some numerical results of system (1.2) to verify the analytical predictions obtained in the previous section. Let us consider the following system:
(3.1)s˙(t)=0.5s[1-(s+i)]-si,i˙(t)=-0.2i+si-6iy(t-τ1),y˙(t)=-0.3y+5yi(t-τ1),
which has a positive equilibrium E0(0.82,0.06,0.1033). We can easily obtain that (H1)–(H7) are satisfied. When τ1=0, using Matlab 7.0, we obtain ω0≈0.4742, τ20≈0.32. The positive equilibrium E0(0.82,0.06,0.1033) is asymptotically stable for τ2<τ20≈0.32 and unstable for τ2>τ20≈0.32 which is shown in Figure 1. When τ2=τ20≈0.32, (3.1) undergoes a Hopf bifurcation at the positive equilibrium E0(0.82,0.06,0.1033), that is, a small amplitude periodic solution occurs around E0(0.82,0.06,0.1033) when τ1=0 and τ2 is close to τ20=0.32 which is shown in Figure 2.

Trajectory portrait and phase portrait of system (3.1) with τ1=0, τ2=0.2<τ20≈0.32. The positive equilibrium E0(0.82,0.06,0.1033) is asymptotically stable. The initial value is (0.8,0.1,0.05).

Trajectory portrait and phase portrait of system (3.1) with τ1=0, τ2=0.415>τ20≈0.32. Hopf bifurcation occurs from the positive equilibrium E0(0.82,0.06,0.1033). The initial value is (0.8,0.1,0.05).

Let τ2=0.25∈(0,0.32) and choose τ1 as a parameter. We have τ10≈0.16, Then the positive equilibrium is asymptotically when τ1∈[0,τ10). The Hopf bifurcation value of (3.1) is τ10≈0.16 (see Figures 3 and 4).

Trajectory portrait and phase portrait of system (3.1) with τ2=0.25, τ1=0.13<τ10≈0.16. The positive equilibrium E0(0.82,0.06,0.1033) is asymptotically stable. The initial value is (0.8,0.1,0.05).

Trajectory portrait and phase portrait of system (3.1) with τ2=0.25, τ1=0.21>τ10≈0.16. Hopf bifurcation occurs from the positive equilibrium E0(0.82,0.06,0.1033). The initial value is (0.8,0.1,0.05).

When τ2=0, using Matlab 7.0, we obtain ω*0≈0.7745, τ10≈0.16. The positive equilibrium E0(0.82,0.06,0.1033) is asymptotically stable for τ1<τ10≈0.16 and unstable for τ1>τ10≈0.16 which is shown in Figure 5. When τ1=τ10≈0.16, (3.1) undergoes a Hopf bifurcation at the positive equilibrium E0(0.82,0.06,0.1033), that is, a small amplitude periodic solution occurs around E0(0.82,0.06,0.1033) when τ2=0 and τ1 is close to τ10=0.16 which is illustrated in Figure 6.

Trajectory portrait and phase portrait of system (3.1) with τ2=0, τ1=0.28<τ10≈0.33. The positive equilibrium E0(0.82,0.06,0.1033) is asymptotically stable. The initial value is (0.8,0.1,0.05).

Trajectory portrait and phase portrait of system (3.1) with τ2=0, τ1=0.42>τ10≈0.33. Hopf bifurcation occurs from the positive equilibrium E0(0.82,0.06,0.1033). The initial value is (0.8,0.1,0.05).

Let τ1=0.25∈(0,0.32) and choose τ2 as a parameter. We have τ20≈0.33. Then the positive equilibrium is asymptotically stable when τ2∈[0,τ20). The Hopf bifurcation value of (3.1) is τ20≈0.33 (see Figures 7 and 8).

Trajectory portrait and phase portrait of system (3.1) with τ1=0.12, τ2=1.8<τ20≈2.1. The positive equilibrium E0(0.82,0.06,0.1033) is asymptotically stable. The initial value is (0.8,0.1,0.05).

Trajectory portrait and phase portrait of system (3.1) with τ1=0.12, τ2=2.5>τ20≈2.1. Hopf bifurcation occurs from the positive equilibrium E0(0.82,0.06,0.1033). The initial value is (0.8,0.1,0.05).

4. Conclusions

In this paper, we have investigated local stability of the positive equilibrium E0(s*,i*,y*) and local Hopf bifurcation of an ecoepidemiological model with two delays. It is shown that if some conditions hold true, and τ2∈[0,τ20), then the positive equilibrium E0(s*,i*,y*) is asymptotically stable when τ1∈(0,τ10), when the delay τ1 increases, the positive equilibrium E0(s*,i*,y*) loses its stability and a sequence of Hopf bifurcations occur at the positive equilibrium E0(s*,i*,y*), that is, a family of periodic orbits bifurcates from the the positive equilibrium E0(s*,i*,y*). We also showed if a certain condition is satisfied and τ1∈[0,τ10), then the positive equilibrium E0(s*,i*,y*) is asymptotically stable when τ2∈(0,τ20), when the delay τ2 increases, the positive equilibrium E0(s*,i*,y*) loses its stability and a sequence of Hopf bifurcations occur at the positive equilibrium E0(s*,i*,y*). Some numerical simulations verifying our theoretical results is performed. In addition, we must point out that although Song et al. [20] have also investigated the the existence of Hopf bifurcation for system (1.2) with respect to positive equilibrium E0(s*,i*,y*), it is assumed that τ1+τ2=τ. But what effect different time delay has on the dynamical behavior of system (1.2)? Song et al. [20] did not consider this issue. Thus we think that our work generalizes the known results of Song et al. [20]. In addition, we can investigate the Hopf bifurcation nature of system (1.2) by choosing the delay τ1 or τ2 as bifurcation parameter. We will further investigate the topic elsewhere in the near future.

Acknowledgments

This work is supported by National Natural Science Foundation of China (no. 11261010 and no. 11101126), Soft Science and Technology Program of Guizhou Province (no. 2011LKC2030), Natural Science and Technology Foundation of Guizhou Province (J[2012]2100), Governor Foundation of Guizhou Province ([2012]53), and Doctoral Foundation of Guizhou University of Finance and Economics (2010).

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