# Stability and Hopf Bifurcation in a Prey-Predator System with Disease in the Prey and Two Delays

Authors: Juan Liu
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

This paper is concerned with a prey-predator system with disease in the prey and two delays. Local stability of the positive equilibrium of the system and existence of local Hopf bifurcation are investigated by choosing different combinations of the two delays as bifurcation parameters. For further investigation, the direction and the stability of the Hopf bifurcation are determined by using the normal form method and center manifold theorem. Finally, some numerical simulations are given to support the theoretical analysis.


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## Body

## 1. Introduction

The effect of disease in ecological system is an important issue from mathematical as well as ecological point of view. Therefore, the dynamics of epidemiological models have been investigated by many authors in recent years [1–6]. In [6], Jana and Kar proposed and investigated the following predator-prey system with disease in the prey:
(1)dS(t)dt=rS(1-SK)-αSI(t-τ)a+S-βSPb+S,dI(t)dt=αSI(t-τ)a+S-γI,dP(t)dt=mβSPb+S-δP2-εP,
where S(t), I(t) denote the population densities of the susceptible prey and the infected prey at time t, respectively. P(t) denotes the population of the predator at time t. The susceptible prey grows logistically with the intrinsic growth rate r and the carrying capacity K. The conversion from the susceptible prey to the infected prey is governed by the response function αI/(a+S). The consumption of the susceptible prey by the predator is governed by the response function βP/(b+S). γ is the removal rate of the infected prey biomass. δ is the intraspecific competition coefficient of the predator. ε is the removal rate of the predator due to natural death or harvesting. And the constant τ (τ≥0) is the time delay due to susceptible prey which becomes the infected prey. The predator-prey system with single delay has been investigated by many researchers [7–11]. Jana and Kar [6] studied the boundedness of the solutions and stability of the positive equilibrium of system (1). Existence and properties of the Hopf bifurcation were also investigated.In recent years, there are also some papers on the bifurcations of a prey-predator system with two or multiple delays [12–16]. As is known to all, the consumption of the susceptible prey by the predator throughout its past history governs the present birth rate of the predator. Therefore, it is reasonable to incorporate time delay due to the gestation of the predator into system (1). Based on this consideration, we consider the following system with two delays in this paper:
(2)dS(t)dt=rS(1-SK)-αSI(t-τ1)a+S-βSPb+S,dI(t)dt=αSI(t-τ1)a+S-γI,dP(t)dt=P[mβS(t-τ2)b+S(t-τ2)-δP-ε],
where τ1≥0 is the time delay due to susceptible prey which becomes the infected prey and τ2≥0 is the time delay due to the gestation of the predator.This paper is organized as follows. In Section2, we investigate local stability of the positive equilibrium and existence of local Hopf bifurcation of system (2) with respect to both delays. In Section 3, by using the normal form method and center manifold theorem, the properties of the Hopf bifurcation such as direction and stability are determined. Some numerical simulations are given for the support of the analytical findings in Section 4.


## 2. Local Stability and Hopf Bifurcation

According to the analysis in [6], system (2) has a unique positive equilibrium E*(S*,I*,P*) if α>γ, amβγ/(b+(a-b)γ)>ε and (1-S*/K)>βP*/(b+S*), where S*=aγ/(α-γ), I*=(r(1-S*/K)-βP*/(b+S*))(a+S*)/α, P*=(mβS*-ε(b+S*))/(δ(b+S*)).LetS-=S-S*, I-=I-I*, P-=P-P*. Dropping the bars for convenience, system (2) becomes the following form:
(3)dS(t)dt=a1S(t)+a2P(t)+b1I(t-τ1)+F1,dI(t)dt=a3S(t)+a4I(t)+b2I(t-τ1)+F2,dP(t)dt=a5P(t)+c1S(t-τ2)+F3,
where
(4)a1=-rS*K+αS*I*(a+S*)2+βS*P*(b+S*)2,a2=-βS*b+S*,a3=aαI*(a+S*)2,a4=-γ,a5=-δP*,b1=-αS*a+S*,b2=αS*a+S*,c1=mbβP*(b+S*)2,(5)F1=a21S2(t)+a22S(t)I(t-τ1)+a23S(t)P(t)+a31S3(t)+s32S2(t)I(t-τ1)+a33S2(t)P(t)+⋯,F2=b21S2(t)+b22S(t)I(t-τ1)+b31S3(t)+b32S2(t)I(t-τ1)+⋯,F3=c21P2(t)+c22P(t)S(t-τ2)+c23S2(t-τ2)+c31P(t)S2(t-τ2)+c32S3(t-τ2)+⋯,
with
(6)a21=-rK+aαI*(a+S*)3+bβP*(b+S*)3,a22=-aα(a+S*)3,a23=-bβ(b+S*)2,a31=-aαI*(a+S*)4-bβP*(b+S*)4,a32=aα(a+S*)3,a33=bβ(b+S*)3,b21=-aαI*(a+S*)3,b22=aα(a+S*)2,b31=-aαI*(a+S*)4,b32=-aα(a+S*)4,c21=-δ,c22=mbβ(b+S*)2,c23=-mbβP*(b+S*)3,c31=-mbβ(b+S*)3,c32=mbβP*(b+S*)4.
The linearized system of (3) is
(7)dS(t)dt=a1S(t)+a2P(t)+b1I(t-τ1),dI(t)dt=a3S(t)+a4I(t)+b2I(t-τ1),dP(t)dt=a5P(t)+c1S(t-τ2).
Thus, we can get that the characteristic equation of system (7) is
(8)λ3+m2λ2+m1λ+m0+(n2λ2+n1λ+n0)e-λτ1+(p1λ+p0)e-λτ2+q0e-λ(τ1+τ2)=0,
where
(9)m0=-a1a4a5,m1=a1a4+a1a5+a4a5,m2=-(a1+a4+a5),n0=a5(a1b2-a3b1),n1=(a1+a5)b2-a3b1,n2=-b2,p0=a2a4c1,p1=-a2c1,q0=a2b2c1.Case 1 (τ1=τ2=0).
Equation (8) becomes
(10)λ3+m12λ2+m11λ+m10=0,
where
(11)m12=m2+n2,m11=m1+n1+p1,m10=m0+n0+p0+q0,A10=A0+B0+C0+D0+E0+F0.It follows from the Routh-Hurwitz criteria that all roots of (10) have negative real parts if the following condition holds: (H11): m12>0 and m12m11>m10. Then, the positive equilibrium of system (2) without delay is locally asymptotically stable.Case 2 (τ1>0,τ2=0).
On substitutingτ2=0, (8) becomes
(12)λ3+m22λ2+m21λ+m20+(n22λ2+n21λ+n20)e-λτ1=0,
where
(13)m22=m2,m21=m1+p1,m20=m0+p0,n22=n2,n21=n1,n20=n0+q0.
Let λ=iω1(ω1>0) be a root of (12). Then, we have
(14)n21ω1sin⁡ω1τ1+(n20-n22ω12)cos⁡⁡ω1τ1=m22ω12-m20,n21ω1cos⁡⁡ω1τ1-(n20-n22ω12)sin⁡ω1τ1=ω13-m21ω1,
which implies that
(15)ω16+g22ω14+g21ω12+g20=0,
where
(16)g20=m202-n202,g21=m212-n212-2m20m22+2n20n22,g22=m222-n222-2m21.
Denote ω12=v1; then (15) becomes
(17)v13+g22v12+g21v1+g20=0.
Let
(18)f1(v1)=v13+g22v12+g21v1+g20.
In [17], Song et al. obtained the following results on the distribution of roots of (17).Lemma 1.
For (17),(1)
ifg20<0, then (17) has at least one positive root;(2)
ifg20≥0 and g222-3g21≤0, then (17) has no positive roots;(3)
ifg20≥0 and g222-3g21<0, then (17) has positive root if and only if v1*=(-g20+g222+3g21)/3>0 and f1(v1*)≤0.
(
H
21
) Suppose that (17) has at least one positive root.
Without loss of generality, we assume that (17) has three positive roots, which are denoted by v11, v12, and v13. Then (15) has three positive roots ω1k=v1k, k=1,2,3. For every fixed ω1k, one can get
(19)sin⁡(ω1kτ1)=(n22ω1k5+(m22n21-m21n22-n20)ω1k3+(m21n20-m20n21)ω1k)×(n222ω1k4+(n212-2n20n22)ω1k2+n202)-1≜T2s(ω1k),cos⁡(ω1kτ1)=((n21-m22n22)ω1k4+(m20n22+m22n20-m21n21)×ω1k2-m20n20)×(n222ω1k4+(n212-2n20n22)ω1k2+n202)-1≜T2c(ω1k).
Thus,
(20)τ1k(j)={1ω1k(arccos⁡(T2c(ω1k))+2jπ),T2s(ω1k)≥0,1ω1k(2π-arccos⁡(T2c(ω1k))+2jπ),T2s(ω1k)<0,
with k=1,2,3; j=0,1,2,….
Let(21)τ10=min⁡⁡{τ1k(0)},k=1,2,3,ω10=ω1k0.
Differentiating both sides of (12) with respect to τ1 we can get
(22)[dλdτ1]-1=-3λ2+2m22λ+m21λ(λ3+m22λ2+m21λ+m20)+2n22λ+n21λ(n22λ2+n21λ+n20)-τ1λ.
Thus, we have
(23)Re[dλdτ1]τ=τ10-1=f1′(v1*)n222ω104+(n212-2n20n22)ω102+n202,
where v1*=ω102. Obviously, if the condition (H22):f1′(v1*)≠0 holds, then Re[dλ/dτ1]τ1=τ10-1≠0. By the Hopf bifurcation theorem in [18], we have the following results.Theorem 2.
Suppose that conditions(H21)-(H22) hold. The positive equilibrium E*(S*,I*,P*) of system (2) is asymptotically stable for τ1∈[0,τ10) and system (2) undergoes a Hopf bifurcation at E*(S*,I*,P*) when τ1=τ10.Case 3 (τ2>0,τ1=0).
Substituteτ1=0 into (8), then (8) becomes
(24)λ3+m32λ2+m31λ+m30+(p31λ+p30)e-λτ2=0,
where
(25)m32=m2+n2,m31=m1+n1,m30=m0+n0,p31=p1,p30=p0+q0.
Let λ=iω2 (ω2>0) be a root of (24). Then, we get
(26)p31ω2sin⁡ω2τ2+p30cos⁡⁡ω2τ2=m32ω22-m30,p31ω2cos⁡⁡ω2τ2-p30sin⁡ω2τ2=ω23-m31ω2,
which follows that
(27)λ26+g32λ24+g31λ2+g30=0,
where
(28)g32=m322-2m31,g31=m312-2m30m32-p312,g30=m302-p302.
Denote ω22=v2; then (27) becomes
(29)v23+g32v22+g31v2+g30=0.
Let
(30)f2(v2)=v23+g32v22+g31v2+g30=0.
Similarly as in Case2, we suppose that (H31), (29) has at least one positive root. Without loss of generality, we assume that it has three positive roots and we denote them by v21, v22, and v23, respectively. Then (27) has three positive roots ω2k=v2k, k=1,2,3. For every fixed ω2k,
(31)sin(ω2kτ2)=(m32p31-p30)ω2k3+(m31p30-m30p31)ω2kp312ω2k2+p302≜T3s(ω2k),(32)cos⁡(ω2kτ2)=p31ω2k4+(m32p30-m31p31)ω2k2-m30p30p312ω2k2+p302≜T3c(ω2k).
Thus,
(33)τ2k(j)={1ω2k(arccos⁡(T3c(ω2k))+2jπ),T3s(ω2k)≥0,1ω2k(2π-arccos⁡(T3c(ω2k))+2jπ),T3s(ω2k)<0,
with k=1,2,3; j=0,1,2,….
Let(34)τ20=min⁡⁡{τ2k(0)},k=1,2,3,ω20=ω2k0.
Differentiating (24) regarding τ2, we get
(35)[dλdτ2]-1=3λ2+2m32λ+m31λ(λ3+m32λ2+m31λ+m30)+p31λ(p31λ+p30)-τ2λ.
Then, we can get
(36)Re[dλdτ2]τ=τ20-1=f2′(v2*)p312ω202+p302,
where v2*=ω202. Therefore, if the condition (H32): f2′(v2*)≠0 holds, then Re[dλ/dτ2]τ2=τ20-1≠0. Thus, by the Hopf bifurcation theorem in [18], we have the following results.Theorem 3.
Suppose that conditions(H31)-(H32) hold. The positive equilibrium E*(S*,I*,P*) of system (2) is asymptotically stable for τ1∈[0,τ20) and system (2) undergoes a Hopf bifurcation at E*(S*,I*,P*) when τ2=τ20.Case 4 (τ1=τ2=τ>0).
Letτ1=τ2=τ; then (8) becomes
(37)λ3+m42λ2+m41λ+m40+(n42λ2+n41λ+n40)e-λτ+q40e-2λτ=0,
where
(38)m42=m2,m41=m1,m40=m0,n42=n2,n41=n1+p1,n40=n0+p0,q40=q0.
Multiplying (37) by eλτ, then (37) becomes
(39)n42λ2+n41λ+n40+(λ3+m42λ2+m41λ+m40)eλτ+q40e-λτ=0.
Let λ=iω(ω>0) be the root of (39); then we can get
(40)(m40+q40-m42ω2)cos⁡⁡τω+(ω3-m41ω)sinτω=n32ω2-n40,(m40-q40-m42ω2)sin⁡τω-(ω3-m41ω)cos⁡⁡τω=-n41ω,
which follows that
(41)sin⁡(τω)=g5ω5+g3ω3+g1ωω6+h4ω4+h2ω2+h0≜T4s(ω),cos⁡⁡(τω)=g4ω4+g2ω2+g0ω6+h4ω4+h2ω2+h0≜T4c(ω),
where
(42)g0=n40q40-m40n40,g1=m41n40-m40n41-n41q40,g2=m40n42-m41n41+m42n40-n42q40,g3=m42n41-m41n42-n40,g4=n41-m42n42,g5=n42,h0=m402-q402,h2=m412-2m40m42,h4=m422-2m41.
Then, we can obtain
(43)ω12+g45ω10+g44ω8+g43ω6+g42ω4+g41ω2+g40=0,
where
(44)g40=h02-g02,g41=2h0h2-2g0g2-g12,g42=h22+2h0h4-g22-2g0g4-2g1g3,g43=2h0+2h2h4-g32-2g1g5-2g2g4,g44=h42+2h2-g42-2g3g5,g45=2h4-g52.
Let ω2=v3; then (43) becomes
(45)v36+g45v35+g44v34+g43v33+g42v32+g41v3+g40=0.If we know all the coefficients of system (2), then we can get all the coefficients of (45) and then all the roots of (45) can be obtained by Matlab. Therefore, we give the following assumption.Suppose that(H41): (45) has at least one positive root.Without loss of generality, we assume that (45) has six positive roots, which are denoted by v31,v32,…,v36, respectively. Then, (43) has six positive roots ωk=v3k, k=1,2,…,6. For every ωk,
(46)τk(j)={1ωk(arccos⁡(T4c(ωk))+2jπ),T4s(ωk)≥0,1ωk(2π-arccos⁡(T4c(ωk))+2jπ),T4s(ωk)<0,
with k=1,2,3,…,6; j=0,1,2,….Let(47)τ0=min⁡⁡{τk(0)},k=1,2,…,6,ω0=ωk0.
Next, taking the derivative of λ with respect to τ in (39), we have
(48)[dλdτ]-1=(2n42λ+n41+(3λ2+2m42λ+m41)eλτ)×(q40λe-λτ-(λ4+m42λ3+m41λ2hhhhhhhhhhhhhh+m40λλ4)eλτ)-1-τλ.
Then we have
(49)Re[dλdτ]τ=τ0-1=PRQR+PIQIQR2+QI2,
where
(50)PR=(m41-3ω02)cos⁡⁡τ0ω0-2m42ω0sin⁡τ0ω0+n41,PI=(m41-3ω02)sin⁡τ0ω0+2m42ω0cos⁡⁡τ0ω0+2n42ω0,QR=(m41ω02-ω04)cos⁡⁡τ0ω0-(m42ω03-m40ω0-q40ω0)sin⁡τ0ω0,QI=(m41ω02-ω04)sin⁡τ0ω0+(m42ω03-m40ω0+q40ω0)cos⁡⁡τ0ω0.Obviously, if condition(H42): PRQR+PIQI≠0 holds, then Re[dλ/dτ]τ=τ0-1≠0. Thus, by the discussion above and the Hopf bifurcation theorem in [18], we have the following results.Theorem 4.
Suppose that conditions(H41)-(H42) hold. The positive equilibrium E*(S*,I*,P*) of system (2) is asymptotically stable for τ1∈[0,τ0) and system (2) undergoes a Hopf bifurcation at E*(S*,I*,P*) when τ=τ0.Case 5 (τ2>0 and τ1∈(0,τ10)).
Letλ=iω2*(ω2*) be the root of (8). Then, we get
(51)M51sin⁡τ2ω2*+M52cos⁡⁡τ2ω2*=N51,M51cos⁡⁡τ2ω2*-M52sin⁡τ2ω2*=N52,
where
(52)M51=p1ω2*-q0sinτ1ω2*,N52=p0+q0cos⁡τ1ω2*,(53)N51=m2(ω2*)2-m0-(n0-n2(ω2*)2)cos⁡⁡τ1ω2*-n1ω2*sin⁡τ1ω2*,(54)N52=(ω2*)3-m1ω2*+(n0-n2(ω2*)2)sin⁡τ1ω2*-n1ω2*cos⁡⁡τ1ω2*.
Then, we can have
(55)g51(ω2*)+2g52(ω2*)cos⁡⁡τ1ω2*+2g53(ω2*)sin⁡τ1ω2*=0,
where
(56)g51(ω2*)=(ω2*)6+(m22+n22-2m1)(ω2*)4+(m12+n12-p12-2m0m2-2n0n2)(ω2*)2+m02+n02-p02-q02,g52(ω2*)=(m2n2-n1)(ω2*)4+(m1n1-m0n2-m2n0)(ω2*)2+m0n0-p0q0,g53(ω2*)=-n2(ω2*)5+(n0-m2n1+m1n2)(ω2*)3+(m0n1-m1n0+p1q0)ω2*.We suppose that(H51), (55) has at least finite positive roots. And we denote the positive roots of (55) by ω21*,ω22*,…,ω2k*. Then, for every fixed ω2i* (i=1,2,…,k),
(57)sin⁡(τ2ω2i*)=M51N51-M52N52M512+M522|ω2*=ω2i*≜T5s(ω2i*),cos⁡⁡(τ2ω2i*)=M51N52+M52N51M512+M522|ω2*=ω2i*≜T5c(ω2i*).
Thus,
(58)τ2i*(j)={1ω2i*(arccos⁡(T5c(ω2i*))+2jπ),T5s(ωk)≥0,1ω2i*(2π-arccos⁡(T5c(ω2i*))+2jπ),T5s(ωk)<0,
with i=1,2,…,k; j=0,1,2,….Letτ20*=min⁡⁡{τ2i*(0)∣i=1,2,…,k}. When τ2=τ20*, (8) has a pair of purely imaginary roots ±iω2* for τ1∈(0,τ10). Next, in order to give the main results with respect to τ2>0, τ1∈(0,τ10), we give the following assumption: (H52): Re[dλ/dτ2]τ=τ20*-1≠0.Through the analysis above and the Hopf bifurcation theorem in [18], we have the following results.Theorem 5.
If conditions(H51)-(H52) hold and τ1∈(0,τ10), then the positive equilibrium E*(S*,I*,P*) of system (2) is asymptotically stable for τ2∈[0,τ20*) and system (2) undergoes a Hopf bifurcation at E*(S*,I*,P*) when τ2=τ20*.


## 3. Stability of Bifurcating Periodic Solutions

In the previous section, it is shown that system (2) undergoes a Hopf bifurcation for different combinations of τ1 and τ2 under certain conditions. In this section, the properties of Hopf bifurcation such as direction and stability are investigated with respect to τ2 for τ1∈(0,τ10) by using the normal form method and center manifold theorem in [18]. Throughout this section, we assume that τ10*<τ20* where τ10*∈(0,τ10).For convenience, letτ2=μ+τ20*, so that μ=0 is the Hopf bifurcation value of system (2). Let u1(t)=S(t)-S*, u2(t)=I(t)-I*, u3(t)=P(t)-P* and rescale the time delay t→(t/τ2); then system (2) can be rewritten as
(59)u˙(t)=Lμut+F(μ,ut),
where
(60)Lμϕ=(τ20*+μ)(A′ϕ(0)+B′ϕ(-τ10*τ20*)+C′ϕ(-1)),F(μ,ϕ)=(τ20*+μ)(F1,F2,F3)T,
with
(61)ϕ(θ)=(ϕ1(θ),ϕ2(θ),ϕ3(θ))T∈C([-1,0],R3),A′=(a10a2a3a4000a5),B′=(0b100b20000),C′=(000000c100),F1=a21ϕ12(0)+a22ϕ1(0)ϕ2(-τ10*τ20*)+a23ϕ1(0)ϕ3(0)+a31ϕ13(0)+a32ϕ12(0)ϕ2(-τ10*τ20*)+a33ϕ12(0)ϕ3(0)+⋯,F2=b21ϕ12(0)+b22ϕ1(0)ϕ2(-τ10*τ20*)+b31ϕ13(0)+b32ϕ12(0)ϕ2(-τ10*τ20*)+⋯,F3=c21ϕ32(0)+c22ϕ1(-1)ϕ3(0)+c23ϕ12(-1)+c31ϕ12(-1)ϕ3(0)+c32ϕ13(-1)+⋯.Therefore, according to the Riesz representation theorem, there exists a3×3 matrix function η(θ,μ):[-1,0]→R3 whose elements are of bounded variation such that
(62)Lμϕ=∫-10‍dη(θ,μ)ϕ(θ),ϕ∈C([-1,0],R3).
In fact, we choose
(63)η(θ,μ)={(τ20*+μ)(A′+B′+C′),θ=0,(τ20*+μ)(B′+C′),θ∈[-τ10*τ20*,0),(τ20*+μ)C′,θ∈(-1,-τ10*τ20*),0,θ=-1.
For ϕ∈C([-1,0],R3), we define
(64)A(μ)ϕ={dϕ(θ)dθ,-1≤θ<0,∫-10‍dη(θ,μ)ϕ(θ),θ=0,R(μ)ϕ={0,-1≤θ<0,F(μ,ϕ),θ=0.
Then system (59) can be transformed into the following operator equation:
(65)u˙(t)=A(μ)ut+R(μ)ut,
where ut=u(t+θ)=(u1(t+θ),u2(t+θ),u3(t+θ)) for θ∈[-1,0].Forφ∈C1([0,1],(R3)*), where (R3)* is the 3-dimensional space of row vectors, we define the adjoint operator A* of A:
(66)A*(φ)={-dφ(s)ds,0<s≤1,∫-10‍dηT(s,0)φ(-s),s=0,
and a bilinear inner product
(67)〈φ(s),ϕ(θ)〉=φ-(0)ϕ(0)-∫θ=-10‍∫ξ=0θ‍φ-(ξ-θ)dη(θ)ϕ(ξ)dξ,
where η(θ)=η(θ,0).ThenA(0) and A*(0) are adjoint operators. From the discussion above, we know that ±iω2*τ20* are eigenvalues of A(0) and they are also eigenvalues of A*(0).Letq(θ)=(1,q2,q3)Teiω2*τ20*θ be the eigenvectors of A(0) corresponding to the eigenvalue +iω2*τ20* and q*(s)=D(1,q2*,q3*)eiω2*τ20*s the eigenvectors of A*(0) corresponding to the eigenvalue -iω2*τ20*.It is not difficult to verify that(68)q2=c1e-iω2*τ20*iω2*-a5,q3=a3iω2*-a4-b2e-iω2*τ20*,q2*=-b1eiω2*τ20*iω2*+a4+b2eiω2*τ20*,q3*=-a2iω2*+a5.
From (67), we can get
(69)D-=[1+q2q-2*+q3q-3*+b11τ10*e-iω1*τ10*+τ2*e-iω1*τ2*(q-2*(c21+c22q2)+q-3*(c32q2+c33q3))τ10*e-iω1*τ10*]-1,
such that 〈q*,q〉=1, 〈q*,q-〉=0.In the remainder of this section, we obtain the coefficients used to determine the properties of the periodic solution by the algorithms given in [18] and using a computation process similar to that in [11]:
(70)g20=2τ20*D-[a21+a22q(2)(-τ10*τ20*)+a23q(3)(0)+q-2*(b21+b22q(2)(-τ10*τ20*))+q-3*(c21(q(3)(0))2+c22q(1)(-1)q(3)(0)+c23(q(1)(-1))2(q(3)(0))2)(-τ10*τ20*)],g11=τ20*D-[2a21+a22(q-(2)(-τ10*τ20*)+q(2)(-τ10*τ20*))+a23(q(3)(0)+q-(3)(0))+q-2*(2b21+b22(q-(2)(-τ10*τ20*)+q(2)(-τ10*τ20*)))+q-3*(2c21q(3)(0)q-(3)(0)+c22(q(3)(0)q-(1)(-1)+q-(3)(0)q(1)(-1))+2c23q(1)(-1)q-(1)(-1))((-τ10*τ20*)-τ10*τ20*)],g02=2τ20*D-[a21+a22q-(2)(-τ10*τ20*)+a23q-(3)(0)+q-2*(b21+b22q-(2)(-τ10*τ20*))+q-3*((q-(1)(-1))2c21(q-(3)(0))2+c22q-(1)(-1)q-(3)(0)+c23(q-(1)(-1))2)(-τ10*τ20*)],g21=2τ20*D-[(-τ10*τ20*)a21(2W11(1)(0)+W20(1)(0))+a22(W11(1)(0)q(2)(-τ10*τ20*)+12W20(1)(0)q-(2)×(-τ10*τ20*)+W11(2)(-τ10*τ20*)+12W20(2)(-τ10*τ20*))+a23(W11(1)(0)q(3)(0)+12W20(1)(0)q-(3)(0)+W11(3)(0)+12W20(3)(0))+3a31+a32(q-(2)(-τ10*τ20*)+2q(2)(-τ10*τ20*))+a33(q-(3)(0)+2q(3)(0))+q-2*((q-(2)(-τ10*τ20*)+2q(2)(-τ10*τ20*))b21(2W11(1)(0)+W20(1)(0))+b22(W11(1)(0)q(2)(-τ10*τ20*)+12W20(1)(0)q-(2)×(-τ10*τ20*)+W11(2)(-τ10*τ20*)+12W20(2)(-τ10*τ20*))+3b31+b32×(q-(2)(-τ10*τ20*)+2q(2)(-τ10*τ20*)))+q-3*(12W20(1)c21(2W11(3)(0)q(3)(0)+W20(3)(0)q-(3)(0))+c22(W11(1)(-1)q(3)(0)+12W20(1)(-1)×q-(3)(0)+W11(3)(0)q(1)(-1)+12W20(3)(0)q-(1)(-1))+c23(2W11(1)(-1)+W20(1)(-1))+c31((q(1)(-1))2q-(3)(0)+2q(1)(-1)×q(3)(0)q-(1)(-1)(q(1)(-1))2)+3c32(q(1)(-1))2q-(1)(-1)12W20(1))(-τ10*τ20*)],
with
(71)W20(θ)=ig20q(0)ω2*τ20*eiω2*τ20*θ+ig-02q-(0)3ω2*τ20*e-iω2*τ20*θ+E1e2iω2*τ20*θ,W11(θ)=-ig11q(0)ω2*τ20*eiω2*τ20*θ+ig-11q-(0)ω2*τ20*e-iω2*τ20*θ+E2,
where E1 and E2 can be computed as the following equations, respectively:
(72)(2iω2*-a1-b1e-2iω2*τ20*-a2-a32iω2*-a4-b2e-2iω2*τ20*0-c1e-2iω2*τ20*02iω2*-a5)E1=2(E1(1)E1(2)E1(3)),(a1b1a2a3a4+b20c10a5)E2=-(E2(1)E2(2)E2(3)),
with
(73)E1(1)=a21+a22q(2)(-τ10*τ20*)+a23q(3)(0),E1(2)=b21+b22q(2)(-τ10*τ20*),E1(3)=c21(q(3)(0))2+c22q(1)(-1)q(3)(0)+c23(q(1)(-1))2,E2(1)=2a21+a22(q-(2)(-τ10*τ20*)+q(2)(-τ10*τ20*))+a23(q(3)(0)+q-(3)(0)),E2(2)=2b21+b22(q-(2)(-τ10*τ20*)+q(2)(-τ10*τ20*)),E2(3)=2c21q(3)(0)q-(3)(0)+c22(q(3)(0)q-(1)(-1)+q-(3)(0)q(1)(-1))+2c23q(1)(-1)q-(1)(-1).
Therefore, we can calculate the following values:
(74)C1(0)=i2ω2*τ20*(g11g20-2|g11|2-|g02|23)+g212,μ2=-Re{C1(0)}Re{λ′(τ20*)},β2=2Re{C1(0)},T2=-Im⁡{C1(0)}+μ2Im⁡{λ′(τ20*)}ω2*τ20*.Based on the discussion above, we can obtain the following results.Theorem 6.
For system (2),(i)
μ2 determines the direction of the Hopf bifurcation. If μ2>0 (μ2<0), then the Hopf bifurcation is supercritical (subcritical).(ii)
β2 determines the stability of the bifurcating periodic solutions. If β2<0 (β2>0),   then the bifurcating periodic solutions are stable (unstable).(iii)
T2 determines the period of the bifurcating periodic solutions. If T2>0 (T2<0), then the period of the bifurcating periodic solutions increases (decreases).


## 4. Numerical Example

In this section, we give a numerical example to support the theoretical results in Sections2 and 3. We use the same coefficients which are used by Jana and Kar in [6]. They are as follows: a=5.1, b=3.995, m=0.8, r=1.1, K=8.8, α=1.25, β=1.3, γ=0.39, δ=0.29, and ε=0.15. Thus, we get the following particular case of system (2):
(75)dS(t)dt=1.1S(1-S8.8)-1.25SI(t-τ1)5.1+S-1.3SP3.995+S,dI(t)dt=1.25SI(t-τ1)5.1+S-0.39I,dP(t)dt=P[1.04S(t-τ2)3.995+S(t-τ2)-0.29P-0.15],
which has a positive equilibrium E*(2.3128,3.8339,0.7977).Forτ1>0,τ2=0. We get ω10=0.4149<0, τ10=0.5148. Further, we have f1′(v1*)=0.1022>0. Thus, conditions (H21) and (H22) hold. From Theorem 2, the positive equilibrium E* is asymptotically stable when τ1<τ10 as illustrated by Figure 1. When τ1 passes through the critical value τ10, the positive equilibrium E* loses its stability and a Hopf bifurcation occurs and a family of periodic solutions bifurcate from the positive equilibrium E*, which can be shown as in Figure 2. Similarly, we have ω20=0.6010, τ20=1.8829 for τ1=0, τ2>0. The corresponding waveforms and the phase plots are shown in Figures 3 and 4.Figure 1
E
* is asymptotically stable when τ1=0.45<0.5148=τ10.Figure 2
E
* is unstable when τ1=0.575>0.5148=τ10.Figure 3
E
* is asymptotically stable when τ2=1.275<1.8829=τ20.Figure 4
E
* is unstable when τ2=2.25>1.8829=τ20.Forτ1=τ2=τ>0, we can obtain ω0=2.0030 and then we get τ0=0.3652. From Theorem 4, we know that when τ increases from zero to the critical value τ0 the positive equilibrium E* is asymptotically stable; then it will lose its stability and a Hopf bifurcation occurs once τ>τ0. These properties can be shown as in Figures 5 and 6.Figure 5
E
* is asymptotically stable when τ=0.25<0.3652=τ0.Figure 6
E
* is unstable when τ=0.45>0.3652=τ0.Forτ2>0 and τ1=0.15∈(0,τ10), we can obtain ω2*=2.1105, τ20*=1.0782. By Theorem 5, the positive equilibrium E* is asymptotically stable when τ2∈[0,τ20*) and E* is unstable when τ2>τ20* and a Hopf bifurcation occurs, which can be illustrated by Figures 7 and 8. In addition, by complex computations, we obtain C1(0)=-2.2071+1.6159i, and further we have μ2=6.8522>0, β2=-4.4142<0, T2=-3.7367<0. By Theorem 6, we know that the Hopf bifurcation with respect to τ2 with τ1=0.15∈(0,τ10) is supercritical; the bifurcating periodic solutions are stable and decrease. From the viewpoint of ecology, if the periodic solutions bifurcating from the Hopf bifurcation are stable, the species in a prey-predator system may coexist in an oscillatory mode. Therefore, we can conclude that the three species in system (75) can coexist in an oscillatory mode, since the bifurcating periodic solutions are stable.Figure 7
E
* is asymptotically stable when τ2=0.75<1.0782=τ20* and τ1=0.15∈(0,τ10).Figure 8
E
* is unstable when τ2=1.35>1.0782=τ20* and τ1=0.15∈(0,τ10).


## 5. Conclusion

In this present paper a prey-predator system with disease in the prey and two delays is considered. Based on the system proposed in [6], we further incorporate the time delay due to the gestation of the predator. The main purpose of this paper is to investigate the effects of the two delays on the system. We have shown that the two delays play a complicated role in the system. By choosing the possible combinations of the two delays as bifurcation parameters, sufficient conditions for local stability and existence of local Hopf bifurcation are obtained. When the time delay is below the corresponding critical value, we get that the system is local stable. Otherwise, a local Hopf bifurcation occurs at the positive equilibrium. We also find that the delay due to the susceptible prey becoming the infected prey is more marked compared with the delay due to the gestation of the predator, because the critical value of τ1 is much smaller than that of τ2 when we only consider one of the two delays, which can be seen from the numerical simulations. Further, the properties of the bifurcated periodic solutions such as the direction and the stability are determined. And a numerical example is also given to support the theoretical results. From the numerical simulations we can see that the species in the system considered in this paper can coexist under some certain conditions.



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*Source: 624546-2014-04-06.xml*