# Stability and Local Hopf Bifurcation for a Predator-Prey Model with Delay

Authors: Yakui Xue; Xiaoqing Wang
Journal: Discrete Dynamics in Nature and Society
 (2012)

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## Abstract

A predator-prey system with disease in the predator is investigated, where the
discrete delayτ is regarded as a parameter. Its dynamics are studied in terms of local
analysis and Hopf bifurcation analysis. By analyzing the associated characteristic
equation, it is found that Hopf bifurcation occurs when τ crosses some critical values.
Using the normal form theory and center manifold argument, the explicit formulae
which determine the stability, direction, and other properties of bifurcating periodic
solutions are derived.


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## Body

## 1. Introduction

Many models in ecology can be formulated as system of differential equations with time delays. The effect of the past history on the stability of system is also an important problem in population biology. Recently, the properties of periodic solutions arising from the Hopf bifurcation have been considered by many authors [1–4].May [5] first proposed and discussed the delayed predator-prey system
(1.1)dxdt=x(t)[r1-a11x(t-τ)-a12y(t)],dydt=y(t)[-r2+a21y(t)-a22y(t)],
where x(t) and y(t) can be interpreted as the population densities of prey and predator at time t, respectively; τ≥0 is the feedback time delay of the prey to the growth of the species itself; r1>0 denotes the intrinsic growth rate of the prey, and r2>0 denotes the death rate of the predator; the parameter aij(i,j=1,2) are all positive constants. System (1,1) shows that, in the absence of predator species, the prey species are governed by the well-known delayed logistic equation dx/dt=x(t)[r1-a11x(t-τ)] and the predator species will decrease in the absence of the prey species.There has been an extensive literature dealing with system (1,1) or the system similar to (1.1), regarding boundedness of solutions, persistence, local and global stabilities of equilibria, and existence of nonconstant periodic solutions [6–9].Recently, Faria [7] investigated the stability and Hopf bifurcation of the following system with instantaneous feedback control and two different discrete delays:
(1.2)dxdt=x(t)[r1-a11x(t)-a12y(t-τ1)]dydt=y(t)[-r2+a21x(t-τ2)-a22y(t)],
where τ1>0 and τ2>0. But, as pointed out by Kuang [8], in view of the fact that in real situations, instantaneous responses are rare, and thus, more realistic models should consist of delay differential equations without instantaneous feedbacks. Based on this idea, in the present paper, we combine the model (1.1) and (1.2) and consider the following delayed prey-predator system with a single delay:
(1.3)dXdt=X(t)[r1-r2X(t-τ)-pS(t-τ)]dSdt=S(t)[-c1+kpX(t-τ)-σI(t)]+γI(t)dIdt=I(t)[σS(t)-c2-γ],
where X,S,I denote, respectively, the population of prey species, susceptible predator species and infected predator species. In addition, the coefficients r1,r2,p,k,σ,c1,c2 in model (1.3) are all positive constants and their ecological meaning are interpreted as follows: r1 denotes the intrinsic growth rate of prey and r1/r2 denotes the carrying capacity of prey; p,k,c1 and c2 represent the predating coefficient of predator to prey, absorbing rate of predator to prey, and the death rate of susceptible and infected predator, respectively.The main purpose of this paper is to investigate the effects of the delay on the dynamics of model (1.3) with the following initial conditions:
(1.4)X(t)=ϕ1(t)>0,S(t)=ϕ2(t)>0,I(t)=ϕ3(t)>0,t∈[-τ,0](ϕ1(t),ϕ2(t),ϕ3(t))∈C([-τ,0],R0,+3),
where ℜ0,+3={(x,y,z)∣x≥0,y≥0,z≥0}. We will take the delay τ as the bifurcation parameter and show that when τ passes through a certain critical value, the positive equilibrium loses its stability and a Hopf bifurcation will take place. Furthermore, when τ takes a sequence of critical values containing the above critical value, the positive equilibrium of system (1.3) will undergo a Hopf bifurcation. In particular, by using the normal form theory and the center manifold, the formulae determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions are also obtained.The organization of this paper is as follows. In Section2, we discuss the stability of the positive solutions and the existence of the Hopf bifurcations. In Section 3, the direction of the Hopf bifurcation and the stability of bifurcated periodic solutions are obtained by using the normal form theory and the center manifold theorem. In Section 4, we do some numerical simulations to validate our theoretical results.


## 2. Stability of Positive Equilibrium and Hopf Bifurcation

System (1.3) has a unique positive equilibrium E+(X*,S*,I*) provided that the condition(H1)
σr1>(c2+γ)p,kp[σr1-p(c2+γ)]>σc1r2is satisfied, where
(2.1)X*=σr1-p(c2+γ)σr2,S*=c2+γσ,I*=(c2+γ)kp[σr1-p(c2+γ)]-σc1r2σ2r2(c2+2γ).
Linearizing system (1.2) at E+ gives the following linear system:
(2.2)dUdt=-r2X*U(t-τ)-pX*V(t-τ)dVdt=kpS*U(t-τ)-(c1+σI*-kpX*)V(t)-(σS*-γ)WdWdt=σI*V.The characteristic matrix of this system (2.2) is
(2.3)(λ+r2X*e-λτpX*e-λτ0-kpS*e-λτλ+(c1+σI*-kpX*)σS*-γ0-σI*λ).
Thus, the characteristic equation of system (2.2) is given by
(2.4)λ3+(c1+σI*-kpX*)λ2+(σ2S*I*-σγI*)λ+r2X*e-λτλ2+(c1+σI*-kpX*)λr2X*e-λτ+(σ2S*I*-σγI*)r2X*e-λτ+kp2S*X*λe-2λτ=0.
Let
(2.5)d1=c1+σI*-kpX*,d2=σ2S*I*-σγI*,d3=r2X*,d4=(c1+σI*-kpX*)r2X*,d5=(σ2S*I*-σγI*)r2X*,d6=kp2S*X*.
Then we rewrite (2.4) as:
(2.6)(λ3+d1λ2+d2λ)eλτ+d3λ2+d4λ+d5+d6λe-λτ=0.Obviously,iω(ω>0) is a root of (2.6) if and only if ω satisfies
(2.7)(-iω3-d1ω2+d2iω)(cos⁡ωτ+isin⁡ωτ)-d3ω2+d4iω+d5+d6iω(cos⁡ωτ-isin⁡ωτ)=0.
Separating the real and imaginary parts, we have
(2.8)-d1ω2cos⁡ωτ+(ω3-d2ω+d6ω)sin⁡ωτ=d3ω2-d5(-ω3+d2ω+d6ω)cos⁡ωτ-d1ω2sin⁡ωτ=-d4ω.
By calculating, we have obtained
(2.9)sin⁡ωτ=d3ω5+(d1d4-d2d3-d3d6-d5)ω3+(d2d5+d5d6)ωω6+(d12-2d2)ω4+(d22-d62)ω2cos⁡ωτ=(d4-d1d3)ω4+(d1d5+d4d6-d2d4)ω2ω6+(d12-2d2)ω4+(d22-d62)ω2.
Let e1=d12-2d2, e2=d22-d62, e3=d3, e4=d1d4-d2d3-d3d6-d5, e5=d2d5+d5d6, e6=d4-d1d3, e7=d1d5+d4d6-d2d4. Then sinωτ,cosωτ can be written as
(2.10)sin⁡ωτ=ω(e3ω4+e4ω2+e5)ω6+e1ω4+e2ω2(2.11)cos⁡ωτ=e6ω4+e7ω2ω6+e1ω4+e2ω2.
As sin2ωτ+cos2ωτ=1, so we have
(2.12)ω10+f1ω8+f2ω6+f3ω4+f4ω2+f5=0,
where f1=2e1-e32, f2=e12+2e2-2e3e4-e62, f3=2e1e2-2e3e5-2e6e7-e42, f4=e22-2e4e5-e72, f5=-e52.Denotez=ω2, then (2.12) becomes
(2.13)z5+f1z4+f2z3+f3z2+f4z+f5=0.
Let
(2.14)G(z)=z5+f1z4+f2z3+f3z2+f4z+f5.
Since limz→∞G(z)=+∞ and f5<0, then we can get the following conclusion.(H2)
Equation (2.13) has at least one positive real root.Without loss of generality, we assume that it has five positive roots, defined byz1,z2,z3,z4,z5, respectively. Then (2.13) has five positive roots
(2.15)ω1=z1,ω2=z2,ω3=z3,ω4=z4,ω5=z5.
By (2.11), we have
(2.16)cos⁡ωkτ=e6ωk2+e7ωk4+e1ωk2+e2.
Thus, if we denote
(2.17)τk(j)=1ωk{arccos(e6ωk2+e7ωk4+e1ωk2+e2)+2jπ},
where k=1,…,5;j=0,1,…, then ±iωk is a pair of purely imaginary roots of (2.6) with τk(j). Define
(2.18)τ0=τk0(0)=min⁡k∈{1,…,5}{τk(0)},ω0=ωk0.
Note that when τ=0, (2.6) becomes
(2.19)λ3+bλ2+(a+d)λ+c=0.
By Routh-Hurwitz criterion, we know that all the roots of (2.19) have negative real parts, that is, the positive equilibrium E+ is locally asymptotically stable for τ=0.In order to give the main results, it is necessary to make the following assumption:(H3)
[d(Reλ)/dτ]τ=τ0≠0.Differentiating two sides of (2.6) in respect to τ, we get
(2.20)(dλdτ)-1=(3λ2+2d1λ+d2λ)eλτ+τ(λ3+d1λ2+d2λ)eλτ+2d3λ+d4+d6e-λτ-d6λτe-λτ-λ(λ3+d1λ2+d2λ)eλτ+d6λ2e-λτ=(3λ2+2d1λ+d2λ)eλτ+2d3λ+d4+d6e-λτ-λ(λ3+d1λ2+d2λ)eλτ+d6λ2e-λτ-τλ=(3λ2+2d1λ+d2λ)eλτ+2d3λ+d4+d6e-λτd3λ3+d4λ2+d5λ+2d6λ2e-λτ-τλ=(-3ω2+d2+d6)cos⁡ωτ-2d1ωsin⁡ωτ+d4(-d4ω2-2d6ω2cos⁡ωτ)+i(-d3ω3+d5ω+2d6ω2sin⁡ωτ)+[(-3ω2+d2-d6)sin⁡ωτ+2d1ωcos⁡ωτ+2d3ω](-d4ω2-2d6ω2cos⁡ωτ)+i(-d3ω3+d5ω+2d6ω2sin⁡ωτ).
Let
(2.21)Q=(-d4ω2-2d6ω2cos⁡ωτ)2+(-d3ω3+d5ω+2d6ω2sin⁡ωτ)2>0(2.22)QRe(dλdτ)-1=[(-3ω2+d2+d6)cos⁡ωτ-2d1ωsin⁡ωτ+d4][-d4ω2-2d6ω2cos⁡⁡ωτ]+[(-3ω2+d2-d6)sin⁡ωτ+2d1ωcos⁡ωτ+2d3ω][-d3ω3+d5ω+2d6ω2sin⁡ωτ].
Noting that
(2.23)sgn⁡[d(Reλ)dτ]τ=τ0=sgn⁡[Re(dλdτ)-1]τ=τ0.
Now, we can employ a result from Ruan and Wei [10] to analyze (2.6), which is, for the convenience of the reader, stated as follows.Lemma 2.1 (see [2]).
Consider the exponential polynomial(2.24)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,
where τi≥0(i=1,2,…,m) and pj(i)(i=1,2,…,m;j=1,2,…,m) are constants. As (τ1,τ2,…,τm) vary, the sum of the order of the zeros of P(λ,e-λτ1,…,e-λτm) on the open right half plane can change only if a zero appears on or crosses the imaginary axis.Form Lemma2.1, it is easy to obtain the following theorem.Theorem 2.2.
Suppose the condition, (H1), (H2), and (H3) are satisfied, then one has the following results:(i)
ifτ∈[0,τ0), then the positive equilibrium E+ of (1.2) is locally asymptotically stable and unstable when τ>τ0,(ii)
system (1.2) undergoes a Hopf bifurcation at the positive equilibrium E+ when τ=τk(k=0,1,2,…), where τk is defined by (2.17).


## 3. Stability and Direction of Hopf Bifurcation

In this section, we will derive the explicit formulae determining the properties of the Hopf bifurcation at the critical value using the normal form theory and center manifold theorem introduced by Hassard et al. [11].Without loss of generality, letτ=τk+μ, where τk is defined by (2.17), μ∈R, then system (1.3) can be rewritten as
(3.1)u′(t)=Lμ(ut)+f(μ,ut),
where u(t)=(u1(t),u2(t),u3(t))T=(U(τt),V(τt),W(τt))T∈R3, ut(θ)=u(t+θ) and Lμ:C→R3, f:R×C→R3 are given, respectively, by
(3.2)Lμ(ut)=(τk+μ)(0000kpX*-c1-σI*-σS*+γ0σI*0)(u1t(0)u2t(0)u3t(0))+(τk+μ)(-r2X*-pX*0kpS*00000)(u1t(-1)u2t(-1)u3t(-1)),(3.3)f(μ,ut)=(τk+μ)(-r2u1t(0)u1t(-1)-pu1t(0)u2t(-1)kpu2t(0)u1t(-1)-σu2t(0)u3t(0)σu2t(0)u3t(0)).
By the Riesz representation theorem, there exists a function η(θ,μ) of bounded variation for θ∈[-1,0], such that
(3.4)Lμ(ϕ)=∫-10‍dη(θ,μ)ϕ(θ)forϕ∈([-1,0],R3).
In fact, we can choose
(3.5)η(θ,μ)=(τk+μ)(0000kpX*-c1-σI*-σS*+γ0σI*0)δ(θ)-(τk+μ)(-r2X*-pX*0kpS*00000)δ(θ+1),
where δ denote the Dirac delta function. For ϕ∈C([-1,0],R3), define
(3.6)A(μ)ϕ={dϕ(θ)dθ,θ∈[-1,0),∫-10‍dη(s,μ)ϕ(s),θ=0,R(μ)(ϕ)={0,θ∈[-1,0),f(μ,ϕ),θ=0.
Then system (3.1) is equivalent to
(3.7)u′(t)=A(μ)ut+R(μ)ut.
For ψ∈C([0,1],(R3)*), define
(3.8)A*={-dψ(s)ds,s∈(0,1],∫-10‍ψ(-t)dη(t,0),s=0
and a bilinear inner product
(3.9)〈ψ(s),ϕ(θ)〉=ψ¯(0)ϕ(0)-∫-10‍∫ξ=0θ‍ψ¯(ξ-θ)dη(θ)ϕ(ξ)dξ,
where η(θ)=η(θ,0).ThenA(0) and A* are adjoint operators. By the discussion in Section 2, we know that ±iω0τk are eigenvalues of A(0). Hence, they are also eigenvalues of A*. We first need to compute the eigenvectors of A(0) and A* corresponding to iω0τk and -iω0τk, respectively.Supposeq(θ)=(1,a1,a2)Teiω0τkθ is the eigenvector of A(0) corresponding to iω0τk, then A(0)q(θ)=iω0τkq(θ). It follows from the definition of A(0) and (3.2), (3.4) and (3.5), we have
(3.10)τ0(iω0+r2X*e-iω0τ0pX*e-iω0τ00-kpX*e-iω0τ0iω0-kpX*+c1+σI*σS*-γ0-σI*iω0)q(0)=(000).
For q(-1)=q(0)e-iω0τk, then we obtain
(3.11)a1=iω0+r2X*e-iω0τ0pX*e-iω0τ0,a2=σI*(iω0+r2X*e-iω0τ0)iω0(pX*e-iω0τ0).
On the other hand, suppose that q*(s)=J(1,a1*,a2*)eiω0τkS is the eigenvector of A* corresponding to -iω0τk, by the similar method, we have
(3.12)a1*=-iω0+r2X*eiω0τ0kpX*eiω0τ0,a2*=-σI*(-iω0+r2X*eiω0τ0)iω0(kpX*eiω0τ0).
In order to assure 〈q*(s),q(θ)〉=1, we need to determine the value of J. From (3.9), we have
(3.13)〈q*(s),q(θ)〉=J¯(1,a¯1*,a¯2*)(1,a1,a2)T-∫-10‍∫ξ=0θ‍J¯(1,a¯1*,a¯2*)e-iω0τk(ξ-θ)dη(θ)(1,a1,a2)Teiω0τkξdξ=J¯{1+a1a¯1*+a2a¯2*-∫-10‍(1,a¯1*,a¯2*)θeiω0τkθdη(θ)(1,a1,a2)T}=J¯{1+a1a¯1*+a2a¯2*+τk(-r2X*-a1pX*+kpS*a1*¯)e-iω0τk}.
Therefore, we can choose J as
(3.14)J¯=11+a1a¯1*+a2a¯2*+τk(-r2X*-a1pX*+kpS*a1*¯)e-iω0τk.
Next we will compute the coordinate to describe the center manifold C0 at μ=0. Let ut be the solution of (3.1) with μ=0. Define
(3.15)z(t)=〈q*,ut〉,W(t,θ)=ut(θ)-z(t)q(θ)-z¯(t)q*(θ)=ut(θ)-2Re{z(t)q(θ)}.
On the center manifold C0, we have
(3.16)W(t,θ)=W(z(t),z¯(t),θ),
where
(3.17)W(z(t),z¯(t),θ)=W20(θ)z22+W11(θ)zz¯+W02(θ)z¯22+⋯.z and z¯ are local coordinates of center manifold C0 in the direction of q* and q¯*. Note that W is real if ut is real. We consider only real solutions. For solution ut∈C0 of (3.7), since μ=0, we have
(3.18)z′(t)=〈q*,u′(t)〉=〈q*,Aut+Rut〉=〈q*,Aut〉+〈q*,Rut〉=〈A*q*,ut〉+〈q*,Rut〉=iω0τkz+q¯*(0)f(0),W(z,z¯,0)+2Re{zq(0)}≜iω0τkz+q¯*(0)f0(z,z¯).
We rewrite this equation as
(3.19)z′(t)=iω0τkz(t)+g(z,z¯),
where
(3.20)g(z,z¯)=q¯*(0)f0(z,z¯)=g20z22+g11zz¯+g02z¯22+g21z2z¯2+⋯.
Noticing ut(θ)=W(t,θ)+zq(θ)+z¯q¯(θ) and q(θ)=(1,a1,a2)Teiω0τkθ, we have
(3.21)u1t(0)=z+z¯+W20(1)(0)z22+W11(1)(0)zz¯+W02(1)(0)z¯22+⋯,u2t(0)=a1z+a¯1z¯+W20(2)(0)z22+W11(2)(0)zz¯+W02(2)(0)z¯22+⋯,u3t(0)=a2z+a¯2z¯+W20(3)(0)z22+W11(3)(0)zz¯+W02(3)(0)z¯22+⋯,u1t(-1)=e-iω0τkz+eiω0τkz¯+W20(1)(-1)z22+W11(1)(-1)zz¯+W02(1)(-1)z¯22+⋯,u2t(-1)=e-iω0τka1z+eiω0τka¯1z¯+W20(2)(-1)z22+W11(2)(-1)zz¯+W02(2)(-1)z¯22+⋯.
It follows together with (3.3), that
(3.22)g(z,z¯)=q¯*(0)f0(z,z¯)=q¯*(0)f0(0,ut).=τkJ¯(1,a¯1*,a¯2*)(-r2u1t(0)u1t(-1)-pu1t(0)u2t(-1)kpu2t(0)u1t(-1)-σu2t(0)u3t(0)σu2t(0)u3t(0))=-τkJ¯{r2[z+z¯+W20(1)(0)z22+W11(1)(0)zz¯+W02(1)(0)z¯22+⋯]-τkJ¯W*[e-iω0τkz+eiω0τkz¯+W20(1)(-1)z22+W11(1)(-1)zz¯+W02(1)(-1)z¯22+⋯]-τkJ¯W+p[z+z¯+W20(1)(0)z22+W11(1)(0)zz¯+W02(1)(0)z¯22+⋯]-τkJ¯1*[e-iω0τka1z+e-iω0τka¯1z¯+W20(2)(-1)z22+W11(2)(-1)zz¯+W02(2)(-1)z¯22+⋯]}+τkJ¯a¯1*{kp[a1z+a¯1z¯+W20(2)(0)z22+W11(2)(0)zz¯+W02(2)(0)z¯22+⋯]τkJ¯a¯1*8W1*[e-iω0τkz+eiω0τkz¯+W20(1)(-1)z22+W11(1)(-1)zz¯+W02(1)(-1)z¯22+⋯]τkJ¯a¯1*81W-σ[a1z+a¯1z¯+W20(2)(0)z22+W11(2)(0)zz¯+W02(2)(0)z¯22+⋯]τkJ¯a¯1*81W*[a2z+a¯2z¯+W20(3)(0)z22+W11(3)(0)zz¯+W02(3)(0)z¯22+⋯]}+τkJ¯a¯2*{σ[a1z+a¯1z¯+W20(2)(0)z22+W11(2)(0)zz¯+W02(2)(0)z¯22+⋯]τkJ¯a¯1*88W*[a2z+a¯2z¯+W20(3)(0)z22+W11(3)(0)zz¯+W02(3)(0)z¯22+⋯]}.
h (3.20), we have
(3.24)g20=2τkJ¯[-r2e-iω0τk-pa1e-iω0τk+a¯1*kpa1e-iω0τk-a¯1*σa1a2+a¯2*σa1a2],g11=τkJ¯{e-iω0τk(-r2-pa1+a¯1*kpa¯1)+eiω0τk(-r2-pa¯1+a¯1*kpa1)τkJ¯3-a¯1*σa1a¯2-a¯1*σa¯1a2+a¯2*σa1a¯2+a¯2*σa¯1a2e-iω0τk(-r2-pa1+a¯1*kpa¯1)+eiω0τk(-r2-pa¯1+a¯1*kpa1)},g02=2τkJ¯[-r2eiω0τk-pa¯1eiω0τk+a¯1*kpa¯1eiω0τk-a¯1*σa¯1a¯2+a¯2*σa¯1a¯2],g21=τkJ¯{-r2(2W11(1)(-1)+W20(1)(-1)+W20(1)(0)eiω0τk+2W11(1)(0)e-iω0τk)τkJ¯1-p(2W11(2)(-1)+W20(2)(-1)+a¯1W20(1)(0)eiω0τk+2a1W11(1)(0)e-iω0τk)τkJ¯1+a¯1*kp(2a1W11(1)(-1)+a¯1W20(1)(-1)+W20(2)(0)eiω0τk+2W11(2)(0)e-iω0τk)τkJ¯1-a¯1*σ(2a1W11(3)(0)+a¯1W20(3)(0)+a¯2W20(2)(0)+2a2W11(2)(0))τkJ¯1+a¯2*σ(2a1W11(3)(0)+a¯1W20(3)(0)+a¯2W20(2)(0)+2a2W11(2)(0))}.
Since there are W20(θ) and W11(θ) in g21, we need to determine them.From (3.7) and (3.15), we have
(3.25)W′=ut′-z′q-z¯′q¯={A(0)W-2Re{q¯*(0)f0q(θ)},θ∈[-1,0)A(0)W-2Re{q¯*(0)f0q(0)}+f0,θ=0≜A(0)W+G(z,z¯,θ),
where
(3.26)G(z,z¯,θ)=G20(θ)z22+G11(θ)zz¯+G02(θ)z¯22+⋯.
Substituting the corresponding series into (3.25) and comparing the coefficients, we have
(3.27)(A(0)-2iω0τkI)W20(θ)=-G20(θ),A(0)W11(θ)=-G11(θ).
From (3.20) and (3.25), we have, for θ∈[-1,0)(3.28)G(z,z¯,θ)=-q¯*(0)f0q(θ)-q*(0)f¯0q¯(θ)=-g(z,z¯)q(θ)-g¯(z,z¯)q¯(θ).
Comparing the coefficients with (3.26), we have
(3.29)G20(θ)=-g20q(θ)-g¯02q¯(θ),(3.30)G11(θ)=-g11q(θ)-g¯11q¯(θ).
From (3.27), (3.29) and the definition of A(0), we have
(3.31)W20′(θ)=2iω0τkW20(θ)+g20q(θ)+g¯02q¯(θ).
Notice that q(θ)=(1,a1,a2)Teiω0τkθ, hence
(3.32)W20(θ)=ig20ω0τkq(0)eiω0τkθ+ig¯023ω0τkq¯(0)e-iω0τkθ+E1e2iω0τkθ,
where E1=(E1(1),E1(2),E1(3))T∈R3 is a constant vector. Similarly, we obtain
(3.33)W11(θ)=-ig11ω0τkq(0)eiω0τkθ+ig¯11ω0τkq¯(0)e-iω0τkθ+E2,
where E2=(E2(1),E2(2),E2(3))T∈R3 is a constant vector. In the following, we will seek the values of E1 and E2. From the definition of A(0) and (3.27), we have
(3.34)∫-10‍dη(θ)W20(θ)=2iω0τkW20(0)-G20(0),(3.35)∫-10‍dη(θ)W11(θ)=-G11(0),
where η(θ)=η(0,θ).By (3.25), when θ=0, we have
(3.36)G(z,z¯,0)=2Re{q¯*(0)f0q(0)}+f0=-q¯*(0)f0q(0)-q*(0)f¯0q¯(0)+f0=-g(z,z¯)q(0)-g¯(z,z¯)q¯(0)+f0.
So we obtain
(3.37)G20(θ)z22+G11(θ)zz¯+G02(θ)z¯22+⋯=-q(0)(g20z22+g11zz¯+g02z¯22+⋯)-q¯(0)(g¯20z¯22+g¯11zz¯+g¯02z22+⋯)+f0.
By (3.37), we have
(3.38)G20(0)=-g20q(0)-g¯02q¯(0)+2τk(-r2e-iω0τk-pa1e-iω0τkkpa1e-iω0τk-σa1a2σa1a2),(3.39)G11(0)=-g11q(0)-g¯11q¯(0)+2τk(-r2-pRe{a1}kpRe{a1}-σRe{a1a¯2}σRe{a1a¯2}).
Noticing that
(3.40)(iω0τkI-∫-10‍eiω0τkθdη(θ))q(0)=0,(-iω0τkI-∫-10‍e-iω0τkθdη(θ))q¯(0)=0.
So, substituting (3.32) and (3.38) into (3.34), we have
(3.41)(2iω0τkI-∫-10‍e2iω0τkθdη(θ))E1=2τk(-r2e-iω0τk-pa1e-iω0τkkpa1e-iω0τk-σa1a2σa1a2).
That is
(3.42)(2iω0+r2X*e-iω0τkpX*e-iω0τk0-kpS*e-iω0τk2iω0-kpX*+c1+σI*σS*-γ0-σI*2iω0)E1=2(-r2e-iω0τk-pa1e-iω0τkkpa1e-iω0τk-σa1a2σa1a2).
Let
(3.43)L1=|2iω0+r2X*e-iω0τkpX*e-iω0τk0-kpS*e-iω0τk2iω0-kpX*+c1+σI*σS*-γ0-σI*2iω0|.
It follows that
(3.44)E1(1)=2L1|-r2e-iω0τk-pa1e-iω0τkpX*e-iω0τk0kpa1e-iω0τk-σa1a22iω0-kpX*+c1+σI*σS*-γσa1a2-σI*2iω0|,E1(2)=2L1|2iω0+r2X*e-iω0τk-r2e-iω0τk-pa1e-iω0τk0-kpS*e-iω0τkkpa1e-iω0τk-σa1a2σS*-γ0σa1a22iω0|,E1(3)=2L1|2iω0+r2X*e-iω0τkpX*e-iω0τk-r2e-iω0τk-pa1e-iω0τk-kpS*e-iω0τk2iω0-kpX*+c1+σI*kpa1e-iω0τk-σa1a20-σI*σa1a2|.
Similarly, we can get
(3.45)(r2X*e-iω0τkpX*e-iω0τk0-kpS*e-iω0τk-kpX*+c1+σI*σS*-γ0-σI*0)E2=(-r2-pRe{a1}kpRe{a1}-σRe{a1a¯2}σRe{a1a¯2}),
and hence
(3.46)E2(1)=1L2|-r2-pRe{a1}pX*e-iω0τk0kpRe{a1}-σRe{a1a¯2}-kpX*+c1+σI*σS*-γσRe{a1a¯2}-σI*0|,E2(2)=1L2|r2X*e-iω0τk-r2-pRe{a1}0-kpS*e-iω0τkkpRe{a1}-σRe{a1a¯2}σS*-γ0σRe{a1a¯2}0|,E2(3)=1L2|r2X*e-iω0τkpX*e-iω0τk-r2-pRe{a1}-kpS*e-iω0τk0kpRe{a1}-σRe{a1a¯2}0-σI*σRe{a1a¯2}|,
where
(3.47)L2=|r2X*e-iω0τkpX*e-iω0τk0-kpS*e-iω0τk-kpX*+c1+σI*σS*-γ0-σI*0|=r2X*(σ2S*I*-σI*γ)e-iω0τk.
Therefore, we can determine W20(0) and W11(0), hence we can obtain g21. Thus, we can compute these values
(3.48)c1(0)=i2ω0τk(g20g11-2|g11|2-|g02|23)+g212,μ2=-Re{c1(0)}Re{λ′(τk)},β2=2Re{c1(0)},T2=-Im{c1(0)}+μ2Im{λ′(τk)}ω0τk,
which determine the qualities of bifurcating periodic solution in the center manifold at the critical values τk, so we have the following results.Theorem 3.1.
(i)μ2 determines the directions of the Hopf bifurcation: if μ2>0(μ2<0), then the Hopf bifurcation is supercritical (subcritical) and the bifurcating periodic solutions exist for τ>τk(τ<τk).
(ii)β2 determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) if β2<0(β2>0).
(iii)T2 determines the period of the bifurcating periodic solutions: the period increases (decreases) if T2>0(T2<0).


## 4. Discussion and Numerical Example

In this section,we present some numerical results of system (1.3) at different values of τ. Form Section 3, we may determine the direction of a Hopf bifurcation and the stability of the bifurcating periodic solutions. We consider the following system:
(4.1)dXdt=X(t)[0.9-0.2X(t)-0.4S(t)]dSdt=S(t)[-0.2+0.3X(t)-0.6I(t)]+0.2I(t)dIdt=I(t)[0.6S(t)-0.3-0.2],
which has a positive equilibrium E+=(2.8333,0.8333,1.8056). Form (2.13) and (2.14), we are easy to get at least one positive real root 0.5977. In addition, it is easy to show that [d(Reλ)/dτ]τ=τk=2.4377, the hypothesis of H3 holds. Hence, E+ satisfies the condition of Theorem (2.2). When τ=0, the positive equilibrium E+=(2.8333,0.8333,1.8056) is asymptotically stable. According to (2.18), we obtain τ0=1.124,ω0=0.7731,λ′(τ0)=0.3504-0.1448i. Form the formulae (3.48) in Section 3, it follows that c1(0)=-15.6822+2.8655i,μ2=44.7551>0,β2=-31.3764<0 and T2=4.1602>0. Thus, E+ is stable when τ<τ0 as is illustrated by the computer simulations (see Figures 1(a) and 1(b)).Whenτ=1.11<τ0, the positive equilibrium E+=(2.8333,0.8333,1.8056) is asymptotically stable. (a) shows the trajectories graphs of the system (4.1) with initial data X(t)=2,S1(t)=2,I2(t)=2. (b) shows the phase portrait of system (4.1).
(a)
(b)Whenτ passes through the critical value τ0. E+ loses its stability and a Hopf bifurcation occurs, that is, a family of periodic solutions bifurcate from E+. Since μ2>0 and β2<0, the Hopf bifurcation is supercritical and the direction of the bifurcation is τ>τ0 and these bifurcating periodic solutions from E+ at τ0 are stable,which are depicted in Figures 2(a) and 2(b).Whenτ=1.13>τ0, bifurcation periodic solutions form E+. (a) shows the trajectory graphs of system (4.1) with initial data x(t)=2,y1(t)=2,y2(t)=2. (b) shows the phase portrait of system (4.1).
(a)
(b)



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*Source: 252437-2012-07-01.xml*