# Hopf Bifurcation Analysis for a Semiratio-Dependent Predator-Prey System with Two Delays

Authors: Ming Zhao
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

This paper is concerned with a semiratio-dependent predator-prey system with nonmonotonic functional response and two delays. It is shown that the positive equilibrium of the system is locally asymptotically stable when the time delay is small enough. Change of stability of the positive equilibrium will cause bifurcating periodic solutions as the time delay passes through a sequence of critical values. The properties of Hopf bifurcation such as direction and stability are determined by using the normal form method and center manifold theorem. Numerical simulations confirm our theoretical findings.


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## Body

## 1. Introduction

It is well known that there are many factors which can affect dynamical properties of predator-prey systems. One of the important factors is the functional response describing the number of prey consumed per predator per unit time for given quantities of prey and predators. Numerous laboratory experiment, and observations have shown that a more suitable general predator-prey system should be based on the “ratio-dependent” theory, especially when predators have to search, share, or compete for food [1–3]. And ratio-dependent predator-prey systems have been investigated by many scholars [4–11]. In [4], Zhang and Lu considered the following semi-ratio-dependent predator-prey system with the nonmonotonic functional response
(1)dx1(t)dt=x1(t)×[r1(t)-b1(t)x1(t)-a1(t)x2(t)m2+nx1(t)+x12(t)],dx2(t)dt=x2(t)[r2(t)-a2(t)x2(t)x1(t)],
where x1(t) and x2(t) denote the densities of the prey and the predator, respectively. r1(t) and r2(t) denote the intrinsic growth rates of the prey and the predator, respectively. b1(t) is the intraspecific competition rate of the prey. a1(t) is the capturing rate of the predator. a2(t) is a measure of the food quality that the prey provided for conversion into predator birth. The predator grows logistically with the carrying capacity x1(t)/a2(t) proportional to the population size of the prey.m
≠
0,n≥0, and ai(t),ri(t),b1(t),(i=1,2) are assumed to be continuously positive periodic functions with period ω. Zhang and Lu [4] established the existence of positive periodic solutions of system (1), and sufficient conditions for the uniqueness and global stability of the positive solutions of system (1) were also obtained by constructing a Lyapunov function.Time delays of one type or another have been incorporated into predator-prey systems by many researchers since a time delay could cause a stable equilibrium to become unstable and cause the population to fluctuate [5, 6, 12–15]. In [5], Ding et al. incorporated the time delay τ(t)≥0 due to negative feedback of the prey into system (1) and got the special case (n=0) of system (1) with time delay
(2)dx1(t)dt=x1(t)[r1(t)-b1(t)x1(t-τ(t))-a1(t)x2(t)m2+x12(t)],dx2(t)dt=x2(t)[r2(t)-a2(t)x2(t)x1(t)].
Ding et al. [5] established the existence of a positive periodic solution for system (2) by using the continuation theorem of coincidence degree theory. Sufficient conditions for the permanence of system (2) were obtained by Li and Yang in [6]. But studies on predator-prey systems not only involve the persistence and the periodic phenomenon but also involve many patterns of other behavior such as global attractivity [16, 17] and bifurcation phenomenon [18–21]. Starting from this point, we will study the bifurcation phenomenon and the properties of periodic solutions of the following predator-prey system with two delays:
(3)dx1(t)dt=x1(t)[r1-b1x1(t-τ1)-a1x2(t)m2+nx1(t)+x12(t)],dx2(t)dt=x2(t)[r2-a2x2(t-τ2)x1(t-τ2)].
Unlike the assumptions in [4–6], we assume that all the parameters of system (3) are positive constants. τ1≥0 and τ2≥0 are the feedback delays of the prey and the predator, respectively.The rest of this paper is organized as follows. In Section2, sufficient conditions are obtained for the local stability of the positive equilibrium and the existence of Hopf bifurcation for possible combinations of the two delays in system (3). In Section 3, we give the formula determining the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions. Finally, numerical simulations supporting the theoretical analysis are also included.


## 2. Stability of the Positive Equilibrium and Local Hopf Bifurcations

It is not difficult to verify that system (2) has at least one positive equilibrium E*(x1*,x2*), where x2*=r2x1*/a2 and x1* is the root of (4)
(4)b1x13+(nb1-r1)x12+(a1r2a2-nr1+m2b1)x1-m2r1=0.Letx-1(t)=x1(t)-x1*, x-2(t)=x2(t)-x2*. Dropping the bars for the sake of simplicity, system (2) can be rewritten in the following system:
(5)dx1(t)dt=a11x1(t)+a12x2(t)+b11x1(t-τ1)+f1,dx2(t)dt=c21x1(t-τ2)+c22x2(t-τ2)+f2,
where
(6)a11=a1x1*x2*(n+2x1*)(m2+nx1*+(x1*)2)2,a12=-a1x1*m2+nx1*+(x1*)2,b11=-b1x1*,c21=a2(x2*)2(x1*)2,c22=-a2x2*x1*,f1=a13x12(t)+a14x1(t)x2(t)+a15x1(t)x1(t-τ1)+a16x12(t)x2(t)+a17x13(t)+⋯f2=c23x12(t-τ2)+c24x1(t-τ2)x2(t)+c25x2(t-τ2)x2(t)+c26x1(t-τ2)x2(t-τ2)+c27x13(t-τ2)+c28x12(t-τ2)x2(t)+c29x12(t-τ2)x2(t-τ2)+c210x1(t-τ2)×x2(t-τ2)x2(t)+⋯,a13=a1x1*x2*(m2-n2-3nx1*-3(x1*)2)(m2+nx1*+(x1*)2)3,a14=a1((x1*)2-m2)(m2+nx1*+(x1*)2)2,a15=-b1,a16=a1x1*(m2-n2-3nx1*-3(x1*)2)(m2+nx1*+(x1*)2)3,a17=a1x2*(m2-n2-6nx1*-9(x1*)2)3(m2+nx1*+(x1*)2)3-a1x1*x2*(m2-n2-3nx1*-3(x1*)2)(n+2x1*)(m2+nx1*+(x1*)2)4,c23=-a2(x2*)2(x1*)3,c24=a2x2*(x1*)2,c25=-a2x1*,c26=a2x2*(x1*)2,c27=a2(x2*)2(x1*)4,c28=-a2x2*(x1*)3,c29=-a2x2*(x1*)3,c210=a2(x1*)2.
The linearized system of (5) is
(7)dx1(t)dt=a11x1(t)+a12x2(t)+b11x1(t-τ1),dx2(t)dt=c21x1(t-τ2)+c22x2(t-τ2).
The associated characteristic equation of system (7) is
(8)λ2+Aλ+Bλe-λτ1+(Cλ+D)e-λτ2+Ee-λ(τ1+τ2)=0,
where
(9)A=-a11,B=-b11,C=-c22,D=a11c22-a12c21,E=b11c22.Case 1 (τ1=τ2=0).
Equation (8) becomes
(10)λ2+(A+B+C)λ+D+E=0.
All the roots of (10) have negative real parts if and only if (H1)
A+B+C>0 and D+E>0.Thus, the positive equilibrium E*(x1*,x2*) is locally stable when the condition (H1) holds.Case 2 (τ1>0,τ2=0).
Equation (8) becomes
(11)λ2+(A+C)λ+D+(Bλ+E)e-λτ1=0.
Let λ=iω1(ω1>0) be the root of (11), and we have
(12)Bω1sinτ1ω1+Ecosτ1ω1=ω12-D,Bωcosτ1ω1-Esinτ1ω1=-(A+C)ω1.
It follows that
(13)ω14+((A+C)2-B2-2D)ω12+D2-E2=0.
If condition (H2)D-E<0 holds, then (13) has only one positive root
(14)ω10=((((A+C)2-B2-2D)2-4(D2-E2)-((A+C)2-B2-2D)+((A+C)2-B2-2D)2-4(D2-E2))×2-1(((A+C)2-B2-2D)2-4(D2-E2))1/2)1/2.
Now, we can get the critical value of τ1 by substituting ω10 into (12). After computing, we obtain
(15)τ1k=1ω10arccos(E-AB-BC)ω102-DEB2ω102+E2+2kπω10,k=0,1,2….
Then, when τ=τ1k, the characteristic equation (11) has a pair of purely imaginary roots ±iω0.Next, we will verify the transversality condition. Differentiating (11) with respect to τ1, we get
(16)[dλdτ1]-1=-2λ+A+Cλ[λ2+(A+C)λ+D]+Bλ(Bλ+E)-τ1λ.
Thus,
(17)Re[dλdτ1]τ1=τ10-1=2ω102+(A+C)2-B2-2DB2ω102+E2=((A+C)2-B2-2D)2-4(D2-E2)B2ω102+E2.
Obviously, if condition (H2) holds, then Re[dλ/dτ1]τ1=τ0-1>0. Note that sgn{Re[dλ/dτ1]τ1=τ10-1} = sgn{[dRe(λ)/dτ1]τ1=τ10-1}.Therefore, if condition(H2) holds, the transversality condition is satisfied. From the analysis above, we have the following results.Theorem 1.
If conditions(H1) and (H2) hold, then the positive equilibrium E*(x1*,x2*) of system (3) is asymptotically stable for τ1∈[0,τ10) and unstable when τ1>τ10. System (3) undergoes a Hopf bifurcation at the positive equilibrium E*(x1*,x2*) when τ1=τ10.Case 3 (τ1=0,τ2>0).
Equation (8) becomes
(18)λ2+(A+B)λ+(Cλ+D+E)e-λτ2=0.
Let λ=iω2(ω2>0) be the root of  (18), and then we have
(19)Cω2sinτ2ω2+(D+E)cosτ2ω2=ω22,Cω2cosτ2ω2-(D+E)sinτ2ω2=-(A+B)ω2,
which leads to
(20)ω24+((A+B)2-C2)ω22-(D+E)2=0.
Obviously, if the condition (H1) holds, and then (20) has only one positive root
(21)ω20=-((A+B)2-C2)+((A+B)2-C2)2+4(D+E)22.
The corresponding critical value of  τ2 is
(22)τ2k=1ω20arccos(D+E-AC-BC)ω102C2ω202+(D+E)2+2kπω20,k=0,1,2,….
Similar to Case 2, differentiating (18) with respect to τ2, we get
(23)[dλdτ2]-1=-2λ+A+Bλ[λ2+(A+B)λ]+Cλ(Cλ+D+E)-τ2λ.
Then we can get
(24)Re[dλdτ2]τ2=τ20-1=2ω202+(A+B)2-C2C2ω202+(D+E)2=((A+B)2-C2)2-4(D+E)2C2ω202+(D+E)2.Obviously, if condition(H1) holds, then Re[dλ/dτ2]τ2=τ20-1>0. Namely, if condition (H1) holds, then the transversality condition is satisfied. Thus, we have the following results.Theorem 2.
If condition(H1) holds, then the positive equilibrium E*(x1*,x2*) of system (3) is asymptotically stable for τ2∈[0,τ20) and unstable when τ2>τ20. System (3) undergoes a Hopf bifurcation at the positive equilibrium E*(x1*,x2*) when τ2=τ20.Case 4 (τ1=τ2=τ>0).
Equation (8) can be transformed into the following form:
(25)λ2+Aλ+((B+C)λ+D)e-λτ+Ee-2λτ=0.
Multiplying eλτ on both sides of (25), we get
(26)(B+C)λ+D+(λ2+Aλ)eλτ+Ee-λτ=0.
Let λ=iω(ω>0) be the root of (26), then we obtain
(27)Aωsinτω+(ω2-E)cosτω=D,Aωcosτω-(ω2+E)sinτω=-(B+C)ω,
which follows that
(28)sinτω=(B+C)ω3+(AD-BE-CE)ωω4+A2ω2-E2,cosτω=(D-AB-AC)ω2+DEω4+A2ω2-E2.
Then, we have
(29)ω8+c3ω6+c2ω4+c1ω2+c0=0,
where
(30)c0=E4,c3=2A2-(B+C)2,c1=2DE(AB+AC-D)-2A2E2-(AD-BE-CE)2,c2=A4-2E2-2(B+C)(AD-BE-CE)-(D-AB-AC)2.
Denote ω2=v, and then  (29) becomes
(31)v4+c3v3+c2v2+c1v+c0=0.
Next, we give the following assumption.(H3)
Equation (31) has at least one positive real root.Suppose that(H3) holds. Without loss of generality, we assume that (31) has four real positive roots, which are denoted by v1,v2,v3, and v4. Then (29) has four positive roots ωk=vk, k=1,2,3,4.For every fixedωk, the corresponding critical value of time delay is
(32)τk(j)=1ωkarccos(D-AB-AC)ωk2+DEωk4+A2ωk2-E2+2jπωk,k=1,2,3,4,j=0,1,2,….
Let
(33)τ0=min{τk(0)},k=1,2,3,4.ω0=ωk0.
Taking the derivative of λ with respect to τ in (26), we obtain
(34)[dλdτ]-1=(2λ+A)eλτ+B+Cλ[Ee-λτ-(λ2+Aλ)eλτ]-τλ.
Thus,
(35)Re[dλdτ]τ=τ0-1=PRQR+PIQIQR2+QI2,
with
(36)PR=Acosτ0ω0-2ω0sinτ0ω0+B+C,PI=Asinτ0ω0+2ω0cosτ0ω0,QR=Aω02cosτ0ω0+(Eω0-ω03)sinτ0ω0,QI=Aω02sinτ0ω0+(Eω0+ω03)cosτ0ω0.It is easy to see that if condition(H4)PRQR+PIQI≠0, then Re[dλ/dτ]τ=τ0-1≠0. Therefore, we have the following results.Theorem 3.
If conditions(H1) and (H4) holds, then the positive equilibrium E*(x1*,x2*) of system (3) is asymptotically stable for τ∈[0,τ0) and unstable when τ>τ0. System (3) undergoes a Hopf bifurcation at the positive equilibrium E*(x1*,x2*) when τ=τ0.Case 5 (τ1≠τ2,τ1>0 and τ2>0).
Considered the following:(37)λ2+Aλ+Bλe-λτ1+(Cλ+D)e-λτ2+Ee-λ(τ1+τ2)=0.
We consider (8) with τ2 in its stable interval and regard τ1 as a parameter.
Letλ=iω1*(ω1*>0) be the root of (8), and then we can obtain
(38)(Bω1*-Esinτ2ω1*)sinτ1ω1*+Ecosτ2ω1*cosτ1ω1*=(ω1*)2-Dcosτ2ω1*-Cω1*sinτ2ω1*,(Bω1*-Esinτ2ω1*)cosτ1ω1*-Ecosτ2ω1*cosτ1ω1*=-Aω1*+Dsinτ2ω1*-Cω1*cosτ2ω1*.
Then we get
(39)(ω1*)4+(A2-B2+C2)(ω1*)2+D2-E2+2(AC-D2)(ω1*)2cosτ2ω1*-2(C(ω1*)3+ADω1*)sinτ2ω1*=0.
Define
(40)F(ω1*)=(ω1*)4+(A2-B2+C2)(ω1*)2+D2-E2+2(AC-D2)(ω1*)2cosτ2ω1*-2(C(ω1*)3+ADω1*)sinτ2ω1*.
If condition(H1)D+E>0 and (H2)D-E<0 hold, then F(0)=D2-E2<0. In addition, F(+∞)=+∞. Therefore, (39) has at least one positive root. We suppose that the positive roots of (39) are denoted as ω11*,ω12*,…ω1k*. For every fixed ω1i*, the corresponding critical value of time delay
(41)τ1i*(j)=1ω1i*×arccose1(ω1i*)+e2(ω1i*)cosτ2ω1i*+e3(ω1i*)sinτ2ω1i*e4(ω1i*)+e5(ω1i*)sinτ2ω1i*+2jπω1i*i=1,2,3,…,k,j=0,1,2,…
with
(42)e1(ω1i*)=-AB(ω1i*)2-DE,e2(ω1i*)=(E-BC)(ω1i*)2,e3(ω1i*)=(AE+BD)ω1i*,e4(ω1i*)=B2(ω1i*)2+E2,e5(ω1i*)=-2BEω1i*.
Let τ1*=min{τ1i*(0)}, i=1,2,3,…k. When τ1=τ1*, (8) has a pair of purely imaginary roots ±iω1* for τ2∈(0,τ20).Next, in order to give the main results, we give the following assumption(H5)
dReλ(τ10*)/dτ1≠0.Therefore, we have the following results on the stability and bifurcation in system (3).Theorem 4.
If the conditions(H1) and (H5) hold and τ2∈(0,τ20), then the positive equilibrium E*(x1*,x2*) of system (3) is asymptotically stable for τ1∈[0,τ1*) and unstable when τ1>τ1*. System (3) undergoes a Hopf bifurcation at the positive equilibrium E*(x1*,x2*) when τ1=τ1*.


## 3. Direction and Stability of the Hopf Bifurcation

In this section, we will investigate the direction of Hopf bifurcation and stability of the bifurcating periodic solutions of system (3) with respect to τ1 for τ2∈(0,τ20). We assume that τ2*<τ1* where τ2*∈(0,τ20).Letτ1=τ1*+μ,μ∈R so that μ=0 is the Hopf bifurcation value of system (3). Rescale the time delay by t→(t/τ1). Let u1(t)=x1(t)-x1*, u2(t)=x2(t)-x2*, and then system (3) can be rewritten as an PDE in C=C([-1,0],R2) as
(43)u˙(t)=Lμut+F(μ,ut),
where
(44)u(t)=(u1(t),u2(t))T∈R2,Lμ:C⟶R2,F:R×C⟶R2
are given, respectively, by
(45)Lμϕ=(τ1*+μ)(A′ϕ(0)+C′ϕ(-τ2*τ1*)+B′ϕ(-1)),F(μ,ut)=(τ1*+μ)(F1,F2)T
with
(46)ϕ(θ)=(ϕ1(θ),ϕ2(θ))T∈C([-1,0],R2),A′=(a11a1200),B′=(b11000),C′=(00c21c22),F1=a13ϕ12(0)+a14ϕ1(0)ϕ2(0)+a15ϕ1(0)ϕ1(-1)+a16ϕ12(0)ϕ2(0)+a17ϕ13(0)+⋯F2=c23ϕ12(-τ2*τ1*)+c24ϕ1(-τ2*τ1*)ϕ2(0)+c25ϕ2(-τ2*τ1*)ϕ2(0)+c26ϕ1(-τ2*τ1*)ϕ2(-τ2*τ1*)+c27ϕ13(-τ2*τ1*)+c28ϕ12(-τ2*τ1*)ϕ2(0)+c29ϕ12(-τ2*τ1*)ϕ2(-τ2*τ1*)+c210ϕ1(-τ2*τ1*)ϕ2(-τ2*τ1*)ϕ2(0)+⋯.
By the Riesz representation theorem, there is a 2×2 matrix function with bounded variation components η(θ,μ), θ∈[-1,0] such that
(47)Lμϕ=∫-10dη(θ,μ)ϕ(θ),ϕ∈C.
In fact, we have chosen
(48)η(θ,μ)={(τ1*+μ)(A′+B′+C′),θ=0,(τ1*+μ)(B′+C′),θ∈[-τ2*τ1*,0),(τ1*+μ)B′,θ∈(-1,-τ2*τ1*),0,θ=-1.
For ϕ∈C([-1,0]),R2, we define
(49)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 (43) is equivalent to the following operator equation:
(50)u˙(t)=A(μ)ut+R(μ)ut,
where ut=u(t+θ) for θ∈[-1,0].Forφ∈C([-1,0]),(R2)*, we define the adjoint operator A* of A as
(51)A*(μ)s={-dφ(s)ds,0<s≤1,∫-10dηT(s,μ)φ(-s),s=0,
and a bilinear inner product
(52)〈φ,ϕ〉=φ-(0)ϕ(0)-∫θ=-10∫ξ=0θφ-(ξ-θ)dη(θ)ϕ(ξ)dξ,
where η(θ)=η(θ,0).Then,A(0) and A*(0) are adjoint operators. From the discussion above, we can know that ±iω1*τ1* are the eigenvalues of A(0) and they are also eigenvalues of  A*(0).Letq(θ)=(1,q2)Teiω1*τ1*θ be the eigenvector of A(0) corresponding to iω1*τ1*, and q*(θ) = D(1,q2*)eiω1*τ1*s be the eigenvector of A*(0) corresponding to -iω1*τ1*. Then we have
(53)A(0)q(θ)=iω1*τ1*q(θ),A*(0)q*(θ)=-iω1*τ1*q*(θ).
By a simple computation, we can get
(54)q2=iω1*-a11-b11e-iω1*τ1*a12,q2*=iω1*-a11-b11eiω1*τ1*c21eiω1*τ2*D-=[1+q2q-2*+b11τ1*e-iω1*τ1*+q-2*(c21+c22q2)τ2*e-iω1*τ2*]-1.
Then,
(55)〈q*,q〉=1,〈q*,q-〉=0.Following the algorithms explained in the work of Hassard et al. [22] and using a computation process similar to that in [18], we can get the following coefficients which can determine the properties of Hopf bifurcation:
(56)g20=2τ1*D-[(q(1)(-τ2*τ1*))2a13+a14q(2)(0)+a15q(1)(-1)+q-2*(c23(q(1)(-τ2*τ1*))2+c24q(1)(-τ2*τ1*)×q(2)(0)+c25q(2)(-τ2*τ1*)q(2)(0)+c26q(1)(-τ2*τ1*)q(2)(-τ2*τ1*)(q(1)(-τ2*τ1*))2)],g11=τ1*D-[+q-(1)(-τ2*τ1*)q(2)(-τ2*τ1*)))2a13+a14(q(2)(0)+q-(2)(0))+a15(q(1)(-1)+q-(1)(-1))+q-2*(2c23q(1)(-τ2*τ1*)q-(1)(-τ2*τ1*)+c24((-τ2*τ1*)q(1)(-τ2*τ1*)q-(2)(0)+q-(1)(-τ2*τ1*)q(2)(0))+c25(q(2)(-τ2*τ1*)q-(2)(0)+q-(2)(-τ2*τ1*)q(2)(0))+c26(q(1)(-τ2*τ1*)q-(2)(-τ2*τ1*)+q-(1)(-τ2*τ1*)q(2)(-τ2*τ1*)))],g02=2τ1*D-[c23(q-(1)(-τ2*τ1*))2a13+a14q-(2)(0)+a15q-(1)(-1)+q-2*(c23(q-(1)(-τ2*τ1*))2+c24q-(1)(-τ2*τ1*)q-(2)(0)+c25q-(2)(-τ2*τ1*)q-(2)(0)+c26q-(1)(-τ2*τ1*)q-(2)(-τ2*τ1*)c23(q-(1)(-τ2*τ1*))2)],g21=2τ1*D-[×q(2)(-τ2*τ1*)q(2)(0)))a13(W11(1)(0)+W20(1)(0))+a14(W11(1)(0)q(2)(0)+12W20(1)(0)q-(2)(0)+W11(2)(0)+12W20(2)(0))+a15(W11(1)(0)q(1)(-1)+12W20(1)(0)q-(1)(-1)+W11(1)(-1)+12W20(1)(-1))+a16(q-(2)(0)+2q(2)(0))+3a17q-(1)(0)+q-2*(c23(2W11(1)(-τ2*τ1*)q(1)(-τ2*τ1*)+W20(1)(-τ2*τ1*)q-(1)(-τ2*τ1*))+c24(W11(1)(-τ2*τ1*)q(2)(0)+12W20(1)(-τ2*τ1*)q-(2)(0)+W11(2)(0)q(1)(-τ2*τ1*)+12W20(2)(0)q-(1)(-τ2*τ1*))+c25(W11(2)(-τ2*τ1*)q(2)(0)+12W20(2)(-τ2*τ1*)q-(2)(0)+W11(2)(0)q(2)(-τ2*τ1*)+12W20(2)(0)q-(2)(-τ2*τ1*))+c26(W11(1)(-τ2*τ1*)q(2)(-τ2*τ1*)+12W20(1)(-τ2*τ1*)q-(2)(-τ2*τ1*)+W11(2)(-τ2*τ1*)q(1)(-τ2*τ1*)+12W20(2)(-τ2*τ1*)q-(1)(-τ2*τ1*))+3c27(q(1)(-τ2*τ1*))2q-(1)(-τ2*τ1*)+c28((q(1)(-τ2*τ1*))2q-(2)(0)+2q(1)(-τ2*τ1*)q(2)(0)q-(1)(-τ2*τ1*))+c29((q(1)(-τ2*τ1*))2q-(2)(-τ2*τ1*)+2q(1)×(-τ2*τ1*)q(2)(-τ2*τ1*)q-(1)(-τ2*τ1*))+c210(q(1)(-τ2*τ1*)q(2)(-τ2*τ1*)q-(2)(0)+q(1)(-τ2*τ1*)q-(2)(-τ2*τ1*)×q(2)(0)+q-(1)(-τ2*τ1*)×q(2)(-τ2*τ1*)q(2)(0)))],
with
(57)W20(θ)=ig20q(0)ω1*τ1*eiω1*τ1*θ+ig¯02q¯(0)3ω1*τ1*e-iω1*τ1*θ+E1e2iω1*τ1*θ,W11(θ)=-ig11q(0)ω1*τ1*eiω1*τ1*θ+ig¯11q¯(0)ω1*τ1*e-iω1*τ1*θ+E2,
where E1 and E2 can be determined by the following equations, respectively:
(58)(2iω1*-a11-b11e-2iω1*τ1*-a12-c21e-2iω1*τ2*2iω1*-c33e-2iω1*τ2*)E1=2(E1(1)E1(2)),(a11+b11a12c21c22)E2=-(E2(1)E2(2)),
with
(59)E1(1)=a13+a14q(2)(0)+a15q(1)(-1),E1(1)=c23(q(1)(-τ2*τ1*))2+c24q(1)(-τ2*τ1*)q(2)(0)+c25q(2)(-τ2*τ1*)q(2)(0)+c26q(1)(-τ2*τ1*)q(2)(-τ2*τ1*),E2(1)=2a13+a14(q(2)(0)+q-(2)(0))+a15(q(1)(-1)+q-(1)(-1)),E2(2)=2c23q(1)(-τ2*τ1*)q-(1)(-τ2*τ1*)+c24(q(1)(-τ2*τ1*)q-(2)(0)+q-(1)(-τ2*τ1*)q(2)(0))+c25(q(2)(-τ2*τ1*)q-(2)(0)+q-(2)(-τ2*τ1*)q(2)(0))+c26(q(1)(-τ2*τ1*)q-(2)(-τ2*τ1*)+q-(1)(-τ2*τ1*)q(2)(-τ2*τ1*)).
Then, we can get the following coefficients:
(60)C1(0)=i2ω1*τ1*(g11g20-2|g11|2-|g02|23)+g212,μ2=-Re{C1(0)}Re{λ′(τ1*)},β2=2Re{C1(0)},T2=-Im{C1(0)}+μ2Im{λ′(τ1*)}ω1*τ1*.In conclusion, we have the following results.Theorem 5.
Ifμ2>0 (μ2<0), then the Hopf bifurcation is supercritical (subcritical); if β2<0(β2>0), then the bifurcating periodic solutions are stable (unstable); if T2>0 (T2<0), then the period of the bifurcating periodic solutions increases (decreases).


## 4. Numerical Simulation and Discussions

To demonstrate the algorithm for determining the existence of Hopf bifurcation in Section2 and the properties of Hopf bifurcation in Section 3, we give an example of system (3) in the following form:
(61)dx1(t)dt=x1(t)[4-x1(t-τ1)-0.5x2(t)4+x1(t)+x12(t)],dx2(t)dt=x2(t)[3-1.5x2(t-τ2)x1(t-τ2)],
which has a positive equilibrium E*(3.8298,7.6596). By calculation, we have A+B+C=6.5788, D+E=11.2470, and D-E=-11.7318. Namely, conditions (H1) and (H2) hold.Forτ1>0, τ2=0. We can get ω10=3.9074, τ10=0.3788. By Theorem 1, we know that when τ1∈[0,τ10), the positive equilibrium E*(3.8298,7.6596) is locally asymptotically stable and is unstable if τ1>τ10. As can be seen from Figure 1, when we let τ1=0.35∈[0,τ10), the positive equilibrium E*(3.8298,7.6596) is locally asymptotically stable. However, when τ1=0.395>τ10, the positive equilibrium E*(3.8298,7.6596) is unstable and a Hopf bifurcation occurs and a family of periodic solutions bifurcate from the positive equilibrium E*(3.8298,7.6596), which can be illustrated by Figure 2. Similarly, we can get ω20=3.0827,τ20=0.4912. The corresponding waveform and the phase plots are shown in Figures 3 and 4.E
* is asymptotically stable for τ1=0.350<τ10=0.3788.
(a)
(b)
(c)E
* is unstable for τ1=0.395>τ10=0.3788.
(a)
(b)
(c)E
* is asymptotically stable for τ2=0.450<τ20=0.4912.
(a)
(b)
(c)E
* is unstable for τ2=0.505>τ20=0.4912.
(a)
(b)
(c)Forτ1=τ2=τ>0. We can obtain ω0=3.4601,τ0=0.3361. By Theorem 3, we know that when the time delay τ increases from zero to the critical value τ0=0.3361, the positive equilibrium E*(3.8298,7.6596) is locally asymptotically stable. It will lose its stability and a Hopf bifurcation occurs once τ>τ0=0.3361. This property can be illustrated by Figures 5 and 6.E
* is asymptotically stable for τ=0.320<τ0=0.3361.
(a)
(b)
(c)E
* is unstable for τ=0.355>τ0=0.3361.
(a)
(b)
(c)Finally, considerτ1 as a parameter and let τ2=0.3∈(0,τ20). We can get ω1*=2.3761,τ1*=0.3490,λ′(τ1*)=8.2230+2.6054i. From Theorem 4, the positive equilibrium E*(3.8298,7.6596) is locally asymptotically stable for τ1∈[0,0.3490) and unstable when τ1>τ1*=0.3490, which can be seen from Figures 7 and 8.E
* is asymptotically stable for τ1=0.325<τ1*=0.3490 at τ2=0.3.
(a)
(b)
(c)E
* is unstable for τ1=0.360>τ1*=0.3490 at τ2=0.3.
(a)
(b)
(c)In addition, from (60), we can get C1(0)=-41.5020-17.8366i. Furthermore, we have μ2=5.0471>0, β2=-83.0040<0,T2=5.6519>0. Therefore, from Theorem 5, we know that the Hopf bifurcation is supercritical. The bifurcating periodic solutions are stable and increase. If the bifurcating periodic solutions are stable, then the prey and the predator may coexist in an oscillatory mode. Therefore, the two species in system (3) may coexist in an oscillatory mode under certain conditions.



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*Source: 495072-2013-09-30.xml*