# Dynamic Analysis of a Delayed Reaction-Diffusion Predator-Prey System with Modified Holling-Tanner Functional Response

Authors: Xinhong Pan; Min Zhao; Chuanjun Dai; Yapei Wang
Journal: Abstract and Applied Analysis
 (2015)

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## Abstract

A predator-prey model with modified Holling-Tanner functional response and time delays is considered. By regarding the delays as bifurcation parameters, the local and global asymptotic stability of the positive equilibrium are investigated. The system has been found to undergo a Hopf bifurcation at the positive equilibrium when the delays cross through a sequence of critical values. In addition, the direction of the Hopf bifurcation and the stability of bifurcated periodic solutions are also studied, and an explicit algorithm is obtained by applying normal form theory and the center manifold theorem. The main results are illustrated by numerical simulations.


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## Body

## 1. Introduction

The dynamic relationship between prey and predators has long been and will continue to be one of the dominant subjects in mathematical ecology due to its universal existence and importance [1–12]. In [13, 14], the author proposed the following predator-prey model based on the model in May [15]:(1)dudt=ru1-uK-puv,dvdt=vs1-hvu,where u and v denote the population of prey and predator, respectively, and r and s are the intrinsic growth rates of prey and predator, respectively. The parameter K represents the carrying capacity of the prey and the ratio u/h represents the carrying capacity of the predator. It has been assumed that both prey and predator populations grow logistically and that the predator consumes the prey according to a functional p(u).In recent years, models with time delay have been extensively studied by many authors [5, 16–26]. The authors of [1] discussed model (1) with a discrete delay:(2)dudt=ru1-ut-τK-muv,dvdt=vs1-hvuand obtained the stability of equilibria, the existence of Hopf bifurcation, and the direction of bifurcating periodic solutions. This paper focuses mainly on the effects of both spatial diffusion and time delay on system (1). It is assumed that the delay affects predation and consumption and that the system has homogeneous Neumann boundary conditions:(3)∂u∂t=d1Δu+ru(1-uK)-mu2vt-τ1a+u2,x∈Ω,t>0,∂v∂t=d2Δv+v[s(1-hvut-τ2)],x∈Ω,t>0,∂u∂n=∂v∂n=0,x=∂Ω,t>0,ux,t=u0x,t,x∈Ω,t∈-τ2,0,vx,t=v0x,t,x∈Ω,t∈-τ1,0,where d1 and d2 are the diffusion coefficients of prey and predator, respectively, Δ=∂2/∂x2 denotes the Laplacian operator, and n is the outward unit normal vector on ∂Ω. For convenience, it is assumed that Ω=(0,lπ), l>0 and that all parameters are positive.The rest of this paper is structured as follows. In Section2, the local stability of equilibria is analyzed using the associated characteristic equations, and the occurrence of the Hopf bifurcation with time delays is presented. In Section 3, the global asymptotical stability of the interior equilibrium for any τ1,τ2≥0 is proved by means of the upper-lower solution method. In Section 4, using normal form theory and the center manifold theorem, the stability and direction of bifurcating periodic orbits are investigated. Finally, numerical simulations and a brief discussion are presented.


## 2. Local Stability and Hopf Bifurcation Analysis

In this section, the local stability of the equilibria of system (3) is analyzed. Denote(4)X=C0,lπ,R2,u,v=u1,v1+u2,v2,τ=τ1+τ2,for u=(u1,u2), v=(v1,v2)∈X; then (X,〈·,·〉) is a Hilbert space. In the abstract space C([-τ,0],X), system (3) can be regarded as an abstract functional differential equation.System (3) has two nonnegative equilibria (K,0) and (u*,v*), where(5)v*=1hu*,and u* is the positive root of the equation(6)rhu3+(mK-rhK)u2+arhu-arhK=0.Let A(u)=rhu3+(mK-rhK)u2+arhu-arhK; then A(0)=-arhK<0, A(K)=mK3>0, which guarantee the existence of u*∈(0,K).From analysis of the characteristic equation of(K,0), it can easily be determined that it always has a saddle point. To analyze the stability of the positive equilibrium (u*,v*), the first step is to linearize system (3) at (u*,v*):(7)∂U(t)∂t=dΔUt+LUt,where dΔ=(d1Δ,d2Δ),(8)domdΔ=∂u∂x(u,v)T:u,v∈C20,lπ,R,hhh∂u∂x=∂v∂x=0,x=0,lπ,and L:C([-τ,0],X)→X is defined as(9)Lϕ=r-2ru*K-2amu*v*a+u*22ϕ10-mu*a+u*2ϕ2-τ1shϕ1-τ2-sϕ20for ϕ=(ϕ1,ϕ2)T∈C([-τ,0],X).The characteristic equation of (7) is(10)λy-dΔy-L(eλy)=0,y∈dom(dΔ),y≠0.Recall that-Δ under the Neumann boundary condition has eigenvalues 0=μ0<μ1≤μ2≤⋯≤μn≤μn+1≤⋯ and limi→∞⁡μi=∞, with the corresponding eigenfunctions ψn(x). Substituting(11)y=∑n=0∞ψn(x)y1ny2ninto (10), the following expression results:(12)r-2ru*K-2amu*v*a+u*22-μnd1-mu*a+u*2e-λτ1she-λτ2-s-μnd2y1ny2n=λy1ny2n.Hence, it can be concluded that the characteristic equation (10) is equivalent to the equation(13)λ2+Anλ+Bn+Ce-λτ=0,n=0,1,2,…,where(14)An=μnd1+d2-r+2ru*K+2amu*v*a+u*22+s,Bn=r-2ru*K-2amu*v*a+u*22-μnd1-s-μnd2,C=shmu*a+u*2,τ=τ1+τ2.The stability of the positive equilibrium(u*,v*) can be determined by the distribution of the roots of (13). It is locally asymptotically stable if all the roots of (13) have negative real parts for all n=0,1,2,…. Obviously, 0 is not a root of (13) for all n=0,1,2,…. When τ=0 as well as τ1=τ2=0, (13) can be simplified as(15)λ2+Anλ+Bn=0.It can be verified that(16)An>0,Bn>0,ifr-2ru*K<0;that is,(17)u*>K2,which requires that(18)A(K2)=m4K3-rh8K3-12arhK<0or(19)2m-rh<0.Therefore, the following theorem can be stated.Theorem 1.
If2m-rh<0 holds, the interior equilibrium (u*,v*) of system (3) with τ1=τ2=0 is asymptotically stable.
Whenτ≠0, assume that λ=iw0(w0>0) is a root of (13). Substituting it into (13) yields:(20)w02-Bn=Ccos⁡w0τ,w0An=Csinw0τ;that is,(21)w04+(An2-2Bn)w02+Bn2-C2=0,where(22)An2-2Bn=-r+2ru*K+2amu*v*a+u*22+μnd12+s+μnd22,Bn2-C2=-r+2ru*K+2amu*v*a+u*22+μnd12×s+μnd22-shmu*a+u*22.This leads to the following theorem.Theorem 2.
If2m-rh<0 and Bn2-C2>0 hold for n=0,1,2,…, then the interior equilibrium (u*,v*) of system (3) is asymptotically stable for all τ1,τ2≥0.Proof.
IfBn2-C2<0, then there exists N0≥0 such that, for 0≤n≤N0, (20) has a unique positive real root:(23)w0=2Bn-An2+An2-2Bn2-4Bn2-C22.From (20), it follows that(24)sinw0τ=w0AnC,cos⁡w0τ=w02-BnC.Then (13) has a pair of pure imaginary roots ±iw0 when(25)τ=τk=τ0+2kπw0,k=0,1,2…,where(26)τ0=1w0arccosw02-BnC.Substituting λ(τ) into (13) and taking the derivatives with respect to τ lead to(27)2λdλ(τ)dτ+Andλ(τ)dτ+Ce-λτ(-λ-τdλτdτ)=0.For τ=τ0, considering Ce-iw0τ=w02-Bn-iw0An, it follows that(28)signdReλτdτλ=iw0=signRedλτdτλ=iw0=signReiCw0e-iw0τ02w0i+An-Cτ0e-iw0τ0=signReiw03-Bnw0i+w02An2w0i+An-τ0w02-Bn-iw0An=signRew02An+iw03-Bnw0-τ0w02-Bn+An+iτ0w0An+2w0=sign+τ0w0An+2w02-1w03-Bnw0τ0w0An+2w0w02An-τ0w02-Bn+Anhhhhhhihh+w03-Bnw0τ0w0An+2w0hhhhhhhh×τ0w0An+2w02-τ0w02-Bn+An2hhhhhhhhhh+τ0w0An+2w02-τ0w02-Bn+An2-1=signw02An2+2w02-2Bn-τ0w02-Bn+An2+τ0w0An+2w02=signw02·2Bn-An2+An2-2Bn2-4Bn2-C22hhhhhhhhh×τ0w0An+2w02-τ0w02-Bn+An2hhhhhhhhhhi+τ0w0An+2w02-τ0w02-Bn+An2-1w02·2Bn-An2+An2-2Bn2-4Bn2-C22=1.In other words, dλ/dττ=τ0>0.From the above discussion, the following theorem can be stated.Theorem 3.
IfBn2-C2<0 holds, then the following statements are true.(1)
If0≤τ1+τ2<τ0, then the interior equilibrium (u*,v*) of system (3) is asymptotically stable.(2)
Ifτ1+τ2>τ0, then the interior equilibrium (u*,v*) of system (3) is unstable.(3)
System (3) undergoes a Hopf bifurcation at the interior equilibrium (u*,v*) for τ1+τ2=τk,k=0,1,2,….


## 3. Global Stability

This section mainly proves that the interior equilibrium(u*,v*) is globally asymptotically stable with the upper-lower solution method in [27, 28]. For simplicity, let (u1(t,x),v1(t,x))>(u2(t,x),v2(t,x)) denote u1(t,x)>u2(t,x) and v1(t,x)>v2(t,x).Lemma 4 (see [29]).
Assume thatu(t,x) is defined by(29)ut-d1Δu=ru(1-uK),x∈Ω,t>0,∂u∂n=0,x∈∂Ω,ux,0=u0x>0,x∈Ω.Then limt→∞⁡u(x,t)=K.Theorem 5.
Ifr-m/2ahK>0, then for system (3), the positive equilibrium (u*,v*) is globally asymptotically stable.Proof.
From the maximum principle of parabolic equations, it is known that for any initial value(u0(t,x),v0(t,x))>(0,0), the corresponding nonnegative solution (u(t,x),v(t,x)) is strictly positive for t>0. Because r-m/2ahK>0, it is possible to choose ε0 satisfying(30)0<ε0<r-m2ahKm2ah+m2a+rK(1+h)-1.Because(31)∂u∂t-d1Δu=ru(1-uK)-mu2v(t-τ1)a+u2≤ru(1-uK)according to Lemma 4 and the comparison principle of parabolic equations, there exists t1>0 such that, for any t>t1, u(x,t)≤K+ε0≜u-. This in turn implies that(32)∂v∂t-d1Δv=sv1-hvut-τ2≤sv1-hvK+ε0for t>t1+τ2.
Hence there existst2>t1 such that, for any t>t2,(33)v(x,t)≤K+ε0h+ε0≜v-.Consequently,(34)∂u∂t-d1Δu=ru1-uK-mu2vt-τ1a+u2≥ru1-uK-mu2vt-τ12au≥ru1-uK-mK+h+1ε02haufor t>t2+τ1.
Becauser-m/2ahK>0, it can be easily verified that(35)K1-mK+(h+1)ε02rha>0,K1-mK+(h+1)ε02rha-ε0>0.Hence, there exists t3>t2 such that, for any t>t3,(36)u(x,t)≥K1-mK+(h+1)ε02rha-ε0≜u_.This implies that(37)∂v∂t-d1Δv=sv1-hvut-τ2≥sv-shv2u_for t>t3+τ2. Again it can be verified that(38)u_h-ε0=Kh1-mK+h+1ε02rha-ε0h-ε0>0,and hence there exists t4>t3 such that, for any t>t4,(39)vx,t≥Kh1-mK+h+1ε02rha-ε0h-ε0≜v_.Therefore, for t>t4, it is possible to obtain(40)u_≤ux,t≤u¯,v_≤vx,t≤v¯,and u_,u¯,v_,v¯ satisfy(41)0≥r1-u¯K-mu¯v_a+u¯2,0≥s1-hv¯u¯,0≤r1-u_K-mu_v¯a+u_2,0≤s1-hv_u_.This implies that (u¯,v¯) and (u_,v_) are a pair of coupled upper and lower solutions of system (3), as in the definition in [29], for the reason that (3) is a mixed quasimonotonic system.
It is clear that there existsK>0 such that, for any (u_,v_)≤(u1,v1),(u2,v2)≤(u¯,v¯),(42)ru11-u1K-mu12v1a+u12-ru21-u2K+mu22v2a+u22≤Ku1-u2+v1-v2,(43)sv11-hv1u1-sv21-hv2u2≤Ku1-u2+v1-v2.Define two iteration sequences (u¯(m),v¯(m)) and (u_(m),v_(m)) as follows: for m≥1,(44)u¯(m)=u¯(m-1)+1Ku¯(m-1)×r1-u¯m-1K-mu¯m-1v_m-1a+u¯m-12,v¯(m)=v¯(m-1)+1Ksv¯(m-1)1-hv¯(m)u¯(m),u_(m)=u_(m-1)+1Ku_(m-1)×r1-u_m-1K-mu_(m-1)v¯(m-1)a+u_(m-1)2,v_(m)=v_(m-1)+1Ksv_(m-1)1-hv_(m-1)u_(m-1),where (u¯(0),v¯(0))=(u¯,v¯), (u_(0),v_(0))=(u_,v_).
Then, form≥1,(45)u_,v_≤u_m,v_m≤u_m+1,v_m+1≤u¯m+1,v¯m+1≤u¯m,v¯m≤u¯,v¯,and there exist (u~,v~)>(0,0) and (u^,v^)>(0,0) such that limm→∞⁡u¯(m)=u~, limm→∞⁡v¯(m)=v~, limm→∞⁡u_(m)=u^, limm→∞⁡v_(m)=v^, and(46)0=u~r1-u~K-mu~v^a+u~2,0=v~s1-hv~u~,0=u^r1-u^K-mu^v~a+u^2,0=v^s1-hv^u^.Because (u*,v*) is the unique positive constant equilibrium of system (3), it must hold that (u~,v~)=(u^,v^)=(u*,v*).
According to the results of [27, 28], the solution (u(x,t),v(x,t)) of system (3) satisfies(47)limt→∞ux,t=u*,limt→∞vx,t=v*,hhhhhhhhhhhhuniformlyforx∈Ω-.Hence, the constant equilibrium (u*,v*) is globally asymptotically stable.


## 4. Direction and Stability of Hopf Bifurcation

Part Two has already shown that system (3) undergoes Hopf bifurcation at the interior equilibrium (u*,v*) for τ1+τ2=τk,k=0,1,2,…. In this section, the direction, stability, and period of the periodic solutions from the steady state will be studied by applying the method introduced by Hassard et al. [30] and the center manifold theorem due to [31–35].For convenience, letu′=u(t-τ2), v′=v(t), τ=τ1+τ2. Then system (3) is equal to a single-delay system:(48)∂u′∂t=d1Δu′+ru(1-u′K)-mu′2v′(t-τ)a+u′2,hhhhhhhhhhhhhhhhhhhx∈(0,lπ),t>0,∂v′∂t=d1Δv′+v′s1-hv′u′,hhhhhhhhhhx∈(0,lπ),t>0,∂u′∂n=∂v′∂n=0,x=0,lπ,t>0.Letx(t)=u′(τt)-u*, y(t)=v′(τt)-v*, τ=τ0+μ. Then μ=0 is the Hopf bifurcation value of system (48), which can be written as follows:(49)∂U(t)∂t=τ0dΔUt+τ0LUt+GUt,where(50)Lϕ=r-2ru*K-2amu*v*a+u*22ϕ10-mu*a+u*2ϕ2-1shϕ10-sϕ20,Gϕ,μ=μdΔϕ0+μLϕ+μ+τ0Fϕ,Fϕ=F1ϕF2ϕ,with(51)F1ϕ=mϕ10+u*2ϕ2-1+v*a+ϕ10+u*2rϕ10+u*1-ϕ10+u*K-mϕ10+u*2ϕ2-1+v*a+ϕ10+u*21-ϕ10+u*Kϕ1(0)-mu*a+u*2ϕ2-1,F2ϕ=sϕ20+v*1-hϕ20+v*ϕ10+u*-shϕ10+sϕ20for ϕ∈C([-1,0],X).From the previous discussion, it is known that±iw0 is a pair of simple purely imaginary eigenvalues of the linear system (7) and the following linear functional differential equation:(52)dz(t)dt=τ0Lzt.By the Riesz representation theorem, there exists a 2 × 2 matrix function η(θ,μ)θ∈[-1,0], whose elements are bounded variables, such that(53)τ0+μLϕ=∫-10dηθ,μϕθ,hhhhhhihhforϕ∈C-1,0,R2.In fact, it is possible to choose(54)ηθ,μ=τ0+μr-2ru*K-2amu*v*a+u*220sh-sδ(θ)hhhhhhhhhhhi+0-mu*a+u*200δθ+1r-2ru*K-2amu*v*(a+u*2)20sh-s,where δ(θ) is a Dirac delta function satisfying(55)δ(θ)=1,θ=0,0,θ≠0.For ϕ∈C1([-1,0],R2), define A(0) as(56)A0ϕθ=dϕθdθ,θ∈-1,0,∫-10dηθ,0ϕθ,θ=0,R0ϕθ=0,θ∈-1,0,Fϕ,μ,θ=0.For ψ∈C1([-1,0],(R2)*), define(57)A*ψs=-dψsds,s∈0,1,∫-10dηθ,0ψ-ξ,s=0.Then A(0) and A* are adjoint operators under the bilinear form:(58)ψ(s),ϕ(θ)0=ψ-0ϕ0-∫-10∫0θψ-ξ-θdη0,θϕξdξ.Therefore, ±iw0 are eigenvalues of A(0) as well as A*. Next, the eigenvectors of A(0) and A* corresponding to the eigenvalues iw0 and -iw0 can be calculated. Let(59)qθ=1Ceiw0τ0θ.Under the condition A(0)q(0)=iw0τ0q(0), that is,(60)τ0r-2ru*K-2amu*v*a+u*22-iw0-mu*a+u*2e-iw0τ0sh-s-iw01C=00,it follows that(61)C=sh(s+iw0).Similarly, let q*(s)=E(1D)eiw0τ0s, and with A*q*(0)=-iw0τ0q*(0), that is,(62)τ0r-2ru*K-2amu*v*a+u*22+iw0sh-mu*a+u*2e-iw0τ0-s+iw01D=00,it is possible to obtain(63)D=-hr-2ru*/K-2amu*v*/a+u*22+iw0s.According to the conditions (q*,q)=1 and (q*,q-)=0,(64)q*,q=q*-0q0-∫-10∫ξ=0θq*-ξ-θdη0,θqξdξ=E-1D-1C-∫-10∫ξ=0θE-(1D-)e-iw0τk(ξ-θ)dη(θ)hhhhhhhhhhhhhhhhhhhhi×1Ceiw0τkξdξ=E-1+D-C-∫-10θ1D-dη0,θ1Ceiw0τkθ=E-1+D-C+1D-τ00-mu*a+u*200hhhhh×1Ce-iw0τk0-mu*a+u*200=E-1+D-C+τ0C-mu*a+u*2e-iw0τk.Therefore,(65)E=11+DC-+τ0C--mu*/a+u*2eiw0τk.Let Φ=(q(θ),q-(θ)), Ψ=(q*(s),q-*(s))T, I=1001; then(66)Ψ,Φ0=I.Therefore, the center subspace of system (52) is P=span⁡{q(θ),q-(θ)}, and the adjoint subspace is P*=span⁡{q*(s),q-*(s)}. Let f0=(f01,f02), where(67)f01=10,f02=01.Using the notion from [30], it is also possible to define(68)c·f0=c1f01+c2f02for c=(c1,c2)T∈C2.Define(ψ·f0)(θ)=ψ(θ)·f0 for ψ(θ)∈[-1,0] and(69)u,v=1lπ∫0lπu1v-1dx+1lπ∫0lπu2v-2dxfor u=(u1,u2), v=(v1,v2)∈C([0,lπ],R2).Hence,(70)ϕ,f0=ϕ,f01,ϕ,f02Tfor ϕ∈C([-1,0],X). Then the center subspace of linear system (7) is given by PCNC, where(71)PCNϕ=ΦΨ,ϕ,f00·f0,ϕ∈C,PCNC=qθz+q-θz-·f0:z∈C,and C=PCNC⊕PSC, where PSC is a stable subspace.According to [30], it is known that the infinitesimal generator AU of linear system (7) satisfies(72)AUψ=ψ˙θ.Moreover, ψ∈dom(AU) if and only if(73)ψ˙θ∈C,ψ0∈domΔ,ψ˙θ0=τ0Δψ0+τ0L0ψ.Setting μ=0 in (49), the center manifold(74)W(z,z-)=W20(θ)z22+W11(θ)zz-+W02(θ)z-22+⋯can be obtained in PSC. The flow of system (49) can be written as (75)ut=Φ(zt,z-t)T·f0+Wzt,z-t,where(76)z˙t=iw0τ0zt+q*-0×GΦ(zt,z-t)T·f0+Wz,z-,0,f0≜iw0τ0zt+gz,z-,(77)gz,z-=q*-0×GΦzt,z-tT·f0+Wz,z-,0,f0=g20(θ)z22+g11(θ)zz-+g02(θ)z-22+g21(θ)z2z-2+⋯.From Taylor’s formula,(78)fu,v=mu+u*2v+v*a+u+u*2=mu*2v*a+u*2+2amu*v*a+u*2u+mu*2a+u*2v+b20u2+b11uv+b30u3+b21u2v+O4,gu,v=shv+v*2u+u*=shv*h-1h2u+2hv+1h2u*u2+1u*v2-2hu*uvhhhhh+2h2u*2u2v-1h2u*2u3+O4v*h-1h2u+2hv,where(79)hib11=2amu*a+u*2,hib20=amv*a2+2au*2+u*4-4amu*2v*a+u*2a+u*24,hib21=ama+u*2-4amu*2a+u*2a+u*23,hib30=-8amu*v*(a+u*2)-8amv*hhhihh×(3u*2-2au*)+64amu*3v*hhhhhh×a+u*2a+u*2-1,O4=Ou,v4.Let G(ϕ,0)=τ0G1,G2T, where(80)G1=-rK-b20ϕ120-b11ϕ10ϕ2-1-b30ϕ130-b21ϕ120ϕ2-1+O4,G2=-shu*ϕ12(0)-shu*ϕ220+2su*ϕ10ϕ20-2shu*2ϕ120ϕ20+shu*2ϕ130+O4.From (77),(81)g20=2E-τ0-rK-b20-b11Ce-iw0τ0+2E-τ0D--shu*-shu*C2+2su*C,g11=E-τ02-rK-b20-b11Ce-iw0τ0+C-eiw0τ0+E-τ0D--2shu*-2shu*CC-+2su*C+C-,g02=2E-τ0(-rK-b20)-b11C-eiw0τ0+2E-τ0D--shu*-shu*C-2+2su*C-,g21=2E-τ0-rK-b20-D-shu*×1lπ∫0lπW2010+2W1110dx-2E-τ0D-shu*1lπ∫0lπW2010C-+2W1110Cdx-Eτ0b111lπ∫0lπW2010C-eiw0τ0+2W1110Ce-iw0τ0hhhhhhhhhhhhhhi+W202-1+2W112-1dx+E-τ0D-2su*1lπ∫0lπW20(1)(0)C-+2W11(1)(0)Chhhhhhhhhhhhhhhi+W2020+2W1120dx-2E-τ0b21C-eiw0τ0+2Ce-iw0τ0-2E-τ0D-2shu*2C-+2C-6E-τ0b30+6E-τ0D-shu*2.In the last formula,W20(0), W20(-1), W11(0), and W11(-1) are still unknown. Therefore, it is necessary to compute W20(θ) and W11(θ), as described below.From (74), it is possible to obtain(82)W˙=W20zz˙+W11z-z˙+W11zz-˙+W02zz-˙+⋯,AUW=AUW20z22+AUW11zz-+W02AUz-22+⋯.In addition, W(z,z-) satisfies(83)W˙=AUW+X0GΦ(z,z-)T·f0+Wz,z-,0-ΦΨ,X0GΦ(z,z-)T·f0+Wz,z-,0,f00·f0≜AUW+H20z22+H11zz-+H02z-22+⋯.Therefore,(84)2iw0τ0W20=AUW20+H20,-AUW11=H11,-2iw0τ0W02=AUW02+H02.For -1≤θ<0,(85)H20z22+H11zz-+H02z-22+⋯=-ΦΨ,X0GΦ(z,z-)T·f0+Wz,z-,0,f00·f0,and therefore(86)H20θ=-g20qθ+g-02q-θ·f0,H11θ=-g11qθ+g-11q-θ·f0.From (84) and (86),(87)W˙20θ=2iw0τ0W20θ+g20qθ+g-02q-θ·f0.Solving for W20(θ),(88)W20θ=ig20qθw0τ0·f0+ig-02q-θ3w0τ0·f0+E1e2iwpτ0θ.Similarly,(89)W11θ=-ig11qθw0τ0·f0+ig-11q-θw0τ0·f0+E2,where E1 and E2 are both two-dimensional vectors and can be determined as follows.Forθ=0, according to the definition of AU and the first two equations of (84),(90)2iw0τ0W200=∫-10dηθ,0W20θ+H200-∫-10dηθ,0W11θ=H110.Then(91)H200=-g20q0+g-02q-0·f0+2τ0×-rK-b20-b11Ce-iw0τ0-shu*-shu*C2+2su*C.Note that(92)∫-10dηθ,0eiw0τ0θq0=iw0τ0q0,∫-10dηθ,0e-iw0τ0θq-0=-iw0τ0q-0.It follows that(93)2τ0-rK-b20-b11Ce-iw0τ0-shu*-shu*C2+2su*C=g20q0+g-02q-0+2iw0τ0W200-∫-10dηθ,0W20θ=-g20q(0)+g-02q-03+2iw0τ0E1-∫-10dηθ,0ig20qθw0τ0+ig-02q-θ3w0τ0+E1e2iwpτ0θ=2iw0τ0I-∫-10dηθ,0e2iwpτ0θE1=r-2ru*K-2amu*v*(a+u*2)2-mu*a+u*2e-2iw0τ0sh-s2iw0τ01001-τ0hhhhh×r-2ru*K-2amu*v*a+u*22-mu*a+u*2e-2iw0τ0sh-sE1=τ02iw0-r+2ru*K+2amu*v*a+u*22mu*a+u*2e-2iw0τ0-sh2iw0+sE1.Therefore, it is possible to obtain(94)E1=E11·E12,where(95)E11=2iw0-r+2ru*K+2amu*v*a+u*22mu*a+u*2e-2iw0τ0-sh2iw0+s-1,E12=2-rK-b20-b11Ce-iw0τ0-shu*-shu*C2+2su*C.Similarly,(96)H110=-g11q0+g-11q-0·f0+τ0×2-rK-b20-b11Ce-iw0τ0+C-eiw0τ0-2shu*-2shu*CC-+2su*C+C-.Then(97)τ02-rK-b20-b11Ce-iw0τ0+C-eiw0τ0-2shu*-2shu*CC-+2su*C+C-=g11q0+g-11q-0-∫-10dηθ,0W11θ=g11q0+g-11q-0-∫-10dηθ,0-ig11qθw0τ0+ig-11q-θw0τ0+E2=-∫-10dηθ,0E2=τ0-r+2ru*K+2amu*v*a+u*22mu*a+u*2-shsE2.Therefore, it is possible to obtain(98)E2=E21·E22,where(99)E21=-r+2ru*K+2amu*v*a+u*22mu*a+u*2-shs-1,E22=2-rK-b20-b11Ce-iw0τ0+C-eiw0τ0-2shu*-2shu*CC-+2su*C+C-.Then g21 is expressed by the parameters. Therefore, the following values can be computed:(100)c10=i2w0τ0g20g11-2g112-g0223+g212,μ2=-Rec10Reλ′τ0,β2=2Rec10,T2=-Imc10+μ2Imλ′τ0w0τ0.Theorem 6.
Ifμ2, β2, and T2 are defined as above, then the following are true:(i)
ifμ2>0 (μ2<0), then the Hopf bifurcation is supercritical (subcritical);(ii)
ifβ2<0 (β2>0), then the periodic solutions are stable (unstable);(iii)
ifT2>0 (T2<0), then the period of the bifurcating periodic solutions of system (3) increases (decreases).Remark 7.
The direction and stability of the Hopf bifurcation have already been computed whenτ=τ0. The other Hopf bifurcation value τk can be analyzed using the same procedure.


## 5. Simulations and Discussion

This paper has considered a delayed predator-prey model with modified Holling-Tanner functional response. The effect of time delays on the positive equilibrium of system (3) has been investigated. The results obtained here show that the time delay plays an important role in determining system stability. Theorem 3 shows that the time delay can cause a stable equilibrium to become unstable or, in other words, that there is a critical value τ0, such that, for 0≤τ1+τ2<τ0, equilibrium (u*,v*) is stable; when τ1+τ2 approaches τ0, equilibrium (u*,v*) bifurcates into periodic solutions; and when τ1+τ2>τ0, the equilibrium becomes unstable.Numerical simulations are performed to illustrate the dynamic behavior of system (3). Assume that r=1, a=1, h=1, K=2, s=1, m=4, d1=0.1, d2=0.2, and l=1. In this case, τ0=2.201. As shown in Figure 1, the positive equilibrium is stable when τ1+τ2<τ0, and the predators and prey tend to a steady state. However, if τ1+τ2>τ0, the positive equilibrium loses its stability and Hopf bifurcation occurs, leading to oscillatory behavior. These results are identical to the numerical results illustrated in Figure 2.Figure 1
Convergence to positive equilibrium. Here the parameters are assumed to beτ1=0, τ2=1.5, and the initial values are u(x,t)=0.1+0.01tsinx, v(x,t)=0.5+0.01tcos⁡x, and t∈[-1.5,0], x∈[0,π].
(a)
(b)Figure 2
Convergence to positive equilibrium. Here the parameters are assumed to beτ1=0, τ2=7, and the initial values u(x,t)=0.1+0.01tsinx, v(x,t)=0.5+0.01tcos⁡x, and t∈[-7,0], x∈[0,π].
(a)
(b)In both figures, the upper diagram represents the prey and the lower diagram represents the predator.



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*Source: 620891-2015-03-19.xml*