# Stability and Hopf Bifurcation in a Diffusive Predator-Prey System with Beddington-DeAngelis Functional Response and Time Delay

Authors: Yuzhen Bai; Xiaopeng Zhang
Journal: Abstract and Applied Analysis
 (2011)

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## Abstract

This paper is concerned with a diffusive predator-prey system with Beddington-DeAngelis
functional response and delay effect. By analyzing the distribution of the eigenvalues, the stability
of the positive equilibrium and the existence of spatially homogeneous and spatially inhomogeneous
periodic solutions are investigated. Also, it is shown that the small diffusion can affect the Hopf
bifurcations. Finally, the direction and stability of Hopf bifurcations are determined by normal form
theory and center manifold reduction for partial functional differential equations.


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## Body

## 1. Introduction

In this paper, we will study the stability and Hopf bifurcations of a diffusive predator-prey system with Beddington-DeAngelis functional response and delay effect as follows:(1.1)ut=d1Δu(t,x)+u(t,x)(1-u(t-τ,x))-sP(u,v),t>0,x∈Ω,vt=d2Δv(t,x)+rP(u,v)-dv(t,x),t>0,x∈Ω,∂νu=∂νv=0,t>0,x∈∂Ω,u(0,x)=u0(x)≥0,v(0,x)=v0(x)≥0,x∈Ω,
where u and v denote the population densities of prey and predator species at time t and space x, respectively; the positive constants d1 and d2 represent the diffusion coefficients of prey and predator species, respectively; s>0 (s is called the capturing rate) and r>0 (r is called the conversion rate) represent the strength of the relative effect of the interaction on the two species; d denotes the death rate of predator species; P(u,v)=uv/(m+u+nv) is the Beddington-DeAngelis functional response function with m and n are positive numbers; τ≥0 denotes the generation time of the prey species; Ω is a bounded domain in RN (N is any positive integer) with a smooth boundary ∂Ω; Δ is the Laplacian operator on Ω; ν is the outward normal to ∂Ω; homogeneous Neumann boundary conditions reflect the situation where the population cannot move across the boundary of the domain.System (1.1) includes the models which have been discussed by many researchers; for examples, when τ=0, the models were considered in [1, 2]; if d1=d2=0 and τ=0, it was discussed in [3]; if P(u,v)=1, it was discussed in [4]. Moreover, when τ=0 and P(u,v)=u2v/(1+u2), system (1.1) can be transformed into Narcisa Apreutesei’s model (see [5]).There has been an increasing interest in the study of diffusive predator-prey system (see [1, 2, 4, 6–14] and references therein) with functional response. As is known to all, the Beddington-DeAngelis functional response, proposed by Beddington [6] and DeAngelis et al. [8], is more general than those the above authors considered, and it has been studied extensively in the literature [1–3, 7, 14–16]. However, to the authors’ best knowledge, few researches have been done on the diffusive predator-prey system with Beddington-DeAngelis functional response and time delay.The aim of this paper is to extend and develop the work in [1, 2]; that is, we will study the stability and Hopf bifurcation of a diffusive predator-prey system with Beddington-DeAngelis functional response and delay. The system we consider here is more general than the system in [1, 2].The rest of the paper is organized as follows. In Section2, we analyze the distribution of the roots of the characteristic equation and give various conditions on the stability of a positive constant steady state and the existence of Hopf bifurcation. In Section 3, we discuss the effect of diffusion on the Hopf bifurcation. In Section 4, by applying the normal form theory and the center manifold reduction of partial functional differential equations by Wu [17], an explicit algorithm for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions is given.


## 2. Analysis of the Characteristic Equations

In this section, by choosing the delayτ as the bifurcation parameter and analyzing the associated characteristic equation of (1.1) at the positive constant steady state, we investigate the stability of the positive constant steady state of (1.1) and obtain the conditions under which (1.1) undergoes Hopf bifurcation.It can be seen that homogeneous Neumann boundary conditions imposed on (1.1) lead to E1(0,0) and E2(1,0), always being two boundary equilibria for any feasible parameters, and (1.1) always having a unique positive constant steady state E(u*,v*) provided that the condition(A1)
0<d<(ru*/(u*+m)), hold, where
(2.1)u*=-(sr-nr-sd)+(sr-nr-sd)2+4smndr2rn,v*=(r-d)u*-dmdn.Under (A1), let u¯=u-u*,v¯=v-v* and drop the bars for simplicity of notations, then (1.1) can be transformed into the following equivalent system:(2.2)ut=d1Δu(t,x)+(u+u*)(1-(u(t-τ,x)+u*))-s(u+u*)(v+v*)m+(u+u*)+n(v+v*),vt=d2Δv(t,x)+r(u+u*)(v+v*)m+(u+u*)+n(v+v*)-d(v+v*).
Let P(u,v)=uv/(m+u+nv). By u*(1-u*)-(su*v*/(m+u*+nv*))=0 and -dv*+(ru*v*/(m+u*+nv*))=0 (2.2) becomes(2.3)ut=d1Δu(t,x)+(1-u*-a1)u(t)-u*u(t-τ,x)-a2v-uu(t-τ,x)-f(u,v),vt=d2Δv(t,x)+b1u(t)+(b2-d)v(t)+g(u,v),
where a1=sP10(u*,v*), a2=sP01(u*,v*), b1=rP10(u*,v*), b2=rP01(u*,v*), and (2.4)f(u,v)=s∑i+j=23Piji!j!uivj+h.o.t.,g(u,v)=r∑i+j=23Piji!j!uivj+h.o.t.,
where ∂i+jP(u,v)/∂ui∂vj|(u,v)=(u*,v*) denoted by Pij and h.o.t. for shorthand of “higher order terms.”Denoteu1(t)=u(t,·),u2(t)=v(t,·), and U=(u1,u2)T.Then (2.3) can be transformed into an abstract differential equation in the phase space ℓ=C([-τ,0],X),(2.5)U̇(t)=DΔU(t)+L(Ut)+F(Ut),
with(2.6)D=(d100d2),L(ϕ)=(1-u*-a1-a2b1b2-d)(ϕ1(0)ϕ2(0))+(-u*000)(ϕ1(-τ)ϕ2(-τ)),F(ϕ)=(-ϕ1(0)ϕ1(-τ)-s∑i+j=23Piji!j!ϕ1i(0)ϕ2j(0)r∑i+j=23Piji!j!ϕ1i(0)ϕ2j(0)),
where ϕ=(ϕ1,ϕ2)T∈ℓ.The linearization of (2.5) is given by(2.7)U̇(t)=DΔU(t)+L(Ut),
and its characteristic equation is(2.8)λy-DΔy-L(eλy)=0,
where y∈dom(Δ) and y≠0, dom(Δ)⊂X.From the properties of the Laplacian operator defined on the bounded domain, the operatorΔ on X has the eigenvalues -k2 with the relative eigenfunctions (2.9)βk1=(γk0),βk2=(0γk),
where γk=coskx,k∈N0={0,1,2,…}. Clearly, (βk1,βk2)k=0∞ construct a basis of the phase space X and therefore any element y in X can be expanded as Fourier series in the following form:(2.10)y=∑k=0∞(〈y,βk1〉βk1+〈y,βk2〉βk2)=∑k=0∞(βk1,βk2)(〈y,βk1〉〈y,βk2〉).
Some simple computations show that(2.11)L(ϕT(βk1βk2))=L(ϕ)T(βk1βk2),k∈N0.
From (2.10)-(2.11), (2.8) is equivalent to(2.12)∑k=0∞(〈y,βk1〉,〈y,βk2〉)[(λI2+Dk2)-(-u*e-λτ+1-u*-a1-a2b1b2-d)](βk1βk2)=0.
Assume that(A2)
1-u*-a1<0,b2-d<0.Let-a3=1-u*-a1, -b3=b2-d, p=a3+b3, r=a3b3+a2b1, s=u*, q=b3u*, then we conclude that the characteristic equation (2.8) is equivalent to the sequence of the characteristic equations:(2.13)λ2+(d1k2+d2k2+p)λ+d1d2k4+b3d1k2+a3d2k2+r+(sλ+d2sk2+q)e-λτ=0.
Obviously, for allk∈N0, λ=0 is not a root of (2.13).Equation (2.13) with τ=0 is equivalent to the following quadratic equations:(2.14)λ2+(d1k2+d2k2+p+s)λ+d1d2k4+b3d1k2+a3d2k2+r+d2sk2+q=0.
Let λ1 and λ2 be the two roots of (2.14). Then, for allk∈N0,(2.15)λ1+λ2=-(d1k2+d2k2+p+s)<0,λ1λ2=d1d2k4+b3d1k2+a3d2k2+r+d2sk2+q>0.
Therefore, we have the following Lemma.Lemma 2.1.
Assume that (A1)  and (A2)  hold. Then the equilibriumE(u*,v*) of (1.1) with τ=0 is asymptotically stable.Assume that(A3)
s2-p2+2r<0,r>q;(A4)
b3d1+a3d2-d2s≥0, or (b3d1+a3d2-d2s)2<4d1d2(r-q), if b3d1+a3d2-d2s<0.Theorem 2.2.
If (A1)–(A4)  hold, then all roots of (2.13) have negative real parts for all τ≥0. Furthermore, the equilibrium E(u*,v*) of the system (1.1) is asymptotically stable for all τ≥0.Proof.
Letλ=iω(ω>0) be a root of the characteristic equation (2.13). Then ω satisfies the following equation for some k∈N0:
(2.16)-ω2+(d1k2+d2k2+p)iω+d1d2k4+b3d1k2+a3d2k2+r+(sωi+d2sk2+q)(cos⁡ωτ-isin⁡ωτ)=0.
Separating the real and imaginary parts of (2.16) leads to
(2.17)-ω2+d1d2k4+b3d1k2+a3d2k2+r+sωsin⁡⁡ωτ+(d2sk2+q)cos⁡⁡ωτ=0,(d1k2+d2k2+p)ω+sωcos⁡⁡ωτ-(d2sk2+q)sin⁡⁡ωτ=0,
which implies that
(2.18)ω4+[(d12+d22)k4+2a3d1k2+2b3d2k2-(s2-p2+2r)]ω2+[(d1d2k4+b3d1k2+a3d2k2+r)2-(d2sk2+q)2]=0.
Let z=ω2, then (2.18) can be rewritten into the following form:
(2.19)z2+[(d12+d22)k4+2a3d1k2+2b3d2k2-(s2-p2+2r)]z+[(d1d2k4+b3d1k2+a3d2k2+r)2-(d2sk2+q)2]=0.
By (A3) and (A4), for all k∈N0, we have
(2.20)-(d12+d22)k4-2a3d1k2-2b3d2k2+(s2-p2+2r)<0,d1d2k4+(b3d1+a3d2-d2s)k2+r-q>0,
which imply that (2.19) has no positive roots. Hence, the characteristic equation (2.13) has no purely imaginary roots. By Lemma 2.1 and the theorem proved by Ruan and Wei [18], all roots of (2.13) have negative real parts.Notice that (2.13) with k=0 is the characteristic equation of the linearization of (1.1) corresponding system without diffusion (ordinary differential equations, ODEs) at the positive equilibrium. And it has been considered under the condition:(B1)
r<q.It is easy to get that when(2.21)τ=τj0=1ω+0arccos{(q((ω+0)2-r)-ps(ω+0)2)s2(ω+0)2+q2+2jπ},(j=0,1,2,…),
Equation (2.13) with k=0 has simple imaginary roots ±iω+0, and Re(dλ/dτ)τ=τj0>0, where λ(τ) is the root of (2.13) with k=0 satisfying λ(τj0)=iω+0, and
(2.22)ω+0=22[s2-p2+2r+(s2-p2+2r)2-4(r2-q2)]1/2,
and τ00=min{τj0,j∈{0,1,2,…}}. Assume that(B2)
d12+d22+2a3d1+2b3d2>s2-p2+2r,d1d2+b3d1+a3d2-d2s+r-q>0.We have the following result.Theorem 2.3.
Assume that (A1), (A2), (B1), and (B2)  are satisfied. Then forτ=τj0(j=0,1,2,…), (2.13) has a pair of simple imaginary roots ±iω+0, and all roots of (2.13), except ±iω+0, have no zero real parts. Moreover, all the roots of (2.13) with τ=τj0, except ±iω+0, have negative real parts.Proof.
Letλ=iω1(ω1>0) be a root of (2.13) with k≥1. By the same way in Theorem 2.2, we can obtain
(2.23)ω14+[(d12+d22)k4+2a3d1k2+2b3d2k2-(s2-p2+2r)]ω12+[(d1d2k4+b3d1k2+a3d2k2+r)2-(d2sk2+q)2]=0,
for all k≥1.
Setz=ω12, then
(2.24)z2+[(d12+d22)k4+2a3d1k2+2b3d2k2-(s2-p2+2r)]z+[(d1d2k4+b3d1k2+a3d2k2+r)2-(d2sk2+q)2]=0.
Let z1 and z2 be the roots of (2.24) with k≥1. We know that if z1+z2<0 and z1z2>0,then (2.13) with k≥1 has no purely imaginary roots.
By (B2), it follows that, for ∀k≥1,
(2.25)-(d12+d22)k4-2a3d1k2-2b3d2k2+(s2-p2+2r)≤-(d12+d22+2a3d1+2b3d2)+(s2-p2+2r)<0,d1d2k4+(b3d1+a3d2-d2s)k2+r-q≥d1d2+b3d1+a3d2-d2s+r-q>0.
Therefore, (2.13) with k≥1 have no purely imaginary roots.
Summarizing the above results and combining Theorem2.3, we have the following theorem on the stability of the positive equilibrium E(u*,v*) of system (1.1) and the existence of Hopf bifurcation at E(u*,v*).Theorem 2.4.
Assume that (A1), (A2), (B1), and (B2)  hold. For system (1.1), the following statements are true:(I)
Ifτ∈[0,τ00), then the equilibrium point E(u*,v*) is asymptotically stable;(II)
Ifτ>τ00, then the equilibrium point E(u*,v*) is unstable;(III)
τ=τj0(j=0,1,2,…) are Hopf bifurcation values of system (1.1), and these Hopf bifurcations are all spatially homogeneous.By the same way in Theorem2.2, let λ=iω(ω>0) be a root of the characteristic equation (2.13), then ω satisfies the following equation:(2.26)ω4+[(d12+d22)k4+2a3d1k2+2b3d2k2-(s2-p2+2r)]ω2+[(d1d2k4+b3d1k2+a3d2k2+r)2-(d2sk2+q)2]=0,
for k∈N0.Now, we make the following assumptions. For a certaink0={1,2,…},(C1)
d1d2+b3d1+a3d2-d2s+r-q>0;(C2)
((s2-p2+2r)/k02)<d12+d22+2a3d1+2b3d2<s2-p2+2r;(C3)
[(d12+d22)k4+2a3d1k2+2b3d2k2-(s2-p2+2r)]2, -4[(d1d24+b3d1k2+a3d2k2+r)2-(d2s2+q)2]≥0,k∈N0∖{k0}.Under the assumptions (C1) and (C2), (2.26) with k=k0 has only a positive solution ω+k0,(2.27)ω+k0=22B+B2-4[(d1d2k04+a3d2k02+b3d1k02+r)2-(d2k02s+q)2],
where B=s2-p2+2r-(d12+d22)k04-2a3d1k02-2b3d2k02<0.Setz=ω2, then (2.26) can be transformed into the following equation:(2.28)z2+[(d12+d22)k4+2a3d1k2+2b3d2k2-(s2-p2+2r)]z+[(d1d2k4+b3d1k2+a3d2k2+r)2-(d2sk2+q)2]=0.
Let z1 andz2 be the roots of (2.28). If the assumptions (C1)–(C3) hold, we have (2.29)z1+z2=-(d12+d22)k4-2a3d1k2-2b3d2k2+(s2-p2+2r)≤s2-p2+2r<0,z1z2=(d1d2k4+a3d2k2+b3d1k2+r)2-(d2k2s+q)2>0,
for k∈N0∖{k0}.Therefore, (2.13) with k∈N0∖{k0} has no solutions with zero real parts.In addition, similar to the proof of Theorem2.2, we have
(2.30)-(ω+k0)2+d1d2k04+b3d1k02+a3d2k02+r+sω+k0sin⁡⁡ω+k0τ+(d2sk02+q)cos⁡⁡ω+k0τ=0,(d1k02+d2k02+p)ω+k0+sω+k0cos⁡⁡ω+k0τ-(d2sk02+q)sin⁡⁡ω+k0τ=0
which implies that(2.31)sin⁡(ω+k0τ)=[(ω+k0)2-d1d2k04-b3d1k02-a3d2k02-r]sω+k0(d2sk02+q)2+s2(ω+k0)2+(d1k02+d2k02+p)(d2sk02+q)ω+k0(d2sk02+q)2+s2(ω+k0)2≜F(ω+k0),cos⁡(ω+k0τ)=[(ω+k0)2-d1d2k04-b3d1k02-a3d2k02-r](d2sk02+q)(d2sk02+q)2+s2(ω+k0)2-(d1k02+d2k02+p)s(ω+k0)2(d2sk02+q)2+s2(ω+k0)2≜E(ω+k0).
Define(2.32)τjk0={1ω+k0(arccos(E(ω+k0))+2jπ)ifF(ω+k0)≥0,1ω+k0(2π-arccos(E(ω+k0))+2jπ)ifF(ω+k0)<0,forj∈{0,1,2,…}.From the above analysis, we have the following Theorem.Theorem 2.5.
Assume that (A1), (A2), and (C1)–(C3)  hold. Then forτ=τjk0(j=0,1,2,⋯), (2.13) has a pair of simple imaginary roots ±iω+k0, and all roots of (2.13), except ±iω+k0, have no zero real parts. Moreover, all the roots of (2.13) with τ=τjk0, except ±iω+k0, have negative real parts.Letλ(τ)=α(τ)+iβ(τ) be the root of (2.13) near τ=τjk0 satisfying(2.33)α(τjk0)=0,β(τjk0)=ω+k0,j=0,1,2,…,
where ω+k0 and τjk0 are given by (2.27) and (2.32), respectively. Then we have the following transversality condition.Lemma 2.6.
Assume that (A1), (A2), and (C1)–(C3)  hold. Then(2.34){Re(dλdτ)}τ=τjk0>0.Proof.
Differentiating the two sides of (2.13) with respect to τ yields
(2.35)(dλdτ)-1=(2λ+d1k02+d2k02+p)eλτ+ssλ2+sλd2k02+λq-τλ.
From (2.30), we have
(2.36)Re(dλdτ)τ=τjk0-1=Re[(2λ+d1k02+d2k02+p)eλτ+ssλ2+sλd2k02+λq-τλ]τ=τjk0=2(ω+k0)2-s2+p2-2r+(d12+d22)k04+2a3d1k02+2b3d2k02s2(ω+k0)2+(sd2k02+q)2=2(ω+k0)2-Bs2(ω+k0)2+(sd2k02+q)2,
according to (2.27), and then
(2.37){Re(dλdτ)}τ=τjk0={Re(dλdτ)-1}τ=τjk0>0.
Applying Lemma 2.6 and Theorem 2.5, we draw the following conclusions.Theorem 2.7.
Assume that (A1), (A2), and (C1)–(C3)  hold. For system (1.1), the following statements are true: (I)
Ifτ∈[0,τ0k0), then the equilibrium point E(u*,v*) is asymptotically stable;(II)
Ifτ>τ0k0, then the equilibrium point E(u*,v*) is unstable;(III)
τ=τjk0(j=0,1,2,…) are Hopf bifurcation values of system (1.1), and these Hopf bifurcations are all spatially inhomogeneous.


## 3. The Effect of Diffusion on Hopf Bifurcations

In the previous section, we have studied the Hopf bifurcations from the positive constant steady-stateE(u*,v*) of (1.1) when τ crosses through the critical value τjk(k=0,k0;j=1,2,3,…) and have the following conclusions.(I)
If (B2) holds, then system (1.1) and the corresponding system without diffusion (ODEs) have the same Hopf bifurcations, containing the existence and properties of Hopf bifurcations. In this case, the diffusion has no effect on the Hopf bifurcations of ODEs.(II)
If (B2) does not hold, then system (1.1) and ODEs have the different Hopf bifurcations. In this case, the diffusion has the effect on the Hopf bifurcations of ODEs.According to Theorems2.4 and 2.7, system (1.1) undergoes Hopf bifurcations under the different conditions. Comparing the conditions of Theorems 2.4 and 2.7, we have the following conclusions.(I)
When system (1.1) undergoes spatially homogeneous Hopf bifurcation, diffusion coefficients satisfy the condition:
(3.1)d12+d22+2a3d1+2b3d2>s2-p2+2r,
and in this case, system (1.1) and ODEs have the same properties of Hopf bifurcation.(II)
When system (1.1) undergoes spatially inhomogeneous Hopf bifurcation, diffusion coefficients satisfy the condition
(3.2)d12+d22+2a3d1+2b3d2<s2-p2+2r.
and in this case, system (1.1) and ODEs have the different properties of Hopf bifurcation.Summarizing the above results, we can obtain the conclusion. The big diffusion has no effect on the Hopf bifurcation of system (1.1), the small diffusion can make system (1.1) undergo the spatially inhomogeneous Hopf bifurcation.


## 4. Direction of Hopf Bifurcation and Stability of the Bifurcating Periodic Orbits

In this section, we will study the directions, stability, and the period of bifurcating periodic solutions by using normal formal theory and center manifold theorem of partial functional differential equations presented in [17]. For fixed j∈{0,1,2,…}, we denote τjk by τ̃. Let(4.1)ũ(t,x)=u(τt,x),ṽ(t,x)=v(τt,x),τ=μ+τk,u1(t)=u(t,⋅),u2(t)=v(t,⋅),U=(u1,u2)T,
and drop the tilde for the sake of simplicity. Then system (1.1) can be written as(4.2)U̇(t)=τ̃DΔU(t)+L(τ̃)(Ut)+f(Ut,μ),
where D=(d100d2),L(μ)(·):ℓ→X, and f:ℓ×R→X are given, respectively, by(4.3)L(μ)(ϕ)=(μ+τ̃)((1-u*-a1-a2b1b2-d)(ϕ1(0)ϕ2(0))+(-u*000)(ϕ1(-1)ϕ2(-1))),(4.4)f(ϕ,μ)=(μ+τ̃)(-ϕ1(0)ϕ1(-1)-s∑i+j=23Piji!j!ϕ1i(0)ϕ2j(0)r∑i+j=23Piji!j!ϕ1i(0)ϕ2j(0))+h.o.t.,
for ϕ=(ϕ1,ϕ2)T∈ℓ, where h.o.t. denotes high order terms.Consider the linear equation(4.5)U̇(t)=τ̃DΔU(t)+L(τ̃)(Ut).
From the discussion of Theorems 2.3 and 2.5 in Section 2, we know that the origin (0,0) is an equilibrium of (4.2), and for τ=τ̃, the characteristic equation of (4.5) has a pair of simple purely imaginary eigenvalues ⋀0={iω+kτ̃,-iω+kτ̃},(k=0,k0).Consider the ordinary functional differential equation(4.6)Ẋ(t)=-τ̃Dk2X(t)+L(τ̃)(Xt).
By the Riesz representation theorem, there exists a 2×2 matrix function η(θ,τ̃)(-1≤θ≤0), whose entry is of bounded variation such that(4.7)-τ̃Dk2ϕ(0)+L(τ̃)(ϕ)=∫-10d[η(θ,τ̃)]ϕ(θ),
for ϕ∈C([-1,0],R2). In fact, we can choose(4.8)η(θ,τ̃)={τ̃(-d1k2+1-u*-a1-a2b1-d2k2+b2-d),θ=0,0,θ∈(-1,0),τ̃(-u*000),θ=-1.
Let A(τ̃) denote the infinitesimal generators of the semigroup induced by the solutions of (4.6) and A* be the formal adjoint of A(τ̃) under the bilinear pairing(4.9)(ψ,ϕ)=ψ(0)ϕ(0)-∫-10∫ξ=0θψ(ξ-θ)d[η(θ,τ̃)]ϕ(ξ)dξ,
for ϕ∈C([-1,0],R2),ψ∈(C[0,1],R2). Then A(τ̃) and A* are a pair of adjoint operators. From the discussion in Section 2, we know that A(τ̃) has a pair of simple purely imaginary eigenvalues ±iω+kτ̃, and they are also eigenvalues of A*. Let P and Q be the center subspaces, that is, the generalized eigenspace of A(τ̃) and A* associated with ⋀0, respectively. Then Q is the adjoint space of P and dimP=dimQ=2, see [17].Direct computations give the following results.Lemma 4.1.
Let(4.10)C=-iω+k+d1k2-1+u*+a1+u*e-iω+kτ̃a2,D=-iω+k+d1k2-1+u*+a1+u*e-iω+kτ̃b1.
Then,
(4.11)p1(θ)=eiω+kτ̃(1,C)T,p2(θ)=p1(θ)¯,-1≤θ≤0
is a basis of P with ⋀0, and
(4.12)q1(s)=(1,D)e-iω+kτ̃,q2(s)=q1(s)¯,0≤s≤1
is a basis of Q with ⋀0.LetΦ=(Φ1,Φ2) and Ψ*=(Ψ1*,Ψ2*)T with (4.13)Φ1(θ)=p1(θ)+p2(θ)2=(Re{eiω+kτ̃θ}Re{Ceiω+kτ̃θ}),Φ2(θ)=p1(θ)-p2(θ)2i=(Im⁡{eiω+kτ̃θ}Im⁡{Ceiω+kτ̃θ}),
for θ∈[-1,0], and (4.14)Ψ1*(s)=q1(s)+q2(s)2=(Re{e-iω+kτkθ}Re{De-iω+kτ̃θ}),Ψ2*(s)=q1(s)-q2(s)2i=(Im⁡{e-iω+kτ̃θ}Im⁡{De-iω+kτ̃θ}),
for s∈[0,1].Define(4.15)Ψ=(Ψ1,Ψ2)T=((Ψ1*,Φ1)(Ψ1*,Φ2)(Ψ2*,Φ1)(Ψ2*,Φ2))-1Ψ*,
then Ψ construct a new basis for Q and (Ψ,Φ)=I2, see [17]. In addition, fk≜(βk1,βk2), where (4.16)βk1=(cos⁡kx0),βk1=(0cos⁡kx).Letc·fk be defined by c·fk=c1βk1+c2βk2,c=(c1,c2)T∈C([-1,0],X).Then the center subspace of linear equation (4.5) is given by PCNℓ, where(4.17)PCNl(ϕ)=Φ(Ψ,〈ϕ,fk〉)⋅fk,ϕ∈l,
and ℓ=PCNℓ⊕Psℓ, and Psℓ denotes the complement subspace of PCNℓ in ℓ.LetAτ̃ be the infinitesimal generator induced by the solution of (4.5). Then (4.2) can be rewritten as the abstract form(4.18)U̇t=Aτ̃Ut+R(μ,Ut),
where
(4.19)R(μ,Ut)={0,θ∈[-1,0),F(Ut,μ),θ=0.
Using the decomposition ℓ=PCNℓ⊕Psℓ and (4.17), the solution of (4.2) can be written as(4.20)Ut=Φ(x1(t)x2(t))⋅fk+h(x1,x2,μ),
where (x1(t)x2(t))=(Ψ,〈Ut,fk〉), and h(x1,x2,μ)∈Psℓ,h(0,0,0)=0,Dh(0,0,0)=0. In particular, the solution of (4.2) on the center manifold is given by(4.21)Ut=Φ(x1(t)x2(t))⋅fk+h(x1,x2,0).
Let z=x1-ix2 and Ψ(0)=(Ψ1(0),Ψ2(0))T, and notice that p1=Φ1+iΦ2, then (4.22)Φ(x1(t)x2(t))⋅fk=(Φ1,Φ2)(z+z¯2i(z-z¯)2)⋅fk=12(p1z+p¯1z¯)⋅fk.
Equation (4.21) can be transformed into(4.23)Ut=12(p1z+p¯1z¯)⋅fk+W(z,z¯),
where W(z,z¯)=h(((z+z¯)/2),(i(z-z¯)/2),0). Moreover, by [17], z satisfies(4.24)ż=iω+kτ̃z+g(z,z¯),
where(4.25)g(z,z¯)=(Ψ1(0)-iΨ2(0))〈f(Ut,0),fk〉.
Let(4.26)W(z,z¯)=W20z22+W11zz¯+W02z¯22+h.o.t.,(4.27)g(z,z¯)=g20z22+g11zz¯+g02z¯22+g21z2z¯2+h.o.t.
From (4.4) and (4.23), it follows that
(4.28)〈f(Ut,0),fk〉=τ̃2(-e-iω+kτ̃-sP11C-sP202-sP02C22rP11C+rP202+rP02C22)1π∫0πcos⁡3kxdxz22+τ̃4(-2cos⁡⁡ω+kτ̃-sP11(C+C¯)-sP20-sP02CC¯rP11(C+C¯)+rP20+rP02CC¯)1π∫0πcos⁡3⁡kxdxzz¯+τ̃2(-eiω+kτ̃-sP11C¯-sP202-sP02C¯22rP11C¯+rP202+rP02C¯22)1π∫0πcos⁡3kxdxz¯22+τ̃(-〈(W20(1)(0)2eiω+kτ̃+W11(1)(0)e-iω+kτ̃+W20(1)(-1)2+W11(1)(-1))cos⁡kx,cos⁡kx〉-sP11〈(W20(1)(0)2C¯+W11(1)(0)C+W20(2)(0)2+W11(2)(0))cos⁡kx,cos⁡kx〉-s〈[P20(W20(1)(0)2+W11(1)(0))+P02(W20(2)(0)2C¯+W11(2)(0)C)]cos⁡kx,cos⁡kx〉-s〈[P21(C¯8+C4)+P12(C28+CC¯4)]cos⁡3⁡kx,cos⁡kx〉-s〈(P308+P03C2C¯8)cos⁡3⁡kx,cos⁡kx〉rP11〈(W20(1)(0)2C¯+W11(1)(0)C+W20(2)(0)2+W11(2)(0))cos⁡kx,cos⁡kx〉+r〈[P20(W20(1)(0)2+W11(1)(0))+P02(W20(2)(0)2C¯+W11(2)(0)C)]cos⁡kx,cos⁡kx〉+r〈[P21(C¯8+C4)+P12(C28+CC¯4)]cos⁡3kx,cos⁡kx〉+r〈(P308+P03C2C¯8)cos⁡3kx,cos⁡kx〉)×z2z¯2+h.o.t.,
where 〈Wij(n)(θ),coskx〉=1/π∫0πWij(n)(θ)(x)coskxdx,i+j=2,n=1,2.Notice that∫0πcos3kxdx=0, for allk∈N={1,2,…}.Let(Ψ1,Ψ2)=Ψ1(0)-iΨ2(0). Then we can obtain the following quantities:
(4.29)g20={0,k∈N,τ̃2[(-e-iω+kτ̃-sP11C-sP202-sP02C22)Ψ1+(rP11C+rP202+rP02C22)Ψ2],k=0.g11={0,k∈N,τ̃4[(-2cos⁡ω+kτ̃-sP11(C+C¯)-sP20-sP02CC¯)Ψ1+(rP11(C+C¯)+rP20+rP02CC¯)Ψ2],k=0.g02=g20¯,g21=τ̃[-〈(W20(1)(0)2eiω+kτ̃+W11(1)(0)e-iω+kτ̃+W20(1)(-1)2+W11(1)(-1))cos⁡kx,cos⁡kx〉-sP11〈(W20(1)(0)2C¯+W11(1)(0)C+W20(2)(0)2+W11(2)(0))cos⁡kx,cos⁡kx〉-s〈[P20(W20(1)(0)2+W11(1)(0))+P02(W20(2)(0)2C¯+W11(2)(0)C)]cos⁡kx,cos⁡kx〉-s〈[P21(C¯8+C4)+P12(C28+CC¯4)]cos⁡3kx,cos⁡kx〉-s〈(P308+P03C2C¯8)cos⁡3⁡kx,cos⁡kx〉]Ψ1+τ̃[rP11〈(W20(1)(0)2C¯+W11(1)(0)C+W20(2)(0)2+W11(2)(0))cos⁡kx,cos⁡kx〉+r〈[P20(W20(1)(0)2+W11(1)(0))+P02(W20(2)(0)2C¯+W11(2)(0)C)]cos⁡kx,cos⁡kx〉+r〈[P21(C¯8+C4)+P12(C28+CC¯4)]cos⁡3kx,cos⁡kx〉+r〈(P308+P03C2C¯8)cos⁡3kx,cos⁡kx〉]Ψ2.
Since W20(θ) and W11(θ) for θ∈[-1,0] appear in g21, we need to compute them. It follows from (4.26) that(4.30)Ẇ(z,z¯)=W20zż+W11żz¯+W11z¯z¯̇+W02z¯z¯̇+h.o.t.,(4.31)Aτ̃W=Aτ̃W20z22+Aτ̃W11zz¯+Aτ̃W02z¯22+h.o.t.
In addition, by [17], W(z,z¯) satisfies(4.32)Ẇ=Aτ̃W+H(z,z¯),
where(4.33)H(z,z¯)=H20z22+H11zz¯+H02z¯22+h.o.t.=X0f(Ut,0)-Φ(Ψ,〈X0f(Ut,0),fk〉)⋅fk.
Thus, from (4.24), (4.27)–(4.33), we can obtain that(4.34)(2iω+kτ̃-Aτ̃)W20=H20,-Aτ̃W11=H11,(-2iω+kτ̃-Aτ̃)W02=H02.
Noticing that Aτ̃ has only two eigenvalues ±iω+kτ̃; therefore, (4.34) has a unique solution Wij in Q given by(4.35)W20=(2iω+kτ̃-Aτ̃)-1H20,W11=-Aτ̃-1H11,W02=(-2iω+kτ̃-Aτ̃)-1H02.
From (4.33), we know that for -1≤θ<0,(4.36)H(z,z¯)=-Φ(θ)Ψ(0)〈f(Ut,0),fk〉⋅fk=-(p1(θ)+p2(θ)2,p1(θ)-p2(θ)2i)(Ψ1(0),Ψ2(0))〈f(Ut,0),fk〉⋅fk=-12(p1(θ)(Ψ1(0)-iΨ2(0)))+p2(θ)(Ψ1(0)+iΨ2(0))〈f(Ut,0),fk〉⋅fk=-12(p1(θ)g20+p2(θ)g02¯)⋅fkz22-12(p1(θ)g11+p2(θ)g11¯)⋅fkzz¯+h.o.t.Therefore, for-1≤θ<0,
(4.37)H20(θ)={0,k∈N,-12[p1(θ)g20+p2(θ)g02¯]⋅f0,k=0.(4.38)H11(θ)={0,k∈N,-12[p1(θ)g11+p2(θ)g11¯]⋅f0,k=0.(4.39)H(z,z¯)(0)=f(Ut,0)-Φ(Ψ,〈f(Ut,0),fk〉)⋅fk,H20(0)={τ̃2(-e-iω+kτ̃-sP11C-sP202-sP02C22rP11C+rP202+rP02C22)cos⁡2kx,k∈N,τ̃2(-e-iω+kτ̃-sP11C-sP202-sP02C22rP11C+rP202+rP02C22)-12[p1(θ)g20+p2(θ)g02¯]⋅f0,k=0.H11(0)={τ̃4(-2cos⁡ω+kτ̃-sP11(C+C¯)-sP20-sP02CC¯rP11(C+C¯)+rP20+rP02CC¯)cos⁡2kx,k∈N,τ̃4(-2cos⁡ω+kτ̃-sP11(C+C¯)-sP20-sP02CC¯rP11(C+C¯)+rP20+rP02CC¯)-12[p1(θ)g11+p2(θ)g11¯]⋅f0,k=0.
By the definition of Aτ̃, we have from (4.34),
(4.40)Ẇ20(θ)=2iω+kτ̃W20(θ)+12[p1(θ)g20+p2(θ)g02¯]⋅fk,-1≤θ≤0.
Note that p1(θ)=p1(0)eiω+kτ̃θ,-1≤θ≤0. Hence,(4.41)W20(θ)=12[ig20p1(θ)ω+kτ̃+ig02¯p2(θ)3ω+kτ̃]⋅fk+E1e2iω+kτ̃θ,(4.42)E1={W20,k∈N,W20(0)-12[ig20p1(0)ω+0τ̃+ig02¯p2(0)3ω+0τ̃]⋅f0,k=0.
Using the definition of Aτ̃, and combining (4.34) and (4.41), we have(4.43)2iω+0τ̃[-12(ig20p1(0)ω+0τ̃+ig02¯p2(0)3ω+0τ̃⋅f0)+E1]-τ̃DΔ[-12(ig20p1(0)ω+0τ̃+ig02¯p2(0)3ω+0τ̃⋅f0)+E1]-L(τ̃)[-12(ig20p1(θ)ω+0τ̃+ig02¯p2(θ)3ω+0τ̃⋅f0)+E1e2iω+0τ̃θ]=τ̃2(-e-iω+kτ̃-sP11C-sP202-sP02C22rP11C+rP202+rP02C22)-12[p1(θ)g20+p2(θ)g02¯]⋅f0.
Notice that (4.44)τ̃DΔ[p1(0)⋅f0]+L(τ̃)[p1(θ)⋅f0]=iω+0τ̃p1(0)⋅f0,τ̃DΔ[p2(0)⋅f0]+L(τ̃)[p2(θ)⋅f0]=-iω+0τ̃p2(0)⋅f0.
Then, for k∈N0,
(4.45)2iω+0τ̃E1-τ̃DΔE1-L(τ̃)(E1)e2iω+0τ̃θ=τ̃2(-e-iω+kτ̃-sP11C-sP202-sP02C22rP11C+rP202+rP02C22)cos⁡2kx.
From the above expression, we obtain that(4.46)E1=12E(-e-iω+kτ̃-sP11C-sP202-sP02C22rP11C+rP202+rP02C22)cos⁡2kx,
where
(4.47)E=(2iω+k+d1k2-1+u*+a1+u*e-2iω+kτ̃a2-b12iω+k+d2k2-b2+d)-1.
By the same way, we have(4.48)-Ẇ11(θ)=-12[p1(θ)g11+p2(θ)g11¯]⋅fk,-1≤θ<0,W11(θ)=12[-ig11p1(θ)ω+kτ̃+ig11¯p2(θ)ω+kτ̃]+E2.
Similar to the above, we can obtain that(4.49)E2=14E0(-2cos⁡ω+kτ̃-sP11(C+C¯)-sP20-sP02CC¯rP11(C+C¯)+rP20+rP02CC¯)cos⁡2⁡kx,
where
(4.50)E0=(d1k2-1+2u*+a1a2-b1d2k2-b2+d)-1.
So far, W20(θ) and W11(θ) have been expressed by the parameters of the system (1.1). And, hence, g21 can be expressed also. Thus, we can compute the following values:
(4.51)c1(0)=i2ω+kτ̃(g20g11-2|g11|2-13|g02|2)+g212,μ2=-Re{c1(0)}Re{λ′(τjk)},β2=2Re{c1(0)},T2=-Im⁡{c1(0)}+μ2Im⁡{λ′(τjk)}ω+kτ̃,
where μ2 determines the direction of Hopf bifurcation, β2 determines the stability of bifurcating periodic solution, and T2 determines the period of the bifurcating periodic solution. Hence, we have the following result.Theorem 4.2.
The signs ofμ2,β2,T2 determine the properties of Hopf bifurcation described in Theorems 2.4 and 2.7. If μ2>0(μ2<0), then the Hopf bifurcation is supercritical (subcritical), and the bifurcating periodic solutions exist (nonexist) for τ>τk(τ<τk). 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 of system (1.1) increases (decreases).Remark 4.3.
From the previous computable results, the expressions ofE1 and E2 contain the diffusion coefficient. According to the definition of g21, the values of g21 have related on E1 and E2. Therefore, the signs of β2 and μ2 which determine the stability and direction of spatially inhomogeneous periodic solutions strictly depend on the diffusion coefficient of d1 and d2.



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*Source: 463721-2011-11-02.xml*