# Spatially Nonhomogeneous Periodic Solutions in a Delayed Predator-Prey Model with Diffusion Effects

Authors: Jia-Fang Zhang
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

This paper is concerned with a delayed predator-prey diffusion model with Neumann boundary conditions. We study the asymptotic stability of the positive constant steady state and the conditions for the existence of Hopf bifurcation. In particular, we show that large diffusivity has no effect on the Hopf bifurcation, while small diffusivity can lead to the fact that spatially nonhomogeneous periodic solutions bifurcate from the positive constant steady-state solution when the system parameters are all spatially homogeneous. Meanwhile, we study the properties of the spatially nonhomogeneous periodic solutions applying normal form theory of partial functional differential equations (PFDEs).


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## Body

## 1. Introduction

Functional differential equations have merited a great deal of attention due to its theoretical and practical significance; they are often used in population dynamics, epidemiology, and other important areas of science; see [1–6]. In particular, Lu and Liu [7] proposed the following modified Holling-Tanner delayed predator-prey model:
(1.1)du(t)dt=ru(t)(1-u(t)K)-αu(t)v(t)a+bu(t)+cv(t),dv(t)dt=v(t)[s(1-hv(t-τ)u(t-τ))],
where u(t) and v(t) denote the densities of prey species and predator species, respectively. The first equation states that the prey grows logistically with carrying capacity K and intrinsic growth rate r in absence of predation. The second equation shows that predators grow logistically with intrinsic growth rate s and carrying capacity proportional to the prey populations size u(t). The parameter h is the number of prey required to support one predator at equilibrium, when v(t) equals u(t)/h. The term hv(t)/u(t) of this equation is called the Leslie-Gower term. This interesting formulation for the predator dynamics has been discussed by Leslie and Gower in [8, 9]. τ is incorporated in the negative feedback of the predator density. αuv/(a+bu+cv) is Beddington-DeAngelis functional response. It is known that the Beddington-DeAngelis form of functional response has desirable qualitative features of ratio-dependent form but takes care of their controversial behaviors at low densities [10]. For more details on the background of this functional response, we refer to [10–12].For convenience, a nondimensional form of system (1.1) will be useful. By defining t~=rt, u~=u(t)/K, v~=αv(t)/rK, and dropping the tildes for the sake of simplicity, model (1.1) becomes the following model:(1.2)du(t)dt=u(t)(1-u(t))-u(t)v(t)a1+bu(t)+c1v(t),dv(t)dt=v(t)[δ-βv(t-τ)u(t-τ)],
where δ=s/r, β=sh/α, a1=a/K, c1=cr/α, τ~=rτ. Lu and Liu [7] proved the system (1.2) is permanent under some appropriate conditions and investigated the local and global stability of the equilibria.In the earlier literature, most population models are often formulated by ordinary differential equations with or without time delays [1, 2, 13–18]. It is well known that the distribution of species is generally heterogeneous spatially, and therefore the species will migrate towards regions of lower population density to add the possibility of survival. Thus, partial differential equations with delay became the subject of a considerable interest in recent years. For a detailed theory and applications of delay equations with diffusion arising in biological and ecological problems, we refer to [19–23]. Therefore, time delays and spatial diffusion should be considered simultaneously in modeling biological interactions. Thus, the growth dynamics of two species corresponding to system (1.2) should be described by the following diffusion system with delay:
(1.3)∂u(t,x)∂t=d1Δu(t,x)+u(t,x)[1-u(t,x)-v(t,x)a1+bu(t,x)+c1v(t,x)],t>0,x∈(0,π),∂v(t,x)∂t=d2Δv(t,x)+v(t,x)[δ-βv(t-τ,x)u(t-τ,x)],t>0,x∈(0,π),∂u(t,x)∂x=∂v(t,x)∂x=0,t≥0,x=0,π,u(t,x)=ϕ(t,x)≥0,v(t,x)=ψ(t,x)≥0,(t,x)∈[-τ,0]×(0,π),
where u(t,x) and v(t,x) can be interpreted as the densities of prey and predator populations at time t and space x, respectively; d1>0, d2>0 denote the diffusion coefficients of prey and predator two species, respectively; Δ is the Laplacian operator; Neumann boundary conditions in (1.3) imply that two species have zero flux across the domain boundary. (ϕ,ψ)∈ℂ=C([-τ,0],X), and X defined by
(1.4)X={(u,v):u,v∈W2,2(0,π):dudx=dvdx=0,x=0,π}
with the inner product <·,·>.In the remaining part of this paper, we focus on system (1.3). The main purpose of this paper is to consider the effects of the delay and diffusion on the dynamics of system (1.3).The organization of this paper is as follows. In Section2, we consider the stability of the positive constant steady-state solutions and the existences of Hopf bifurcations of surrounding the positive constant steady-state solutions. In particular, we show the existence of spatially nonhomogeneous periodic solutions while the system parameters are all spatially homogeneous. In Section 3, we present that the emergence of these spatially nonhomogeneous periodic solutions is clearly due to the effect of the small diffusivity. Finally, we study the properties of the spatially nonhomogeneous periodic solutions applying normal form theory of PFDEs.


## 2. Stability and Hopf Bifurcations

In this section, we investigate the stability of the positive constant steady state of (1.3) and obtain the conditions under which (1.3) undergoes a Hopf bifurcation.It is easy to see that the solutions of system (1.2) have a unique boundary equilibrium E1(1,0) and a unique positive equilibrium E*(u*,v*), where
(2.1)u*=-(a1-b-c1δ/β+δ/β)+Λ2(b+c1δ/β),v*=δβx*,Λ=(a1-b-c1δ/β+δ/β)2+4a1(b+c1δ/β).
Obviously, E1(1,0) and E*(u*,v*) are also the spatially homogeneous steady-state solutions of system (1.3). From the point of view of biology, we should consider system (1.3) in the closed first quadrant in the (u,v) plane, that is, the positive constant steady-state solutions E*(u*,v*) of system (1.3).Letu¯(t,x)=u(t,x)-u*;  v¯(t,x)=v(t,x)-v*, for convenience, we use u(t,x) and v(t,x) to replace u¯(t,x) and v¯(t,x), respectively; then system (1.3) can be transformed into
(2.2)∂u(t,x)∂t=d1Δu(t,x)+α11u(t,x)+α12v(t,x)+∑i+j≥2‍1i!j!fij(1)ui(t,x)vj(t,x),∂v(t,x)∂t=d2Δv(t,x)+α21u(t-τ,x)+α22v(t-τ,x)+∑i+j+l≥2‍1i!j!l!fijl(2)ui(t-τ,x)vj(t-τ,x)vl(t,x),
where
(2.3)α11:=f10(1)=-u*+bu*v*(a1+bu*+c1v*)2,α12:=f01(1)=-u*(a1+bu*)(a1+bu*+c1v*)2<0,α21:=f100(2)=δ2β>0,α22:=f010(2)=-δ<0,fij(1)=∂i+jf(1)∂ui∂vj|(u*,v*),fijl(2)=∂i+j+lf(2)∂ui∂vj∂v1l|(u*,v*,v*),i,j,l≥0,f(1)=u(1-u)-uva1+bu+c1v,f(2)=v1(δ-βvu).
Therefore, the positive constant stationary solution E*(u*,v*) of system (1.3) can be transformed into the origin of system (2.2).Letu1(t)=u(t,·),u2(t)=v(t,·),U(t)=(u1(t),u2(t))T; therefore, system (2.2) can be rewritten as an abstract form in the phase space ℂ=C([-τ,0],X):
(2.4)U˙(t)=dΔU(t)+L(Ut)+f(Ut),
where d=(d1,d2)T,Δ=(∂/∂x200∂/∂x2),Ut(θ)=U(t+θ), -τ≤θ≤0, L:ℂ→X and f:ℂ→X are given, respectively, by
(2.5)L(φ)=(α11φ(0)+α12φ(0)α21φ1(-τ)+α22φ2(-τ)),f(φ)=(∑i+j≥2‍1ijfij(1)φ1iφ2j∑i+j+l≥2‍1ijlfijl(2)φ1i(-τ)φ2j(-τ)φ2l(0)),
for φ=(φ1,φ2)T∈ℂ, φ(θ)=Ut(θ),   -τ≤θ≤0.Linearizing (2.4) at (0,0) gives the linear equation
(2.6)U˙(t)=dΔU(t)+L(Ut),
whose characteristic equation is
(2.7)λy-dΔy-L(eλ⋅y)=0,
where y∈dom(Δ)∖{0} and dom(Δ)⊂X.It is well known that the linear operatorΔ on (0,π) with homogeneous Neumann boundary conditions has the eigenvalues -k2(k∈ℕ0={0,1,2,…}), and the corresponding eigenfunctions are
(2.8)βk1=(γk0),βk2=(0γk),γk=cos⁡(kx)‖cos⁡(kx)‖2,2,k∈N0.
Notice that (βk1,βk2)k=0∞ construct an orthogonal basis of the Banach space X. Therefore L(βk1,βk2)⊂span{βk1,βk2}, and thus any element y in X can be expanded a Fourier series in the form
(2.9)y=∑k=0∞‍YkT(βk1βk2),(2.10)YkT=(〈y,βk1〉〈y,βk2〉).In addition, some easy computations can show that(2.11)L(φT(βk1βk2))=[L(φ)]T(βk1βk2),
for φ=(φ1,φ2)T∈ℂ.From (2.9) and (2.11), (2.7) is equivalent to
(2.12)∑k=0∞‍YkT[(λ+d1k200λ+d2k2)-(α11α12α21e-λτα22e-λτ)](βk1βk2)=0.
Thus λ is a characteristic root of (2.7) if and only if for k∈ℕ0, λ satisfies
(2.13)λ2+Akλ+Bk+(-α22λ+Ck)e-λτ=0,
where
(2.14)Ak=d1k2+d2k2-α11,Bk=d1d2k4-α11d2k2,Ck=α11α22-α12α21-d1k2α22>0,α11α22-α12α21=δu*+a1δ2u*β(a1+bu*+c1v*)2>0.
When τ=0, (2.13) reduces to the following quadratic equation with respect to λ:
(2.15)λ2+(Ak-α22)λ+Bk+Ck=0.
If α11<0, then Ak>0,Bk≥0, and
(2.16)Ak-α22>0,Bk+Ck>0,
since Ck>0.Therefore, it is obvious that all roots of equations (2.15) have negative real parts, and we can conclude that the positive constant steady state E*(u*,v*) of system (2.2) is locally asymptotically stable in the absence of delay when α11<0. Thus, we can have the following conclusions.Theorem 2.1.
Suppose that the conditionα11<0 is satisfied. Then  (i)
all roots of each equation in (2.15) have negative real parts for any wave number k,(ii)
for any wave numberk, the positive constant steady-state solution E*(u*,v*) of system (1.3) is locally asymptotically stable in the absence of delay.In the following, we discuss the effects of delayτ on the stability of the trivial solution of (2.2). Notice that iω(ω>0) is a root of (2.13) if and only if for a certain k∈ℕ0, ω satisfies the following equation:
(2.17)-ω2+Akωi+Bk+(-α22ωi+Ck)(cos⁡ωτ-isin⁡ωτ)=0.
Thus
(2.18)ω4+(Ak2-2Bk-α222)ω2+Bk2-Ck2=0,k∈N0.Lettingω2=z, then (2.18) can be written as
(2.19)z2+(Ak2-2Bk-α222)z+Bk2-Ck2=0,k∈N0.
Equation (2.19) with k=0 has only one positive real root:
(2.20)z0=(A02-α222)2+4C02-(A02-α222)2>0,
where C0=α11α22-α12α21=δu*+(a1δ2u*/β(a1+bu*+c1v*)2)>0.In addition, from (2.17) and (2.20), we have
(2.21)cos⁡(ω0τ)=A0α22(ω0)2+C0(ω0)2α222(ω0)2+C02,
where ω0=z0. Thus
(2.22)τj0=1ω0{arccosA0α22(ω0)2+C0(ω0)2α222(ω0)2+C02+2jπ},j∈N0.
Denote(H)
(d12+d22)-2d1α11>α222-α112 and d1d2+(d1α22-d2α11)>α11α22-α12α21.Theorem 2.2.
Assume that the conditions(H) and α11<0 hold. For τ=τj0, (2.13) with k=0 has a pair of purely imaginary eigenvalues ±ω0i and there are no other roots of (2.13) with zero real parts.Proof.
Assumingλ=iω*, ω*>0 is a solution of (2.13) with k≥1. From (2.17), (2.18), and (2.19), we get
(2.23)ω*4+(Ak2-2Bk-α222)ω*2+Bk2-Ck2=0,k∈N={1,2,…},
so we have
(2.24)ω*2=-(Ak2-2Bk-α222)±(Ak2-2Bk-α222)2-4(Bk2-Ck2)2,k∈N.
Clearly, if Ak2-2Bk-α222>0 and Bk2-Ck2>0, there are no ω* such that (2.13) with k≥1 has purely imaginary roots ±ω*i.
By computing, we have(2.25)Ak2-2Bk-α222=(d12+d22)k4-2d1α11k2+α112-α222,Bk=d1d2k4-α11d2k2,Ck=α11α22-α12α21-d1k2α22>0.
In addition, according toα11<0, we have Bk≥0. It is clear that (d12+d22)k4−2d1α11k2+α112-α222≥(d12+d22)-2d1α11+α112-α222 when k≥1(α11<0). Furthermore, if d1d2+(d1α22-d2α11)>α11α22-α12α21, we can get Bk2-Ck2>0 when k≥1. Therefore, (2.13) with k≥1 has no purely imaginary roots when the conditions (H) and α11<0 hold. Thus the proof of Theorem 2.2 is accomplished.Let(2.26)λ(τ)=σ(τ)+iω(τ)
be a root of (2.13) with k=0 near τ=τj0 satisfying σ(τj0)=0, ω(τj0)=ω0,j∈N0.Then the following result holds.Lemma 2.3.
The following transversality conditions hold:(2.27)dReλ(τj0)dτ>0,j∈N0.From the previous discussions, we have the following theorem on the stability of positive steady-state solution(u*,v*) of system (1.3) and the existence of Hopf bifurcation near (u*,v*).Theorem 2.4.
Assume that the conditions(H) and α11<0 hold. Then  (i)
ifτ∈[0,τj0), the positive constant steady state (u*,v*) of (1.3) is asymptotically stable;(ii)
ifτ>τj0, the positive constant steady state (u*,v*) of (1.3) is unstable;(iii)
τ=τj0 are Hopf bifurcation values of system (1.3), and these Hopf bifurcations are all spatially homogeneous.


## 3.   Effect of Small Diffusivity

In the previous section, we have obtained the conditions under which spatially homogeneous Hopf bifurcations bifurcate from the positive steady-state solutionsE*=(u*,v*) of system (1.3) when the parameter τ crosses through the critical value τj0. In this sense, we say that the diffusion terms do not have effect on the Hopf bifurcations. In this section, we discuss the effect of small diffusivity on Hopf bifurcations for system (1.3) when the condition (H) is not satisfied. For the simplicity of discussion which follows, throughout this section, we always suppose that the condition (H1): d1d2+(d1α22-d2α11)>α11α22-α12α21 holds.Assumeλ=iωk(ωk>0) is a solution of (2.13) with k≥1. From the discussion in Section 2, we have
(3.1)ωk4+(Ak2-2Bk-α222)ωk2+Bk2-Ck2=0,k∈N.
If the condition (H) is not satisfied, and  (H2)
(Ak2-2Bk-α222)2-4(Bk2-Ck2)≥0, then (2.13) with k≥1 has roots ±iωk, where
(3.2)ωk=-(Ak2-2Bk-α222)±(Ak2-2Bk-α222)2-4(Bk2-Ck2)2.
From the discussion in Section 2, we know that there exists k0>0, k0∈ℕ such that (2.13) with k≥1 has only characteristic roots with negative real parts when k>k0 [24].In addition, from (2.17), we have
(3.3)cos⁡(ωkτ)=Akα22(ωk)2-BkCk+Ck(ωk)2α222(ωk)2+Ck2,k∈N.
Thus
(3.4)τjk=1ωk(arccosAkα22(ωk)2-BkCk+Ck(ωk)2α222(ωk)2+Ck2+2jπ),j∈N0,k∈N.In particular, it is easy to know from(H1) that Bk>Ck when α11<0. By the same way in Theorem 2.2, we can see if
(3.5)(d12+d22)16-8d1α11>α222-α112,
then (2.13) with k≥2 has no purely imaginary eigenvalues.Suppose the condition(H) is not satisfied, that is, (d12+d22)-2d1α11<α222-α112, assume further (H2) satisfy. Then (2.13) with k=1 has a pair of purely imaginary eigenvalues iω1, and all other zeros have negative real parts, where ω1 is given by
(3.6)ω1=-(A12-2B1-α222)±(A12-2B1-α222)2-4(B12-C12)2.
Therefore, we can obtain the following.Lemma 3.1.
Suppose thatα11<0 and (d12+d22)16-8d1α11>α222-α112>(d12+d22)-2d1α11. If the condition (H2) holds, then (2.13) with k=1 has a simple pair of purely imaginary roots ±iω1 and all other roots except ±iω1 have strictly negative real parts, where ω1 is defined by (3.6).For system (1.3), by the similar discussion to that of Theorem 2.2, when τ crosses through the critical values τj1, where
(3.7)τj1=1ω1(arccosA1α22(ω1)2-B1C1+C1(ω1)2α222(ω1)2+C12+2jπ),j∈N0,
it can give rise to Hopf bifurcation at the positive constant steady state (u*,v*). By the results in [22], bifurcating periodic solutions of (1.3) at τ=τj1 are spatially nonhomogeneous.Therefore, we have the following conclusion.Theorem 3.2.
If the conditions in Lemma3.1 are satisfied, then τ=τj1 are Hopf bifurcation values of system (1.3), and these Hopf bifurcations are all spatially nonhomogeneous, where τj1 is defined by (3.7).In general, we have the following.Theorem 3.3.
Suppose thatα11<0, if there exist k0>0, k0∈ℕ such that
(3.8)(d12+d22)k04-d1α11k02>α222-α112>(d12+d22)-2d1α11,
and (H2) holds, then (2.13) with k=k0 has purely imaginary roots iωk0 and system (1.3) has a family of spatially nonhomogeneous periodic solutions bifurcating from the spatially homogeneous steady state (u*,v*), when τ crosses through the critical values τjk0, where ωk0 and τjk0 are defined by (3.2) and (3.4) with k=k0, k0∈ℕ, respectively.From Theorems2.2 and 3.3, we can know that large diffusivity has no effect on the Hopf bifurcation, while small diffusivity can lead to the fact that the system bifurcates spatially nonhomogeneous periodic solutions at the positive constant steady state under which the system parameters are all spatially homogeneous. These exhibit that the emergence of these spatially nonhomogeneous periodic solutions is clearly due to the effect of the small diffusivity.


## 4. Properties of Hopf Bifurcation

In Theorem3.2, we have obtained the conditions under which a family of spatially nonhomogeneous periodic solutions bifurcates from the spatially homogeneous steady-state solutions E*=(u*,v*) of system (1.3) when the parameter τ crosses through the critical value τj1. In this section, we redefine an inner product to study the properties of the spatially nonhomogeneous Hopf bifurcation applying normal form theory of PFDEs by developed [22, 25].Normalizing the delayτ in system (2.2) by the time-scaling t→t/τ, (2.2) is transformed into
(4.1)∂u(t,x)∂t=τ{d1Δu(t,x)+α11u(t,x)+α12v(t,x)+∑i+j≥2‍1i!j!fij(1)ui(t,x)vj(t,x)},∂v(t,x)∂t=τ{∑i+j+l≥2‍d2Δv(t,x)+α21u(t-1,x)+α22v(t-1,x)+∑i+j+l≥2‍1i!j!l!fijl(2)ui(t-1,x)vj(t-1,x)vl(t,x)},
where f(1), f(2) are defined by (2.2). Letting τ=τj1+α,j∈ℕ0, then, (4.1) can be written in abstract form in 𝒞=C([-1,0]:X) as
(4.2)ddtU(t)=τj1dΔU(t)+L(τj1)(Ut)+F(Ut,α),
where d=(d1,d2)T, L(τj1)(·):𝒞→X, F(·,α):𝒞×ℝ+→X are given by
(4.3)L(τj1)(φ)=τj1(α11φ1(0)+α12φ2(0)α21φ1(-1)+α22φ2(-1)),F(φ,α)=αdΔφ(0)+L(α)φ+f(φ,α),f(φ,τj1)=(τj1+α)(∑i+j≥2‍1i!j!fij(1)φ1i(0)φ2j(0)∑i+j+l≥2‍1i!j!l!fijl(2)φ1i(-1)φ2j(-1)φ2l(0)),
for φ=(φ1,φ2)T∈𝒞.Linearizing (4.2) at (0,0) leads to the following linear equation:
(4.4)ddtU(t)=τj1dΔU(t)+L(τj1)(Ut).LetΛ1={-iω1,iω1}; consider the following FDE on C([-1,0],X):
(4.5)z˙(t)=τj1dΔz(t)+L(τj1)(zt),
that is,
(4.6)(z˙1(t)z˙2(t))=τj1{(α11-d1α120-d2)(z1(t)z2(t))+(00α21α22)(z1(t-1)z2(t-1))}.Obviously,L(τj1) is a continuous linear function mapping C([-1,0],X) into X. According to the Riesz representation theorem, there exists a 2×2 matrix function η(θ,τ). -1≤θ≤0, whose elements are of bounded variation such that
(4.7)L(τj1)(ϕ)=∫-10‍dη(θ,τj1)ϕ(θ)forϕ∈C.
Thus, we can choose
(4.8)η(θ,τj1)={τj1(α11-d1α120-d2),θ∈(-1,0],τj1(00α21α22),θ=-1,
then (4.7) is satisfied.LettingA(τj1) denote the infinitesimal generator of strongly continuous semigroup, according to [2], then,
(4.9)A(τj1)ϕ(θ)={dϕ(θ)dθ,θ∈[-1,0),L(τj1)(ϕ)=def∫-10dη(t,τj1)ϕ(t),θ=0,
where ϕ∈C1([-1,0],X).Forψ∈C1([0,1],(X)*), define
(4.10)A*ψ(s)={-dψ(s)ds,s∈(0,1],∫-10‍ψ(-t)dη(t,τj1),s=0
and a bilinear inner product of the Sobolev space W2.2(0,π):
(4.11)(ψ(s),ϕ(θ))=ψ(0)ϕ(0)-∫-10‍∫ξ=0θ‍ψ(ξ-θ)dη(θ)ϕ(ξ)dξ=ψ(0)ϕ(0)-τj1∫-10‍ψ(θ+1)(00α21α22)ϕ(θ)dθ,
where η(θ)=η(θ,τj1) and A* are the formal adjoint of A(τj1).It is easy to see from Section2 that A(τj1) has a pair of simple purely imaginary eigenvalues ±iω1 and they are also eigenvalues of A* since A(τj1) and A* are adjoint operators. Let 𝒫 and 𝒫* be the center spaces, that is, the generalized eigenspaces, of A(τj1) and A* associated with Λ1, respectively. Then 𝒫* is the adjoint space of 𝒫 and dim𝒫=dim𝒫*=2.In addition, according to [22, 25], by a few simple calculations, we can choose Φ and Ψ be the bases for 𝒫 and 𝒫*, respectively. It is known that Φ˙=ΦB, where B is the I×I diagonal matrix B=(iτj100iτj1).LetΦ=(Φ1,Φ2) and Ψ=(Ψ1,Ψ2)T, where
(4.12)Φ1(θ)=(1,ξ)Teiω1τj1θ,Φ2(θ)=Φ1(θ)¯,-1≤θ≤0,Ψ1(s)=1ρ(1,ζ)e-iω1τj1s,Ψ2(s)=Ψ1(s)¯,0≤s≤1,ξ=iω1-α11+d1α12,ζ=-iω1+α11-d1α21e-iω1τj1,ρ¯=(1+ξζ)-τj1(-d1+α11+ζα21+ξα12-d2ξζ+ξζα22)e-iω1τj1.From the above expression, we can easily see that(Ψ1,Φ1)=1, (Ψ1,Φ¯1)=0.Letf1=(β11,β12),  c·f1 be defined by c·f1=c1β11+c2β12 for c=(c1,c2)T∈ℝ2 and (ψ·f1)(θ)=ψ(θ)·f1 for θ∈[-1,0]. Then the center space of linear equation (4.4) is given by 𝒫CN𝒞, where
(4.13)PCNφ=Φ(Ψ,<φ,f1>)⋅f1,φ∈C
and 𝒞=𝒫CN𝒞⊕𝒫Q𝒞; here 𝒫Q𝒞 denotes the complementary subspace of 𝒫CN𝒞 in 𝒞.LetAτj1 be defined by
(4.14)Aτj1φ(θ)=φ˙(θ)+X0(θ)[τj1Δφ(0)+L*(τj1)(φ(θ))-φ˙(0)],φ∈C,
where X0: [-1,0]→B(X,X) is given by
(4.15)X0={0,θ∈[-1,0),I,θ=0.ThenAτj1 is the infinitesimal generator induced by the solution of (4.4) and (4.2) and can be rewritten as the following operator differential equation:
(4.16)U˙t=Aτj1Ut+X0F(Ut,α).
Using the decomposition 𝒞=PCN𝒞⊕𝒫Q𝒞 and (4.13), the solution of (4.16) can be written as
(4.17)Ut=Φ(x1(t)x2(t))⋅f1+h(x1,x2,α),
where (x1,x2)T=(Ψ,<Ut,f1>), and h(x1,x2,α)∈𝒫Q𝒞 with h(0,0,0)=Dh(0,0,0)=0.Thus, we describe the flow on the center manifold for (4.2) as
(4.18)Ut*=Φ(x1(t)x2(t))⋅f1+h(x1,x2),
where h(x1,x2)=h(x1,x2,0).Lettingz=x1-ix2 and Ψ(0)=(Ψ1(0),Ψ2(0))T, when α=0, then z satisfies
(4.19)z˙=iω1τj1z+g(z,z¯),
where
(4.20)g(z,z¯)=(Ψ1(0)-iΨ2(0))〈F(Ut,0),f1〉,(4.21)w(z,z¯)=h(z+z¯2,(z-z¯)i2,0),(4.22)w(z,z¯)=w20z22+w11zz¯+w02z¯22+w21z2z¯2+⋯.Noticing thatp1=Φ1+iΦ2, therefore, solutions of (4.16) can be rewritten as
(4.23)Ut*=12Φ((z+z¯)i(z-z¯))⋅f1+w(z,z¯)=12(p1z+p2z¯)⋅f1+w(z,z¯).
In addition, (4.19) can be rewritten as the following form:
(4.24)z˙=iω1τj1z+g20z22+g11zz¯+g02z¯22+g21z2z¯2+⋯.
Let
(4.25)g(z,z¯)=g20z22+g11zz¯+g02z¯22+g21z2z¯2+⋯.
From (4.20), we have
(4.26)〈F(Ut,0),f1〉=τj14((ξf11(1)+12f20(1)+12ξ2f02(1))z2e-2iω1τj1(ξf110(2)+eiω1τj1ξf101(2)+eiω1τj1ξ2f011(2)+12f200(2)+12ξ2f020(2))z2)+τj14([(ξ¯+ξ)f11(1)+f20(1)+ξξ¯f02(1)]zz¯[(ξ¯+ξ)f110(2)+e-iω1τj1ξ¯(f101(2)+ξf011(2))+eiω1τj1ξ(f101(2)+ξ¯f011(2))+f200(2)+ξξ¯f020(2)]zz¯)+τj14((ξ¯f11(1)+12f20(1)+12ξ¯2f02(1))z¯2e2iω1τj1(ξ¯f110(2)+e-iω1τj1ξ¯f101(2)+e-iω1τj1ξ¯2f011(2)+12f200(2)+12ξ¯2f020(2))z¯2)+τj12(〈f11(1)(w112(0)+w202(0)2+w111(0)ξ+w201(0)2ξ¯)+f20(1)(w111(0)+w201(0)2)+f02(1)(w111(0)ξ+w201(0)2ξ¯),1〉〈f110(2)e-iω1τj1(w112(-1)+e2iω1τj1w202(-1)2+w111(-1)ξ+e2iω1τj1w201(-1)2ξ¯)+f101(2)(e-iω1τj1w112(0)+eiω1τj1w202(0)2+w111(-1)ξ+w201(-1)2ξ¯)+f011(2)(e-iω1τj1w112(0)ξ+eiω1τj1w202(0)2ξ¯+w112(-1)ξ+w202(-1)2ξ¯)+12f200(2)(2e-iω1τj1w111(-1)+eiω1τj1w201(-1))+12f020(2)(2e-iω1τj1w112(-1)ξ+eiω1τj1w202(-1)ξ),1w201(-1)2ξ¯〉)z2z¯+⋯.Noting thatΨ1(0)-iΨ2(0)=(2(1-iω1)/(1+ω12)(1+ξζ))(1,ζ), therefore,
(4.27)g20=τj1(1-iω1)(1+ω12)(1+ξζ)[(ξf11(1)+12f20(1)+12ξ2f02(1))+e-2iω1τj1(ξf110(2)+eiω1τj1ξf101(2)+eiω1τj1ξ2f011(2)+12f200(2)+12ξ2f020(2))ζ],g11=τj1(1-iω1)(1+ω12)(1+ξζ){[(ξ¯+ξ)f11(1)+f20(1)+ξξ¯f02(1)]+[(ξ¯+ξ)f110(2)+e-iω1τj1ξ¯(f101(2)+ξf011(2))+eiω1τj1ξ(f101(2)+ξ¯f011(2))+f200(2)+ξξ¯f020(2)]ζ},g02=g20¯g21=2τj1(1-iω1)(1+ω12)(1+ξζ)×[〈f11(1)(w112(0)+w202(0)2+w111(0)ξ+w201(0)2ξ¯)+f20(1)(w111(0)+w201(0)2)+f02(1)(w111(0)ξ+w201(0)2ξ¯),1〉+〈f110(2)e-iω1τj1(w112(-1)+e2iω1τj1w202(-1)2+w111(-1)ξ+e2iω1τj1w201(-1)2ξ¯)+f101(2)(e-iω1τj1w112(0)+eiω1τj1w202(0)2+w111(-1)ξ+w201(-1)2ξ¯)+f011(2)(e-iω1τj1w112(0)ξ+eiω1τj1w202(0)2ξ¯+w112(-1)ξ+w202(-1)2ξ¯)+12f200(2)(2e-iω1τj1w111(-1)+eiω1τj1w201(-1))+12f020(2)(2e-iω1τj1w112(-1)ξ+eiω1τj1w202(-1)ξ),1(w112(-1)+e2iω1τj1w202(-1)2+w111(-1)ξ+e2iω1τj1w201(-1)2ξ¯)〉ζ].
Since w20(θ) and w11(θ) for (θ∈[-1,0]) appear in g21, we still need them.It follows easily from (4.22) that
(4.28)w˙(z,z¯)=w20zz˙+w11(z˙z¯+zz˙¯)+w02z¯z˙¯+⋯,(4.29)Aτj1w=Aτj1w20z22+Aτj1w11zz¯+Aτj1w02z¯22+⋯.
According to [22] we can know,
(4.30)w˙=Aτj1w+H(z,z¯),
where
(4.31)H(z,z¯)=H20z22+H11zz¯+H02z¯22+⋯=X0F(Ut*,0)-Φ(Ψ,〈X0F(Ut*,0),f1〉)⋅f1
and Hij∈𝒫Q𝒞,i+j=2.Thus, by using the chain rule(4.32)w˙=∂w(z,z¯)∂zz˙+∂w(z,z¯)∂z¯z¯˙.From (4.23) and (4.30), we can obtain
(4.33)(2iω1τj1-Aτj1)w20=H20,-Aτj1w11=H11,(-2iω1τj1-Aτj1)w02=H02.
Noticing that Aτj1 has only two eigenvalues ±iω1, therefore, (4.33) has the unique solution wij(i+j=2) in 𝒫Q𝒞 and
(4.34)w20=(2iω1τj1-Aτj1)-1H20,w11=-Aτj1-1H11,w02=(-2iω1τj1-Aτj1)-1H02.
Note that for -1≤θ<0,
(4.35)H(z,z¯)=-Φ(θ)Ψ(0)〈F(Ut,0),f1〉⋅f1=-(p1(θ)+p2(θ)2,p1(θ)-p2(θ)2i)(Ψ1(0)Ψ2(0))〈F(Ut,0),f1〉⋅f1=-12[p1(θ)(Ψ1(0)-iΨ2(0))+p2(θ)(Ψ1(0)+iΨ2(0))]〈F(Ut,0),f1〉⋅f1=-14[g20p1(θ)+g¯02p2(θ)]z2⋅f1-12[g11p1(θ)+g¯11p2(θ)]zz¯⋅f1.
So, for -1≤θ<0,
(4.36)H20(θ)=-12[g20p1(θ)+g¯02p2(θ)]⋅f1,H11(θ)=-12[g11p1(θ)+g¯11p2(θ)]zz¯⋅f1,H20(0)=τj12(ξf11(1)+12f20(1)+12ξ2f02(1)e-2iω1τj1(ξf110(2)+eiω1τj1ξf101(2)+eiω1τj1ξ2f011(2)+12f200(2)+12ξ2f020(2)))-12[g20p1(0)+g¯02p2(0)]⋅f1,H11(0)=τj12((ξ¯+ξ)f11(1)+f20(1)+ξξ¯f02(1)(ξ¯+ξ)f110(2)+e-iω1τj1ξ¯(f101(2)+ξf011(2))+eiω1τj1ξ(f101(2)+ξ¯f011(2))+f200(2)+ξξ¯f020(2))-12[g11p1(0)+g¯11p2(0)]⋅f1.
By the definition of Aτj1, for -1≤θ<0, we have
(4.37)w˙20(θ)=2iω1w20(θ)+12[g20p1(θ)+g¯02p2(θ)]⋅f1,-1≤θ<0.
Note that p1(θ)=p1(0)eiω1θ,-1≤θ≤0; hence
(4.38)w20(θ)=[ig20ω1τj1p1(θ)+ig¯023ω1τj1p2(θ)]⋅f1+e2iω1θE1,-1≤θ<0,w11(θ)=[g11p1(θ)+g¯11p2(θ)]⋅f1+E2,-1≤θ<0,E1=w20(0)-[ig20ω1τj1p1(0)+ig¯023ω1τj1p2(0)]⋅f1,E2=w11(0)-[ig20ω1τj1p1(0)+ig¯02ω1τj1p2(0)]⋅f1.Using the definition ofAτj1 again and combining (4.29) and (4.33), we get
(4.39)2iω1τj1[ig20ω1τj1p1(0)⋅f1+ig¯023ω1τj1p2(0)⋅f1+E1]-τj1Aτj1[ig20ω1p1(0)⋅f1+ig¯023ω1p2(0)⋅f1+E1]-L(τj1)[ig202ω1p1(θ)⋅f1+ig¯026ω1p2(θ)⋅f1+E1e2iω1θ]=τj12(ξf11(1)+12f20(1)+12ξ2f02(1)e-2iω1τj1(ξf110(2)+eiω1τj1ξf101(2)+eiω1τj1ξ2f011(2)+12f200(2)+12ξ2f020(2)))-12[g20p1(0)+g¯02p2(0)]⋅f1.
As
(4.40)τj1Aτj1p1(0)⋅f1+L(τj1)(p1(θ)⋅f1)=iω1τj1p1(0)⋅f1,τj1Aτj1p2(0)⋅f1+L(τj1)(p2(θ)⋅f1)=-iω1τj1p2(0)⋅f1,
then
(4.41)2iω1τj1E1-τj1Aτj1-L(τj1)(E1e2iω1τj1θ)=τj12(ξf11(1)+12f20(1)+12ξ2f02(1)e-2iω1τj1(ξf110(2)+eiω1τj1ξf101(2)+eiω1τj1ξ2f011(2)+12f200(2)+12ξ2f020(2))).
From the above expression, we can see easily that
(4.42)E1=12(-2iω1+α11+d1α12α21-2iω1+α22+d2)-1×(ξf11(1)+12f20(1)+12ξ2f02(1)e-2iω1τj1(ξf110(2)+eiω1τj1ξf101(2)+eiω1τj1ξ2f011(2)+12f200(2)+12ξ2f020(2))).
Similarly
(4.43)E2=12(α11+d1α12α21α22+d2)-1×((ξ¯+ξ)f11(1)+f20(1)+ξξ¯f02(1)(ξ¯+ξ)f110(2)+e-iω1τj1ξ¯(f101(2)+ξf011(2))+eiω1τj1ξ(f101(2)+ξ¯f011(2))+f200(2)+ξξ¯f020(2)).Thus,g21 can be determined by the parameters and delay; we get
(4.44)c1(0)=i2ω1(g11g20-2|g11|2-|g022|3)+g212.Then, we can compute the following values:(4.45)σ2=Re(c1(0))Re(λ′(τj1)),ϵ2=2Re(c1(0)),T2=Im⁡(c1(0))+σ2Im⁡(λ′(τj1))w1.
Therefore, we have the following result.Theorem 4.1.
(i)σ2 determines the directions of the spatially nonhomogeneous Hopf bifurcation. If σ2 < 0 (>0), then the spatially nonhomogeneous Hopf bifurcation is subcritical (supercritical).
(ii)ϵ2 determines the stability of bifurcated periodic solutions. If ϵ2 < 0 (>0), then the bifurcated periodic solutions are stable (unstable).
(iii)T2 determines the period of the bifurcating periodic solutions; if T2 < 0 (>0), the period decreases (increases).


## 5. Conclusions

In this paper, we considered a delayed predator-prey system with diffusion effects. By investigating the linearized system of the original system, the distribution of the roots of the characteristic equations at the positive constant steady-state solution was obtained and its stability was discussed. The obtained results indicate that the positive constant steady-state solution of the system is asymptotically stable whenτ∈[0,τj0). As the delay τ crosses through each τj0, there exist a sequence of critical values τj0(j=0,1,2,…) of τ such that the system undergoes a Hopf bifurcation at the positive constant steady-state solution. Besides, we show that large diffusivity has no effect on the Hopf bifurcation, while small diffusivity can lead to the fact that the system can bifurcate a spatially nonhomogeneous periodic solutions at the positive constant steady-state solution. Furthermore, we study the properties of the spatially nonhomogeneous periodic solutions. The conclusions demonstrate that system (1.3) may have more complex and richer dynamics than system (1.1).



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*Source: 856725-2012-07-18.xml*