# Analysis of Effects of Delays and Diffusion on a Predator-Prey System

Authors: Gaoxiang Yang; Fuchen Zhang
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

A reaction-diffusion predator-prey system with two delays is investigated. It is found that the spatially homogeneous periodic solution will occur when the sum of two delays crosses some critical values and Hopf bifurcation takes place. For the fixed domain and diffusion, some numerical simulations are also given to illustrate the theoretical analysis. In addition, special attention is paid to effects of diffusion on the bifurcating periodic solution. It is found that the diffusion would lead to the bifurcating period solution to destabilize by calculating the relevant expression of the Floquet exponent.


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## Body

## 1. Introduction

It is well known that the reaction-diffusion equations with delays are usually used to describe the biological system. Some results have proved that diffusion and delay take the very important role in the biological systems and can induce many spatiotemporal patterns (see monographs by Wu [1], Arino [2], and Murray [3]). Spatiotemporal patterns of predator-prey population models with diffusion and delay have been studied by many authors [4–9] in recent years. In particular, when the biological system with a single delay has the different boundary conditions such as Neumann boundary and Dirichlet boundary, the spatial homogeneous Hopf bifurcation and the spatially nonhomogeneous Hopf bifurcation near the spatially uniform equilibrium of the predator-prey system have been studied by some authors [10–13].Faria [14] and Chen [15] have studied Lotka-Volterra type prey-predator model with two delays as the single delay varies (1)∂ut,x∂t=d1Δut,x+ut,xr1-a11ut,x-a12vt-τ2,x,∂vt,x∂t=d2Δvt,x+vt,x-r2+a21ut-τ1,x-a22vt,x,where r1, r2, a12, a21, τ1, τ2, a11, a22, d1, d2 are positive constants and have the following biological interprets, respectively. r1 is the birth rate of the prey; r2 is the death rate of the predator; a12 and a21 represent the strength of the relative effects of the interaction on the two interspecies; a11 and a22 denote the strength of the interaction of the intraspecies; τ1 is reaction time of the prey to the predator and τ2 is capture time of the predator.d i i = 1,2 are the diffusion coefficients of prey and predator species, respectively. The variables ut,x and vt,x are densities of population of the prey and the predator at time t and space x, respectively. Δ denotes the Laplacian operator in RNN≥1. System (1) with a single delay varying has the existence of the spatially homogeneous and nonhomogeneous Hopf bifurcation. However, in order to investigate joint effects of the two delays, we will mainly consider effects of the sum of two delays and diffusion on the species in system (1). For convenience, the new variables are introduced as follows:(2)u~t,x=ut-τ1,x,v~t,x=vt,x,τ=τ1+τ2.System (1) can be transformed into the following form and meanwhile dropping the tildes(3)∂ut,x∂t=d1Δut,x+ut,xr1-a11ut,x-a12vt-τ,x,∂vt,x∂t=d2Δvt,x+vt,x-r2+a21ut,x-a22vt,x,which has the following initial conditions:(4)ut,x=ϕt,x≥0,t,x∈-τ,0×Ω-,vt,x=φt,x≥0,t,x∈-τ,0×Ω-,where Ω is a bounded open domain in RNN≥1 with a smooth boundary ∂Ω and the following no flux boundary condition: (5)∂ut,x∂n=∂vt,x∂n=0,t≥0,x∈∂Ω,where n is the outward unit normal vector on the boundary.In what follows, we investigate effects of the delayτ and diffusion on the dynamics of (3) with initial conditions (4) and boundary conditions (5), respectively. We also assume that ϕ,φ∈C=C[-τ,0],X and X is defined by (6)X=u,v∈W2,2Ω:∂ut,x∂t=∂vt,x∂t=0,x∈∂Ω,with the inner product ·,·. In addition, for convenience we restrict ourselves to the one-dimensional spatial domain Ω=0,π throughout this paper.In this paper, we not only consider the bifurcation phenomenon of system (1) as the sum of the two delays varies but also investigate effects of diffusion on the bifurcating periodic solutions. We find that as the sum of two delays crosses some critical values, the bifurcating periodic solution would occur through the spatially homogeneous Hopf bifurcation. In addition, once we change the value of diffusion and fix the domain, we also find that the diffusions of the species can destabilize the bifurcating stable periodic solution under certain conditions.The rest of the paper is organized as follows. In Section2, the local analysis is given by taking the sum of two delays as parameter to discuss stability of the positive constant equilibrium and the existence of local Hopf bifurcation. In Section 3, by employing the theory of the center manifold and normal formal theory about the partial functional differential equation developed by Wu in [1], the bifurcation direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are pointed out. In order to testify theoretical analysis results, some numerical simulation figures are also given. In Section 4, by using the Fredholm alternative theory about the periodic solution, we also investigate the effects of diffusion on the bifurcating periodic solution and obtain the conditions which determine the stability of the bifurcating periodic solution. Finally, some discussions and conclusions are drawn in Section 5.


## 2. Linear Stability Analysis and the Existence of Hopf Bifurcation

Obviously, ifr1a21-r2a11>0, system (1) or (3) has a positive equilibrium point E∗=u∗,v∗, where(7)u∗=r1a22+r2a12a11a22+a12a21,v∗=r1a21-r2a11a11a22+a12a21.Let u=u∗+u-,v=v∗+v-, substitute them into system (3), and meanwhile drop the bars for simplicity of notations, then system (3) can be transformed into the following vector form:(8)∂U∂t=dΔu+BU+CUt-τ+N,where(9)U=uv,d=d100d2,B=b11b12b21b22,C=c11c12c21c22,N=N1N2,and(10)b11=r1-2a11u∗-a12v∗=-a11u∗,b12=0,b21=a21v∗,b22=-r2-2a22v∗+a21u∗=-a22v∗,c11=0,c12=-a12u∗,c21=0,c22=0,N1=-a11u2-a12vt-τu,N2=a21uv-a22v2.The vector form of the corresponding linearization system of system (8) is as follows:(11)∂U∂t=dΔu+BU+CUt-τ.The characteristic equation of linearization system (11) of system (8) at the equilibrium point E∗ has the following form:(12)λ+k2d1-b11-c11e-λτ-b12-c12e-λτ-b21-c21e-λτλ+k2d2-b22-c22e-λτ=0,i.e.,(13)Dkλ,τ=λ2+pkλ+rk+qke-λτ=0,where(14)pk=d1+d2k2+a11u∗+a22v∗,rk=d1d2k4+d1a22v∗+d2a11u∗k2+a11a22u∗v∗,qk=a12a21u∗v∗.Case 1.
Whenτ=0, the associated characteristic equation (13) becomes(15)Dkλ,0=λ2+pkλ+rk+qk=0If qk+rk>0, the corresponding characteristic equation (15) has two roots with negative part for any k∈N0.Case 2.
Whenτ≠0, let us get the conditions of taking place spatially homogeneous Hopf bifurcation. It is known that when k=0 the associated characteristic equation D0λ,τ=0 becomes the following form:(16)λ2+p0λ+r0+q0e-λτ=0,(17)i.e.Dkλ,τ=λ2+pλ+q+ne-λτ=0.Let ±iωω>0 be roots of characteristic equation (13), then(18)-ω2+p0ωi+r0+q0cosωτ-iq0sinωτ=0.Separating the real and imaginary parts, we have(19)ω2-r0=q0cos⁡ωτ,p0ω=q0sin⁡ωτ,which leads to (20)ω4+p20-2r0ω2+r20-q20=0.By the simple analysis, we can immediately obtain the following results:
(a) Ifp20-2r0>0,r20-q20>0 or Δ=p20-2r02-4r20-q20<0, then (20) has no positive real root.
(b) Ifp20-2r0<0,r20-q20>0 and Δ>0, then (20) has two positive roots as follows:(21)ω±∗=-p20-2r0±p20-2r02-4r20-q202.(c) If r20-q20<0 or p20-2r0<0, and Δ=0, then (20) has only one positive root ω+∗=-p20-2r0/2.
From the direct calculations and testing the above conditions, we can get that ifa11a22-a12a21<0 and d1d2+d1a22v∗+d2a11u∗+a11a22u∗v∗-a12a21u∗v∗≥0, then (20) has only one positive root ω∗ defined by(22)ω∗2=-a112u∗2+a222v∗2+a112u∗2-a222v∗22+4a122a212u∗2v∗22,and the corresponding (23)τj∗=1ω∗arccos⁡ω∗2-a11a22u∗v∗a12a21u∗v∗+2jπω∗,where j=0,1,2,....Then when τ=τj∗, the characteristic equation has a pair of purely imaginary roots ±iω∗.Lemma 1.
(1) Ifa11a22-a12a21≥0 holds, then (15) and (17) have the same number roots with positive real parts for all τ≥0 for any k≥0.
(2) Ifa11a22-a12a21<0 and d1d2+d1a22v∗+d2a11u∗+a11a22u∗v∗-a12a21u∗v∗≥0 hold, then (15) and (17) have the same number roots with positive real parts for τ∈0,τ0∗ for any k≥0.Lemma 2.
Suppose that(24)a11a22-a12a21<0,d1d2+d1a22v∗+d2a11u∗+a11a22u∗v∗-a12a21u∗v∗≥0hold and τ=τj∗, then signRedλ/dττ=τj∗>0.Proof.
Denote byλτ=ατ+iωτ the root of characteristic equation such that (25)ωτj∗=ω∗.Substituting λτ into the characteristic equation and taking the derivative with respect to τ, we have (26)dλdτ-1=2λ+p0eλτλq0-τλ,which leads to (27)signRedλdτ-1τ=τj∗=signRe2λ+p0eλτλq0τ=τj∗=sign1q202ω∗2-2r0+p20=signa112u∗2-a222v∗2+4a122a212u∗2v∗2a122a212u∗2v∗2>0.This proof is completed.Therefore, employing Lemma 1, Lemma 2, and Hopf bifurcation theorem for partial functional differential equations in [1], we can obtain the following conclusions.Theorem 3.
(1) Ifa11a22-a12a21≥0 holds, then the coexistence positive equilibrium E∗of system (8) is locally asymptotically stable for all τ≥0.
(2) Ifa11a22-a12a21<0 and d1d2+d1a22v∗+d2a11u∗+a11a22u∗v∗-a12a21u∗v∗≥0 hold, then the coexistence positive equilibrium E∗of system (8) is locally asymptotically stable for τ∈0,τ0∗ and unstable when τ>τ0∗.
(3) Ifa11a22-a12a21<0 and d1d2+d1a22v∗+d2a11u∗+a11a22u∗v∗-a12a21u∗v∗≥0 hold, then system (8) undergoes the Hopf bifurcation at the coexistence positive equilibrium E∗for τ=τj∗.Remark 4.
Fork0=0, this bifurcation is called spatially homogeneous Hopf bifurcation. For k0≠0, this bifurcation is called spatially nonhomogeneous Hopf bifurcation. This spatially homogeneous Hopf bifurcation is the same as the nondispersal equation in [16] if we impose the restrictive diffusion condition as follows:(28)d1d2+d1a22v∗+d2a11u∗+a11a22u∗v∗-a12a21u∗v∗≥0.


## 3. Direction and Stability of the Bifurcating Periodic Solution

In this section, we mainly devote our interests to the properties of the bifurcating periodic solution, including the direction of bifurcation and stability of the bifurcating periodic solution.Lett=τt~, and α=τ-τ∗, substitute them into system (8), and meanwhile drop the tildes above t, then system (8) can be denoted in the form of abstract partial functional differential equations(29)U˙t=α+τ∗dΔU+α+τ∗LUt+α+τ∗NUt=τ∗dΔU+τ∗LUt+αdΔU+αLUt+α+τ∗NUt,where(30)LUt=b11b12b21b22utvt+c11c12c21c22ut-1vt-1,NUt=N1tN2t=-a11u2-a12vt-1ua21uv-a22v2.For convenience, we rewrite system (29) in the following form:(31)U˙t=τ∗dΔU+L∗Ut+N∗Utwhere (32)L∗Ut=τ∗LUt,N∗Ut=αdΔU+αLUt+α+τ∗NUt.And the corresponding linearization system of system (29) becomes(33)U˙t=τ∗dΔU+L∗Ut.From the discussion of Section 2, we know that 0,0 is an equilibrium point of system (29), and the characteristic equation of (33) has a pair of simple purely imaginary eigenvalues ±iω∗τ∗ when τ=τ∗, k=0.Consider the ordinary functional differential equation:(34)u˙t=-τ∗k2dut+L∗ut,and by the Riesz representation theorem, there exists a 2×2 matrix function ηθ,τ∗-1≤θ≤0, whose elements are of bounded variation such that(35)-τ∗k2dϕ0+L∗ϕ=∫-10dηθϕθforϕ∈C-1,0,R2.In fact, we can choose(36)ηθ,τ∗=τ∗-d1k2+b11b12b21-d2k2+b22,θ=0,0,θ∈-1,0,-τ∗c11c12c21c22,θ=-1.Let Aτ∗ denote the infinitesimal generators of the semigroup induced by the solutions of (34) and A∗τ∗ be the formal adjoint operator of Aτ∗ under the following bilinear functional:(37)φθ,ψθ=∑i=12ψi0¯φi0-∫-10∫0θψξ+1¯dηθφξdξ=∑i=12ψi0¯φi0+τ∗∫-10ψξ+1¯Cφξdξfor φ∈C-1,0,R2,ψ∈C0,1,R2. From the discussion of Section 2, we know that Aτ∗ and A∗τ∗ have a pair of simple pure imaginary roots ±iω∗τ∗. Let p1θ and p2θ be eigenvector of Aτ∗ about eigenvalues ±iω∗τ∗, respectively, and q1θ and q2θ be eigenvector of A∗τ∗ about eigenvalues ±iω∗τ∗, respectively.Lemma 5.
(38) Let ξ = a 21 v ∗ i ω ∗ τ ∗ + a 22 v ∗ , η = a 11 u ∗ - i ω ∗ τ ∗ a 21 v ∗ ,then(39)p1θ=eiωτ∗θ1ξ,p2θ=p1θ¯,-1≤θ≤0,is a basis of P with Λ0 and(40)q1θ=1D-q1∗,q1∗=eiωτ∗θ1,η,q2θ=q1θ¯,0≤θ≤1,is a basis of Q with Λ0, where D=p1θ,q1∗s, P and Q are the center subspace. That is to say, P and Q are the generalized eigenspace of operators Aτ∗ and A∗τ∗ associated with Λ0, respectively, dim⁡P=dim⁡Q=2.Proof.
According to the definition of operatorsAτ∗ and A∗τ∗, we have the following equations: (41)iω∗τ∗1ξ=b11b12b21b221ξ+e-iω∗τ∗c11c12c21c221ξ,-iω∗τ∗1ηT=1ηTb11b12b21b22+eiω∗τ∗1ηTc11c12c21c22.We can obtain the values of ξ and η,(42)ξ=a21v∗iω∗τ∗+a22v∗,η=a11u∗-iω∗τ∗a21v∗.Let(43)Φ=p1θ,p2θ,Ψ=q1θq2θ,such that (44)Ψ,Φ=p1θq1θp1θq2θp2θq1θp2θq2θ=I2×2.The proof is completed.In addition,fk=βk1βk2 and c·fk=c1βk1+c2βk2, for any c=c1c2∈C2. The center subspace of linear equation (33) is given by PCNI, where(45)PCNIϕ=ΦΨ,ϕ,fk·fk,ϕ∈I,and I=PCNI⊕PSI, where PSI denotes the complement subspace of PCNI in I.LetAτ∗ be the infinitesimal generator induced by the solution of (33). Then (29) can be rewritten as the abstract form (46)u˙t=Aτ∗ut+Rα,ut,where Rα,ut=0,θ∈-1,0,Nut,α,θ=0.Using the decompositionI=PCNI⊕PSI and (45), the solution of (31) can be written as(47)ut=Φx1tx2t·fk+hx1,x2,α,where x1tx2t=Ψ,ut,fk and hx1,x2,α∈PSI,h0,0,0=0,Dh0,0,0=0.In particular, the solution of (31) on the center manifold is given by(48)ut=Φx1tx2t·fk+hx1,x2,0.Let z=x1-ix2, Ψ0=Ψ10Ψ20, and notice that p1=Φ1+iΦ2. Then (49)Φx1tx2t·fk=Φ1,Φ2z+z-2iz-z-2·fk=12p1z+p-1z-·fk.Then system (48) can be transformed into (50)ut=12p1z+p-1z-·fk+wz,z-,where (51)wz,z-=hz+z-2,iz-z-2,0=w20θz22+w11θzz-+w02θz-22+w30θz36+⋯,wijθ∈C-1,0,Q.Moreover by [1], z satisfies (52)z˙=iωτ∗z+gz,z-,where (53)gz,z-=1D-1,ηNut,0,fk=g20z22+g11zz-+g02z-22+g21z2z-2+⋯.From (31) and (48), it follows that (54)Ut=Φztz-t·fk+Wz,z-=12p1z+p2z-·fk+W20θz22+W20θzz-+W02θz-22+W30θz36+⋯=12eiωτ∗θz+e-iωτ∗θz-ξeiωτ∗θz+ξ-e-iωτ∗θz-·fk+W20θz22+W20θzz-+W02θz-22+W30θz36+⋯=12eiωτ∗θz+e-iωτ∗θz-βk1+12ξeiωτ∗θz+ξ-e-iωτ∗θz-βk2+W20θz22+W20θzz-+W02θz-22+W30θz36+⋯=12eiωτ∗θz+e-iωτ∗θz-γk0+12ξeiωτ∗θz+ξ-e-iωτ∗θz-0γk+W201θW202θz22+W111θW112θzz-+W021θW022θz-22+⋯We can obtain the following results:(55)ut0=12γkz+12γkz-+w2010z22+w1110zz-+⋯,vt0=12γkξz+12γkξ-z-+w2020z22+w1120zz-+⋯,vt-1=12e-iωτ∗γkξz+12eiωτ∗γkξ-z-+w202-1z22+w112-1zz-+⋯,ut20=14γk2z2+12γk2zz-+14γk2z-2+w1110γk+12w2010γkz2z-+⋯,ut0ut2-1=14e-iωτ∗ξγk2z2+14e-iωτ∗ξγk2+14eiωτ∗ξ-γk2zz-+14eiωτ∗ξ-γk2z-2+12w112-1γk+12w1110e-iωτ∗ξγk+14w202-1γk+14w2010eiωτ∗ξ-γkz2z-+⋯,ut0vt0=14ξγk2z2+14ξγk2+14ξ-γk2zz-+14ξ-γk2z-2+12γkw1120+14γkw2020+12ξγkw1110+14ξ-γkw2010z2z-+⋯vt20=14ξ2γk2z2+12ξξ-γk2zz-+14ξ-2γk2z-2+ξγkw1120+12ξ-γkw2020z2z-+⋯,N1Ut=-a11ut20-a12ut0vt-1,N2Ut=a21ut0vt0-a22vt20,NUt=N1UtN2Ut=-a11ut20-a12ut0vt-1a21ut0vt0-a22vt20,NUt,0,fk=NUt,0,βk1NUt,0,βk2=1π∫0πN1Ut,0γkdx∫0πN2Ut,0γkdx.According to the following equality,(56)gz,z-=q10N∗Ut,0,fk=q10NUt,0,fk,we can obtain the following results:(57)g20=1πtermz2∫0πN1Ut,0γkdx+η∫0πN2Ut,0γkdx,g11=1πtermzz-∫0πN1Ut,0γkdx+η∫0πN2Ut,0γkdx,g02=1πtermz-2∫0πN1Ut,0γkdx+η∫0πN2Ut,0γkdx,g21=1πtermz2z-∫0πN1Ut,0γkdx+η∫0πN2Ut,0γkdx,where termz2,termzz-termz-2termz2z- denote the coefficients of the z2, zz-, z-2, z2z- in the polynomial 1/D-∫0πN1Ut,0γkdx+η/D-∫0πN2Ut,0γkdx about z and z-, respectively.Next, in order to get the value ofg21, we need to compute the expression of W20θ,W11θ. It follows from (50) that (58)w˙z,z-=w˙zz˙+w˙z-z-˙=w20zz˙+w11z˙z-+w11zz-˙+w02z-z-˙+⋯,(59)Aτ∗w=Aτ∗w20θz22+Aτ∗w11θzz-+Aτ∗w02θz-22+Aτ∗w30θz36+⋯.In addition, by [1], wz,z- satisfies (60)w˙t=Aτ∗w+Hz,z-,θ=Aτ∗w+X0N∗Ut,0-ΦΨ,X0N∗Ut,0,fk·fk,(61)whereHz,z-,θ=H20z22+H11θzz-+H02z-22+⋯.Thus from (50), (52), (58), (59), (60), and (61) we can obtain the following results:(62)2iω∗τ∗-Aτ∗W20θ=H20θ,-Aτ∗W11θ=H11θ,-2iω∗τ∗-Aτ∗W02θ=H02θ.From (60)–(62), when -1≤θ<0, we have(63)W20θ=ig20ω∗τ∗qθ·fk+ig-023ω∗τ∗qθ¯·fk+E1e2iω∗τ∗θ,W11θ=-ig11ω∗τ∗qθ·fk+ig-11ω∗τ∗qθ¯·fk+E2.Meanwhile when θ=0, employing the definition of operator Aτ∗and (60), we have (64)2iω∗τ∗-Aτ∗W20θ=H20θ,-Aτ∗W11θ=H11θ,-2iω∗τ∗-Aτ∗W02θ=H02θ.Combining (63) and (64), we can get the following results:(65)2iω∗τ∗ig20ω∗τ∗q0·fk+ig-023ω∗τ∗q0¯·fk+E1-Δig20ω∗τ∗q0·fk+ig-023ω∗τ∗q0¯·fk+E1-L∗ig20ω∗τ∗qθ·fk+ig-023ω∗τ∗qθ¯·fk+E1e2iω∗τ∗θ=-g20q0+g02¯q0¯·fk+N1z2N2z2,and(66)-Δ-ig11ω∗τ∗q0·fk+ig-11ω∗τ∗q0¯·fk+E2-L∗-ig11ω∗τ∗qθ·fk+ig-11ω∗τ∗qθ¯·fk+E2=-g11q0+g11¯q0¯·fk+N1zz-N2zz-.Meanwhile note the following equalities:(67)τ∗dΔp10·fk+Lτ∗p10·fk=iωτ∗p10·fk,τ∗dΔp20·fk+Lτ∗p20·fk=-iωτ∗p20·fk.Associating with (65)–(67), we can obtain the following two equalities:(68)-τ∗dΔE2-L∗E2=N1zz-N2zz-,(69)2iωτ∗E1-τ∗dΔE1-L∗E1e2iωτ∗θ=N1z2N2z2.Thus from (68) and (69), we can get the values of E1and E2, (70)E1=2iωτ∗+d1k2+a11u1∗a12u1∗e-2ω∗τ∗-a21u2∗2iωτ∗+d2k2+a22u2∗-1N1z2N2z2,E2=d1k2+a11u1∗a12u1∗-a21u2∗d2k2+a22u2∗-1N1zz-N2zz-.Summarizing the above analysis, we can obtain the expression of W20θ and W11θ and get the value of g21. According to the normal form theory developed by Wu [1], we can get the normal form of system (8).Theorem 6 ([1]).
(71) ξ ˙ = λ μ ξ + C μ ξ ξ 2 + o ξ ξ , μ 3 ,where(72)C0=i2ω∗τ∗g20g11-2g11-13g022+g212.Further, the following several terms which determine the properties of bifurcating periodic solution are given as follows:μ2=-ReC0/τ∗Reλ′τ∗ which determines the direction of bifurcation.β2=2ReC0 which determines the stability of the bifurcating periodic solution.T2=-ImC0+μ2Imλ′τ∗/ω∗τ∗ which determines the variation of period of the bifurcating periodic solution.For system (8), employing the above discussion, we can obtain the following further results.Theorem 7.
(1) IfReC0>0, then the bifurcating periodic solution exists in the side of τ<τ0∗and is unstable.
(2) IfReC0<0, then the bifurcating periodic solution exists in the side of τ>τ0∗and is stable.Next we give the corresponding numerical results and set the parameters as follows:(73)d1=d2=1,r1=0.6,r2=0.5,a11=0.5,a12=0.5,a21=0.8,a22=0.2.We can get the equilibrium E∗=0.74,0.46 and τ0∗=2.725. Figures 1 and 2 present that the equilibrium of system (3) is locally asymptotically stable when the sum of two delays is less than τ0∗. But when the sum of two delays is greater than τ0∗, the equilibrium point of system (3) will be unstable and lead to the spatially homogeneous periodic solution to occur; see Figures 3 and 4.Figure 1
The trajectory of prey densities versus time t and position x with the initial conditionu=2,v=1 when τ=2<τ0∗.Figure 2
The trajectory of predator densities versus time t and position x with the initial conditionu=2,v=1 when τ=2<τ0∗.Figure 3
The trajectory of prey densities versus time t and position x with the initial conditionu=2,v=1 when τ=3>τ0∗.Figure 4
The trajectory of prey densities versus time t and position x with the initial conditionu=2,v=1 when τ=3>τ0∗.


## 4. Effects of Diffusion on the Bifurcating Periodic Solution

In the previous section, we mainly discuss the spatially homogeneous Hopf bifurcation and the relevant properties of the bifurcating periodic solution for the fixed domain and diffusion. However, in this present section, we will employ the method based on Fredholm alternative to investigate the effects of diffusions on the bifurcating periodic solution. For simplicity, we take the same notations of [1].From the previous section, we know thatp1θ(-1≤θ≤0) is the eigenfunction of Nϕ0=A(0) corresponding to the simple eigenvalue iω∗τ∗, that is, A0p1=iω∗τ∗p1; and q1θ(0≤θ≤1) is the eigenfunction of A∗(0) corresponding to the simple eigenvalue -iω∗τ∗, that is, A∗0q1=-iω∗τ∗q1, where (74)p1=p10eiω∗τ∗θ=1ξeiω∗τ∗θ-1≤θ≤0,(75)q1=q10eiω∗τ∗θ=1D-1ηTeiω∗τ∗θ0≤θ≤1.Now we defineζ, ζ^∈C-1,0;R2 by(76)2iω∗τ∗-A0ζ=12Nϕϕ0p1,p1,-A0ζ^=Nϕϕ0p1,p-1,where ζ has the form(77)ζθ=ζ0e2iω∗τ∗θ-1≤θ≤0,and ζ0 is determined by (78)2iω∗τ∗I-L0e2iω∗τ∗.ζ0=12Nϕϕ0p1,p1,and ζ^ is a constant 2-vector satisfying (79)-L0ζ^=Nϕϕ0p1,p-1.With the above preparations, we can now discuss the stability of pt,ε as a solution of (29). First of all, we note that the linear variation equation of (29) about pt,ε is given by(80)dztdt=dΔzt+Nϕμε,pt.,εzt.Adopting the new variables(81)s=ωεt,ωs=zsωε,we can transform (80) into (82)ωεddsωs=dΔωs+Nϕμε,ys,ωε.,εωs,ωs,where ys,ε=ps/ωε;ε, ωs,ωεθ=ωs+ωεθ,-1≤θ≤0.Letλm be the mth eigenvalue of the vector -Δ and let φm be the eigenfunction corresponding to λm, i.e.,(83)Δφm=-λmφm,m=1,2,…,0=λ1≤λ2≤λ3≤⋯.Clearly, γ is a Floquet exponent of the linear 2π-periodic system (80) if and only if there exists a function qs≠0 and a positive integer m such that (84)ωsx=eγs/ωεqsφmxsatisfies (80). So calculating the Floquet exponent γ is reduced to finding a solution ys to (80) in the form (84). Substituting (84) into (80) and comparing the coefficients of φm on both sides of (80), we obtain (85)ωsddsqs=-γ+λmdqs+Nϕμε,ys,ωε·,εqs,ωε·eγ·.Equation (85) is independent of the spatial variable x.Whenm=1, (85) coincides with the one induced from the linearized equation of the diffusion-free equation, that is, (86)ωsddsqs=-γqs+Nϕμε,ys,ωε·,εqs,ωε·eγ·.From the above equations, we know (86) has Floquet exponents 0 and β<0.Next we mainly consider the casem≠1 in (85). Take any positive integer m, greater than 1, and fix it. Let us parameterize the deformation of a domain and diffusion coefficients in terms of the same parameter ε as above. In what follows, we assume the parameterization is taken in such a way that (87)λmd=ε2E,for some matrix E. Then (85) can be written as (88)ωsddsqs=-γ+ε2Eqs+Nϕμε,ys,ωε·;εqs,ωε·eγ·.Now in (88), we will find a real value function qs and a real number γ in the following form:(89)γ=γ2ε2+γ^εε2,γ^0=0,qs;ε=q0s;ε+q1s;εε+q^s;εε,where γ is the Floquet exponent of the linear periodic system (80) whose sign determines the stability of the periodic solution pt;ε. We regard (89) as a perturbation of (86) with perturbation term (87).According to the theory in [1] developed by Wu, after a slightly lengthy but simple calculation, we can obtain the following form expression about γ2: (90)γ22+2ReEp10,q10-B1γ2+Ep10,q102-2ReB-1Ep10,q10=0,which determines the stability of the bifurcating periodic solution as the diffusion coefficients vary, where(91)B1=Nϕϕ0p1θ,ζ^,q10+Nϕϕ0p-1θ,ζ,q10+12Nϕϕϕ0p1θ,p1θ,p-1θ,q10.Combining the above formulas (74), (75), (77), (78), (79), (87), we can obtain the expression of (90) whose coefficients are determined by the parameters from the original system (29).


## 5. Discussions

In this paper, we mainly investigate effects of the sum of two delays and diffusion on the dynamical behaviors of the Lotka-Volterra type predator-prey model with two delays. By employing theories of Hopf bifurcation for some fixed diffusions and spatial domain, we find that delays can cause the spatially homogeneous equilibrium point to destabilize and lead to the spatially homogeneous periodic solution to occur. On the other hand, we also investigate effects of diffusions on the bifurcation periodic solution and find that once the value of diffusion varies the bifurcating stable periodic solution may be unstable. As Morita [17] had pointed out, diffusion would lead to other spatiotemporal patterns to occur such as the periodic traveling wave, even spatiotemporal chaos, which will be studied in future work. So it is very important for us to consider the interaction of delay and diffusion for biological system (1) and explore the mechanism of all kinds of spatiotemporal patterns.



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