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

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).

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: 𝑑𝑢(𝑡)(𝑡)𝑢(𝑡)𝑑𝑡=𝑟𝑢1−𝐾−𝛼𝑢(𝑡)𝑣(𝑡)(𝑡)(𝑡),(𝑡)𝑎+𝑏𝑢+𝑐𝑣𝑑𝑣(𝑡𝑠𝑣()𝑑𝑡=𝑣)1−ℎ𝑡−𝜏𝑢(),𝑡−𝜏 (1.1) 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 𝑡̃=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:𝑑𝑢(𝑡)(𝑡(𝑡−𝑢(𝑡)𝑣(𝑡)𝑑𝑡=𝑢)(1−𝑢))𝑎₁+𝑏𝑢(𝑡)+𝑐₁𝑣(𝑡),𝑑𝑣(𝑡)(𝑡𝑣()𝑑𝑡=𝑣)𝛿−𝛽𝑡−𝜏𝑢(),𝑡−𝜏 (1.2) 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: 𝜕𝑢(𝑡,𝑥)𝜕𝑡=𝑑₁()(()−𝑣()Δ𝑢𝑡,𝑥+𝑢𝑡,𝑥)1−𝑢𝑡,𝑥𝑡,𝑥𝑎₁+𝑏𝑢(𝑡,𝑥)+𝑐₁𝑣()(),()𝑡,𝑥,𝑡>0,𝑥∈0,𝜋𝜕𝑣𝑡,𝑥𝜕𝑡=𝑑₂()(𝑣()Δ𝑣𝑡,𝑥+𝑣𝑡,𝑥)𝛿−𝛽𝑡−𝜏,𝑥𝑢()(),()𝑡−𝜏,𝑥,𝑡>0,𝑥∈0,𝜋𝜕𝑢𝑡,𝑥=()𝜕𝑥𝜕𝑣𝑡,𝑥𝑢()()()()()∈[]×(),𝜕𝑥=0,𝑡≥0,𝑥=0,𝜋,𝑡,𝑥=𝜙𝑡,𝑥≥0,𝑣𝑡,𝑥=𝜓𝑡,𝑥≥0,𝑡,𝑥−𝜏,00,𝜋 (1.3) 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 𝑋=(𝑢,𝑣)∶𝑢,𝑣∈𝑊^2,2(0,𝜋)∶𝑑𝑢=𝑑𝑥𝑑𝑣𝑑𝑥=0,𝑥=0,𝜋 (1.4) 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 Section 2, 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 𝑢^∗=−𝑎₁−𝑏−𝑐₁𝛿/𝛽+𝛿/𝛽+√Λ2𝑏+𝑐₁𝛿/𝛽,𝑣^∗=𝛿𝛽𝑥^∗,Λ=𝑎₁−𝑏−𝑐₁𝛿/𝛽+𝛿/𝛽²+4𝑎₁𝑏+𝑐₁𝛿/𝛽. (2.1) 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).

Let 𝑢(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 𝜕𝑢(𝑡,𝑥)𝜕𝑡=𝑑₁Δ𝑢(𝑡,𝑥)+𝛼₁₁𝑢(𝑡,𝑥)+𝛼₁₂𝑣(𝑡,𝑥)+_{𝑖+𝑗≥2}1𝑓𝑖!𝑗!^(1)𝑖𝑗𝑢^𝑖(𝑡,𝑥)𝑣^𝑗(),()𝑡,𝑥𝜕𝑣𝑡,𝑥𝜕𝑡=𝑑₂Δ𝑣(𝑡,𝑥)+𝛼₂₁𝑢(𝑡−𝜏,𝑥)+𝛼₂₂𝑣(𝑡−𝜏,𝑥)+_{𝑖+𝑗+𝑙≥2}1𝑓𝑖!𝑗!𝑙!^{(2)𝑖𝑗𝑙}𝑢^𝑖(𝑡−𝜏,𝑥)𝑣^𝑗(𝑡−𝜏,𝑥)𝑣^𝑙(𝑡,𝑥), (2.2) where 𝛼₁₁∶=𝑓⁽¹⁾₁₀=−𝑢^∗+𝑏𝑢^∗𝑣^∗𝑎₁+𝑏𝑢^∗+𝑐₁𝑣^∗2,𝛼₁₂∶=𝑓⁽¹⁾₀₁𝑢=−^∗𝑎₁+𝑏𝑢^∗𝑎₁+𝑏𝑢^∗+𝑐₁𝑣^∗2𝛼<0,₂₁∶=𝑓⁽²⁾₁₀₀=𝛿²𝛽>0,𝛼₂₂∶=𝑓⁽²⁾₀₁₀𝑓=−𝛿<0,_(1)𝑖𝑗=𝜕^𝑖+𝑗𝑓⁽¹⁾𝜕𝑢^𝑖𝜕𝑣^𝑗_{(𝑢∗,𝑣∗)},𝑓⁽²⁾_𝑖𝑗𝑙=𝜕^{𝑖+𝑗+𝑙}𝑓⁽²⁾𝜕𝑢^𝑖𝜕𝑣^𝑗𝜕𝑣^𝑙_{1(𝑢∗,𝑣∗,𝑣∗)}𝑓,𝑖,𝑗,𝑙≥0,₍₁₎=𝑢(1−𝑢)−𝑢𝑣𝑎₁+𝑏𝑢+𝑐₁𝑣,𝑓⁽²⁾=𝑣₁𝛿−𝛽𝑣𝑢. (2.3) Therefore, the positive constant stationary solution 𝐸^∗(𝑢^∗,𝑣^∗) of system (1.3) can be transformed into the origin of system (2.2).

Let 𝑢₁(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): 𝑈̇(𝑡)=𝑑Δ𝑈(𝑡)+𝐿𝑈_𝑡+𝑓𝑈_𝑡, (2.4) where 𝑑=(𝑑₁,𝑑₂)^𝑇,Δ=^{𝜕/𝜕𝑥20}_{0𝜕/𝜕𝑥2},𝑈_𝑡(𝜃)=𝑈(𝑡+𝜃), −𝜏≤𝜃≤0, 𝐿∶ℂ→𝑋 and 𝑓∶ℂ→𝑋 are given, respectively, by 𝐿𝜑=𝛼₁₁𝜑(0)+𝛼₁₂𝜑(0)𝛼₂₁𝜑₁(−𝜏)+𝛼₂₂𝜑₂(),𝑓𝜑=−𝜏_{𝑖+𝑗≥2}1𝑓𝑖𝑗^(1)𝑖𝑗𝜑^𝑖₁𝜑^𝑗_{2𝑖+𝑗+𝑙≥2}1𝑓𝑖𝑗𝑙^{(2)𝑖𝑗𝑙}𝜑^𝑖₁(−𝜏)𝜑^𝑗₂(−𝜏)𝜑^𝑙₂(0), (2.5) for 𝜑=(𝜑₁,𝜑₂)^𝑇∈ℂ, 𝜑(𝜃)=𝑈_𝑡(𝜃),   −𝜏≤𝜃≤0.

Linearizing (2.4) at (0,0) gives the linear equation 𝑈̇(𝑡)=𝑑Δ𝑈(𝑡)+𝐿𝑈_𝑡, (2.6) whose characteristic equation is 𝜆𝑦−𝑑Δ𝑦−𝐿𝑒^𝜆⋅𝑦=0, (2.7) where 𝑦∈dom(Δ)⧵{0} and dom(Δ)⊂𝑋.

It is well known that the linear operator Δ on (0,𝜋) with homogeneous Neumann boundary conditions has the eigenvalues −𝑘²(𝑘∈ℕ₀={0,1,2,…}), and the corresponding eigenfunctions are 𝛽¹_𝑘=𝛾_𝑘0,𝛽²_𝑘=0𝛾_𝑘,𝛾_𝑘=cos(𝑘𝑥)‖cos(𝑘𝑥)‖_2,2,𝑘∈ℕ₀. (2.8) Notice that (𝛽¹_𝑘,𝛽²_𝑘)^∞_𝑘=0 construct an orthogonal basis of the Banach space 𝑋. Therefore 𝐿(𝛽¹_𝑘,𝛽²_𝑘)⊂span{𝛽¹_𝑘,𝛽²_𝑘}, and thus any element 𝑦 in 𝑋 can be expanded a Fourier series in the form 𝑦=^∞_𝑘=0𝑌^𝑇_𝑘𝛽¹_𝑘𝛽²_𝑘𝑌_𝑇𝑘,=𝑦,𝛽¹_𝑘𝑦,𝛽²_𝑘. (2.9)(2.10)

In addition, some easy computations can show that 𝐿𝜑^𝑇𝛽¹_𝑘𝛽²_𝑘=𝐿𝜑^𝑇𝛽¹_𝑘𝛽²_𝑘, (2.11) for 𝜑=(𝜑₁,𝜑₂)^𝑇∈ℂ.

From (2.9) and (2.11), (2.7) is equivalent to ^∞_𝑘=0𝑌^𝑇_𝑘𝜆+𝑑₁𝑘²00𝜆+𝑑₂𝑘²−𝛼₁₁𝛼₁₂𝛼₂₁𝑒^−𝜆𝜏𝛼₂₂𝑒^−𝜆𝜏𝛽¹_𝑘𝛽²_𝑘=0. (2.12) Thus 𝜆 is a characteristic root of (2.7) if and only if for 𝑘∈ℕ₀, 𝜆 satisfies 𝜆²+𝐴_𝑘𝜆+𝐵_𝑘+−𝛼₂₂𝜆+𝐶_𝑘𝑒^−𝜆𝜏=0, (2.13) where 𝐴_𝑘=𝑑₁𝑘²+𝑑₂𝑘²−𝛼₁₁,𝐵_𝑘=𝑑₁𝑑₂𝑘⁴−𝛼₁₁𝑑₂𝑘²,𝐶_𝑘=𝛼₁₁𝛼₂₂−𝛼₁₂𝛼₂₁−𝑑₁𝑘²𝛼₂₂𝛼>0,₁₁𝛼₂₂−𝛼₁₂𝛼₂₁=𝛿𝑢^∗+𝑎₁𝛿²𝑢^∗𝛽𝑎₁+𝑏𝑢^∗+𝑐₁𝑣^∗2>0. (2.14) When 𝜏=0, (2.13) reduces to the following quadratic equation with respect to 𝜆: 𝜆²+𝐴_𝑘−𝛼₂₂𝜆+𝐵_𝑘+𝐶_𝑘=0. (2.15) If 𝛼₁₁<0, then 𝐴_𝑘>0,𝐵_𝑘≥0, and 𝐴_𝑘−𝛼₂₂𝐵>0,_𝑘+𝐶_𝑘>0, (2.16) since 𝐶_𝑘>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 condition 𝛼₁₁<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 number 𝑘, 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 delay 𝜏 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: −𝜔²+𝐴_𝑘𝜔𝑖+𝐵_𝑘+−𝛼₂₂𝜔𝑖+𝐶_𝑘(cos𝜔𝜏−𝑖sin𝜔𝜏)=0. (2.17) Thus 𝜔⁴+𝐴²_𝑘−2𝐵_𝑘−𝛼²₂₂𝜔²+𝐵²_𝑘−𝐶²_𝑘=0,𝑘∈ℕ₀. (2.18)

Letting 𝜔²=𝑧, then (2.18) can be written as 𝑧²+𝐴²_𝑘−2𝐵_𝑘−𝛼²₂₂𝑧+𝐵²_𝑘−𝐶²_𝑘=0,𝑘∈ℕ₀. (2.19) Equation (2.19) with 𝑘=0 has only one positive real root: 𝑧₀=𝐴²₀−𝛼²₂₂²+4𝐶²₀−𝐴²₀−𝛼²₂₂2>0, (2.20) where 𝐶₀=𝛼₁₁𝛼₂₂−𝛼₁₂𝛼₂₁=𝛿𝑢^∗+(𝑎₁𝛿²𝑢^∗/𝛽(𝑎₁+𝑏𝑢^∗+𝑐₁𝑣^∗)²)>0.

In addition, from (2.17) and (2.20), we have cos𝜔₀𝜏=𝐴₀𝛼₂₂𝜔₀²+𝐶₀𝜔₀²𝛼²₂₂𝜔₀²+𝐶²₀, (2.21) where 𝜔₀=√𝑧₀. Thus 𝜏⁰_𝑗=1𝜔₀𝐴arccos⁰𝛼₂₂𝜔₀²+𝐶₀𝜔₀²𝛼²₂₂𝜔₀²+𝐶²₀+2𝑗𝜋,𝑗∈ℕ₀. (2.22) Denote(𝐻)(𝑑²₁+𝑑²₂)−2𝑑₁𝛼₁₁>𝛼²₂₂−𝛼²₁₁ and 𝑑₁𝑑₂+(𝑑₁𝛼₂₂−𝑑₂𝛼₁₁)>𝛼₁₁𝛼₂₂−𝛼₁₂𝛼₂₁.

Theorem 2.2. Assume that the conditions (𝐻) and 𝛼₁₁<0 hold. For 𝜏=𝜏⁰_𝑗, (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. Assuming 𝜆=𝑖𝜔_∗, 𝜔_∗>0 is a solution of (2.13) with 𝑘≥1. From (2.17), (2.18), and (2.19), we get 𝜔⁴_∗+𝐴²_𝑘−2𝐵_𝑘−𝛼²₂₂𝜔²_∗+𝐵²_𝑘−𝐶²_𝑘=0,𝑘∈ℕ={1,2,…}, (2.23) so we have 𝜔²_∗=−𝐴²_𝑘−2𝐵_𝑘−𝛼²₂₂±𝐴²_𝑘−2𝐵_𝑘−𝛼²₂₂²−4𝐵²_𝑘−𝐶²_𝑘2,𝑘∈ℕ. (2.24) Clearly, if 𝐴²_𝑘−2𝐵_𝑘−𝛼²₂₂>0 and 𝐵²_𝑘−𝐶²_𝑘>0, there are no 𝜔_∗ such that (2.13) with 𝑘≥1 has purely imaginary roots ±𝜔_∗𝑖.
By computing, we have 𝐴²_𝑘−2𝐵_𝑘−𝛼²₂₂=𝑑²₁+𝑑²₂𝑘⁴−2𝑑₁𝛼₁₁𝑘²+𝛼²₁₁−𝛼²₂₂,𝐵_𝑘=𝑑₁𝑑₂𝑘⁴−𝛼₁₁𝑑₂𝑘²,𝐶_𝑘=𝛼₁₁𝛼₂₂−𝛼₁₂𝛼₂₁−𝑑₁𝑘²𝛼₂₂>0. (2.25)
In addition, according to 𝛼₁₁<0, we have 𝐵_𝑘≥0. It is clear that (𝑑²₁+𝑑²₂)𝑘⁴−2𝑑₁𝛼₁₁𝑘²+𝛼²₁₁−𝛼²₂₂≥(𝑑²₁+𝑑²₂)−2𝑑₁𝛼₁₁+𝛼²₁₁−𝛼²₂₂ when 𝑘≥1(𝛼₁₁<0). Furthermore, if 𝑑₁𝑑₂+(𝑑₁𝛼₂₂−𝑑₂𝛼₁₁)>𝛼₁₁𝛼₂₂−𝛼₁₂𝛼₂₁, we can get 𝐵²_𝑘−𝐶²_𝑘>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 𝜆(𝜏)=𝜎(𝜏)+𝑖𝜔(𝜏) (2.26) be a root of (2.13) with 𝑘=0 near 𝜏=𝜏⁰_𝑗 satisfying 𝜎(𝜏⁰_𝑗)=0, 𝜔(𝜏⁰_𝑗)=𝜔₀,𝑗∈𝑁₀.

Then the following result holds.

Lemma 2.3. The following transversality conditions hold: 𝑑Re𝜆𝜏⁰_𝑗𝑑𝜏>0,𝑗∈𝑁₀. (2.27)

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)if 𝜏∈[0,𝜏⁰_𝑗), the positive constant steady state (𝑢^∗,𝑣^∗) of (1.3) is asymptotically stable;(ii)if 𝜏>𝜏⁰_𝑗, the positive constant steady state (𝑢^∗,𝑣^∗) of (1.3) is unstable;(iii)𝜏=𝜏⁰_𝑗 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 solutions 𝐸^∗=(𝑢^∗,𝑣^∗) of system (1.3) when the parameter 𝜏 crosses through the critical value 𝜏⁰_𝑗. 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.

Assume 𝜆=𝑖𝜔_𝑘(𝜔_𝑘>0) is a solution of (2.13) with 𝑘≥1. From the discussion in Section 2, we have 𝜔⁴_𝑘+𝐴²_𝑘−2𝐵_𝑘−𝛼²₂₂𝜔²_𝑘+𝐵²_𝑘−𝐶²_𝑘=0,𝑘∈ℕ. (3.1) If the condition (𝐻) is not satisfied, and (𝐻₂)(𝐴²_𝑘−2𝐵_𝑘−𝛼²₂₂)²−4(𝐵²_𝑘−𝐶²_𝑘)≥0, then (2.13) with 𝑘≥1 has roots ±𝑖𝜔_𝑘, where 𝜔_𝑘=−𝐴²_𝑘−2𝐵_𝑘−𝛼²₂₂±𝐴²_𝑘−2𝐵_𝑘−𝛼²₂₂²−4𝐵²_𝑘−𝐶²_𝑘2. (3.2) 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 cos𝜔_𝑘𝜏=𝐴_𝑘𝛼₂₂𝜔_𝑘²−𝐵_𝑘𝐶_𝑘+𝐶_𝑘𝜔_𝑘²𝛼²₂₂𝜔_𝑘²+𝐶²_𝑘,𝑘∈ℕ. (3.3) Thus 𝜏^𝑘_𝑗=1𝜔_𝑘𝐴arccos^𝑘𝛼₂₂𝜔_𝑘²−𝐵_𝑘𝐶_𝑘+𝐶_𝑘𝜔_𝑘²𝛼²₂₂𝜔_𝑘²+𝐶²_𝑘+2𝑗𝜋,𝑗∈ℕ₀,𝑘∈ℕ. (3.4)

In particular, it is easy to know from (𝐻₁) that 𝐵_𝑘>𝐶_𝑘 when 𝛼₁₁<0. By the same way in Theorem 2.2, we can see if 𝑑²₁+𝑑²₂16−8𝑑₁𝛼₁₁>𝛼²₂₂−𝛼²₁₁, (3.5) then (2.13) with 𝑘≥2 has no purely imaginary eigenvalues.

Suppose the condition (𝐻) is not satisfied, that is, (𝑑²₁+𝑑²₂)−2𝑑₁𝛼₁₁<𝛼²₂₂−𝛼²₁₁, 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 𝜔₁=−𝐴²₁−2𝐵₁−𝛼²₂₂±𝐴²₁−2𝐵₁−𝛼²₂₂²−4𝐵²₁−𝐶²₁2. (3.6) Therefore, we can obtain the following.

Lemma 3.1. Suppose that 𝛼₁₁<0 and (𝑑²₁+𝑑²₂)16−8𝑑₁𝛼₁₁>𝛼²₂₂−𝛼²₁₁>(𝑑²₁+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 𝜏¹_𝑗, where 𝜏¹_𝑗=1𝜔₁𝐴arccos¹𝛼₂₂𝜔₁²−𝐵₁𝐶₁+𝐶₁𝜔₁²𝛼²₂₂𝜔₁²+𝐶²₁+2𝑗𝜋,𝑗∈ℕ₀, (3.7) 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=𝜏¹_𝑗 are spatially nonhomogeneous.

Therefore, we have the following conclusion.

Theorem 3.2. If the conditions in Lemma 3.1 are satisfied, then 𝜏=𝜏¹_𝑗 are Hopf bifurcation values of system (1.3), and these Hopf bifurcations are all spatially nonhomogeneous, where 𝜏¹_𝑗 is defined by (3.7).

In general, we have the following.

Theorem 3.3. Suppose that 𝛼₁₁<0, if there exist 𝑘₀>0, 𝑘₀∈ℕ such that 𝑑²₁+𝑑²₂𝑘⁴₀−𝑑₁𝛼₁₁𝑘²₀>𝛼²₂₂−𝛼²₁₁>𝑑²₁+𝑑²₂−2𝑑₁𝛼₁₁, (3.8) and (𝐻₂) holds, then (2.13) with 𝑘=𝑘₀ has purely imaginary roots 𝑖𝜔_𝑘0 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 𝜏^𝑘0_𝑗, where 𝜔_𝑘0 and 𝜏^𝑘0_𝑗 are defined by (3.2) and (3.4) with 𝑘=𝑘₀, 𝑘₀∈ℕ, respectively.

From Theorems 2.2 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 Theorem 3.2, 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 𝜏¹_𝑗. 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 delay 𝜏 in system (2.2) by the time-scaling 𝑡→𝑡/𝜏, (2.2) is transformed into 𝜕𝑢(𝑡,𝑥)𝑑𝜕𝑡=𝜏₁Δ𝑢(𝑡,𝑥)+𝛼₁₁𝑢(𝑡,𝑥)+𝛼₁₂𝑣(𝑡,𝑥)+_{𝑖+𝑗≥2}1𝑓𝑖!𝑗!^(1)𝑖𝑗𝑢^𝑖(𝑡,𝑥)𝑣^𝑗(),()𝑡,𝑥𝜕𝑣𝑡,𝑥𝑑𝜕𝑡=𝜏₂Δ𝑣(𝑡,𝑥)+𝛼₂₁𝑢(𝑡−1,𝑥)+𝛼₂₂𝑣(𝑡−1,𝑥)+_{𝑖+𝑗+𝑙≥2}1𝑓𝑖!𝑗!𝑙!^{(2)𝑖𝑗𝑙}𝑢^𝑖(𝑡−1,𝑥)𝑣^𝑗(𝑡−1,𝑥)𝑣^𝑙(𝑡,𝑥), (4.1) where 𝑓⁽¹⁾, 𝑓⁽²⁾ are defined by (2.2). Letting 𝜏=𝜏¹_𝑗+𝛼,𝑗∈ℕ₀, then, (4.1) can be written in abstract form in 𝒞=𝐶([−1,0]∶𝑋) as 𝑑𝑈(𝑡)𝑑𝑡=𝜏¹_𝑗𝑑Δ𝑈(𝑡)+𝐿𝜏¹_𝑗𝑈_𝑡+𝐹𝑈_𝑡,𝛼, (4.2) where 𝑑=(𝑑₁,𝑑₂)^𝑇, 𝐿(𝜏¹_𝑗)(⋅)∶𝒞→𝑋, 𝐹(⋅,𝛼)∶𝒞×ℝ⁺→𝑋 are given by 𝐿𝜏¹_𝑗𝜑=𝜏¹_𝑗𝛼₁₁𝜑₁(0)+𝛼₁₂𝜑₂(0)𝛼₂₁𝜑₁(−1)+𝛼₂₂𝜑₂(),𝐹(0)(𝛼),𝑓−1𝜑,𝛼=𝛼𝑑Δ𝜑+𝐿𝜑+𝑓𝜑,𝛼𝜑,𝜏¹_𝑗=𝜏¹_𝑗+𝛼^{𝑖+𝑗≥2}1𝑓𝑖!𝑗!^(1)𝑖𝑗𝜑^𝑖₁(0)𝜑^𝑗₂(0)_{𝑖+𝑗+𝑙≥2}1𝑓𝑖!𝑗!𝑙!^{(2)𝑖𝑗𝑙}𝜑^𝑖₁(−1)𝜑^𝑗₂(−1)𝜑^𝑙₂(0), (4.3) for 𝜑=(𝜑₁,𝜑₂)^𝑇∈𝒞.

Linearizing (4.2) at (0,0) leads to the following linear equation: 𝑑𝑈(𝑡)𝑑𝑡=𝜏¹_𝑗𝑑Δ𝑈(𝑡)+𝐿𝜏¹_𝑗𝑈_𝑡. (4.4)

Let Λ₁={−𝑖𝜔₁,𝑖𝜔₁}; consider the following FDE on 𝐶([−1,0],𝑋): 𝑧̇(𝑡)=𝜏¹_𝑗𝑑Δ𝑧(𝑡)+𝐿𝜏¹_𝑗𝑧_𝑡, (4.5) that is, 𝑧₁̇(𝑡)𝑧₂̇(𝑡)=𝜏¹_𝑗𝛼₁₁−𝑑₁𝛼₁₂0−𝑑₂𝑧₁(𝑡)𝑧₂(𝑡)+𝛼00₂₁𝛼₂₂𝑧₁()𝑧𝑡−1₂(𝑡−1). (4.6)

Obviously, 𝐿(𝜏¹_𝑗) 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 𝐿𝜏¹_𝑗𝜙=⁰₋₁𝑑𝜂𝜃,𝜏¹_𝑗𝜙(𝜃)for𝜙∈𝐶. (4.7) Thus, we can choose 𝜂𝜃,𝜏¹_𝑗=𝜏¹_𝑗𝛼₁₁−𝑑₁𝛼₁₂0−𝑑₂(],𝜏,𝜃∈−1,0_1𝑗𝛼00₂₁𝛼₂₂,𝜃=−1, (4.8) then (4.7) is satisfied.

Letting 𝐴(𝜏¹_𝑗) denote the infinitesimal generator of strongly continuous semigroup, according to [2], then, 𝐴𝜏¹_𝑗𝜙(𝜃)=𝑑𝜙(𝜃)[),𝐿𝜏𝑑𝜃,𝜃∈−1,0_1𝑗𝜙=^def0₋₁𝑑𝜂𝑡,𝜏¹_𝑗𝜙(𝑡),𝜃=0, (4.9) where 𝜙∈𝐶¹([−1,0],𝑋).

For 𝜓∈𝐶¹([0,1],(𝑋)^∗), define 𝐴^∗𝜓(𝑠)=−𝑑𝜓(𝑠)(],𝑑𝑠,𝑠∈0,1₀₋₁𝜓(−𝑡)𝑑𝜂𝑡,𝜏¹_𝑗,𝑠=0 (4.10) and a bilinear inner product of the Sobolev space 𝑊^2.2(0,𝜋): 𝜓(𝑠),𝜙(𝜃)=𝜓(0)𝜙(0)−⁰₋₁^𝜃_𝜉=0𝜓()(𝜃)𝜙(𝜉)(0)𝜙(0)𝜉−𝜃𝑑𝜂𝑑𝜉=𝜓−𝜏¹_𝑗⁰₋₁𝜓()𝛼𝜃+100₂₁𝛼₂₂𝜙(𝜃)𝑑𝜃, (4.11) where 𝜂(𝜃)=𝜂(𝜃,𝜏¹_𝑗) and 𝐴^∗ are the formal adjoint of 𝐴(𝜏¹_𝑗).

It is easy to see from Section 2 that 𝐴(𝜏¹_𝑗) has a pair of simple purely imaginary eigenvalues ±𝑖𝜔₁ and they are also eigenvalues of 𝐴^∗ since 𝐴(𝜏¹_𝑗) and 𝐴^∗ are adjoint operators. Let 𝒫 and 𝒫^∗ be the center spaces, that is, the generalized eigenspaces, of 𝐴(𝜏¹_𝑗) 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 Φ̇=Φ𝐵, where 𝐵 is the 𝐼×𝐼 diagonal matrix 𝐵=^{𝑖𝜏1𝑗0}_{0𝑖𝜏1𝑗}.

Let Φ=(Φ₁,Φ₂) and Ψ=(Ψ₁,Ψ₂)^𝑇, where Φ₁(𝜃)=(1,𝜉)^T𝑒^{𝑖𝜔1𝜏1𝑗𝜃},Φ₂(𝜃)=Φ₁(𝜃)Ψ,−1≤𝜃≤0,₁(𝑠)=1𝜌(1,𝜁)𝑒^{−𝑖𝜔1𝜏1𝑗𝑠},Ψ₂(𝑠)=Ψ₁(𝑠),0≤𝑠≤1,𝜉=𝑖𝜔₁−𝛼₁₁+𝑑₁𝛼₁₂,𝜁=−𝑖𝜔₁+𝛼₁₁−𝑑₁𝛼₂₁𝑒^{−𝑖𝜔1𝜏1𝑗},𝜌=(1+𝜉𝜁)−𝜏¹_𝑗−𝑑₁+𝛼₁₁+𝜁𝛼₂₁+𝜉𝛼₁₂−𝑑₂𝜉𝜁+𝜉𝜁𝛼₂₂𝑒^{−𝑖𝜔1𝜏1𝑗}. (4.12)

From the above expression, we can easily see that (Ψ₁,Φ₁)=1, (Ψ₁,Φ₁)=0.

Let 𝑓₁=(𝛽¹₁,𝛽²₁), 𝑐⋅𝑓₁ be defined by 𝑐⋅𝑓₁=𝑐₁𝛽¹₁+𝑐₂𝛽²₁ 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 𝒫_CN𝒞, where 𝑃_CN𝜑=ΦΨ,<𝜑,𝑓₁>⋅𝑓₁,𝜑∈𝒞 (4.13) and 𝒞=𝒫_CN𝒞⊕𝒫_𝑄𝒞; here 𝒫_𝑄𝒞 denotes the complementary subspace of 𝒫_CN𝒞 in 𝒞.

Let 𝐴_𝜏¹_𝑗 be defined by 𝐴_𝜏¹_𝑗𝜑(𝜃)=𝜑̇(𝜃)+𝑋₀(𝜃)𝜏¹_𝑗Δ𝜑(0)+𝐿_∗𝜏¹_𝑗𝜑(𝜃)−𝜑̇(0),𝜑∈𝒞, (4.14) where 𝑋₀: [−1,0]→𝐵(𝑋,𝑋) is given by 𝑋₀=0,𝜃∈[−1,0),𝐼,𝜃=0. (4.15)

Then 𝐴_𝜏¹_𝑗 is the infinitesimal generator induced by the solution of (4.4) and (4.2) and can be rewritten as the following operator differential equation: 𝑈̇_𝑡=𝐴_𝜏¹_𝑗𝑈_𝑡+𝑋₀𝐹𝑈_𝑡,𝛼. (4.16) Using the decomposition 𝒞=𝑃_CN𝒞⊕𝒫_𝑄𝒞 and (4.13), the solution of (4.16) can be written as 𝑈_𝑡𝑥=Φ¹(𝑡)𝑥₂(𝑡)⋅𝑓₁+ℎ𝑥₁,𝑥₂,𝛼, (4.17) where (𝑥₁,𝑥₂)^𝑇=(Ψ,<𝑈_𝑡,𝑓₁>), and ℎ(𝑥₁,𝑥₂,𝛼)∈𝒫_𝑄𝒞 with ℎ(0,0,0)=𝐷ℎ(0,0,0)=0.

Thus, we describe the flow on the center manifold for (4.2) as 𝑈^∗_𝑡𝑥=Φ¹(𝑡)𝑥₂(𝑡)⋅𝑓₁+ℎ𝑥₁,𝑥₂, (4.18) where ℎ(𝑥₁,𝑥₂)=ℎ(𝑥₁,𝑥₂,0).

Letting 𝑧=𝑥₁−𝑖𝑥₂ and Ψ(0)=(Ψ₁(0),Ψ₂(0))^𝑇, when 𝛼=0, then 𝑧 satisfies 𝑧̇=𝑖𝜔₁𝜏¹_𝑗𝑧+𝑔(𝑧,𝑧), (4.19) where 𝑔(𝑧,𝑧)=Ψ₁(0)−𝑖Ψ₂(0)𝐹𝑈_𝑡,0,𝑓₁,𝑤(𝑧,𝑧)=ℎ𝑧+𝑧2,(𝑧−𝑧)𝑖2𝑤(,0,𝑧,𝑧)=𝑤₂₀𝑧²2+𝑤₁₁𝑧𝑧+𝑤₀₂𝑧²2+𝑤₂₁𝑧²𝑧2+⋯. (4.20)(4.21)(4.22)

Noticing that 𝑝₁=Φ₁+𝑖Φ₂, therefore, solutions of (4.16) can be rewritten as 𝑈^∗_𝑡=12Φ(𝑧+𝑧)𝑖(𝑧−𝑧)⋅𝑓₁+𝑤(𝑧,𝑧)=12𝑝₁𝑧+𝑝₂𝑧⋅𝑓₁+𝑤(𝑧,𝑧). (4.23) In addition, (4.19) can be rewritten as the following form: 𝑧̇=𝑖𝜔₁𝜏¹_𝑗𝑧+𝑔₂₀𝑧²2+𝑔₁₁𝑧𝑧+𝑔₀₂𝑧²2+𝑔₂₁𝑧²𝑧2+⋯. (4.24) Let 𝑔(𝑧,𝑧)=𝑔₂₀𝑧²2+𝑔₁₁𝑧𝑧+𝑔₀₂𝑧²2+𝑔₂₁𝑧²𝑧2+⋯. (4.25) From (4.20), we have 𝐹𝑈_𝑡,0,𝑓₁=𝜏¹_𝑗4𝜉𝑓⁽¹⁾₁₁+12𝑓⁽¹⁾₂₀+12𝜉²𝑓⁽¹⁾₀₂𝑧²𝑒^{−2𝑖𝜔1𝜏1𝑗}𝜉𝑓⁽²⁾₁₁₀+𝑒^{𝑖𝜔1𝜏1𝑗}𝜉𝑓⁽²⁾₁₀₁+𝑒^{𝑖𝜔1𝜏1𝑗}𝜉²𝑓⁽²⁾₀₁₁+12𝑓⁽²⁾₂₀₀+12𝜉²𝑓⁽²⁾₀₂₀𝑧²+𝜏¹_𝑗4𝜉+𝜉𝑓⁽¹⁾₁₁+𝑓⁽¹⁾₂₀+𝜉𝜉𝑓⁽¹⁾₀₂𝑧𝑧𝜉+𝜉𝑓⁽²⁾₁₁₀+𝑒^{−𝑖𝜔1𝜏1𝑗}𝜉𝑓⁽²⁾₁₀₁+𝜉𝑓⁽²⁾₀₁₁+𝑒^{𝑖𝜔1𝜏1𝑗}𝜉𝑓⁽²⁾₁₀₁+𝜉𝑓⁽²⁾₀₁₁+𝑓⁽²⁾₂₀₀+𝜉𝜉𝑓⁽²⁾₀₂₀𝑧𝑧+𝜏¹_𝑗4𝜉𝑓⁽¹⁾₁₁+12𝑓⁽¹⁾₂₀+12𝜉²𝑓⁽¹⁾₀₂𝑧²𝑒^{2𝑖𝜔1𝜏1𝑗}𝜉𝑓⁽²⁾₁₁₀+𝑒^{−𝑖𝜔1𝜏1𝑗}𝜉𝑓⁽²⁾₁₀₁+𝑒^{−𝑖𝜔1𝜏1𝑗}𝜉²𝑓⁽²⁾₀₁₁+12𝑓⁽²⁾₂₀₀+12𝜉²𝑓⁽²⁾₀₂₀𝑧²+𝜏¹_𝑗2𝑓⁽¹⁾₁₁𝑤²₁₁(0)+𝑤²₂₀(0)2+𝑤¹₁₁(0)𝑤𝜉+¹²⁰(0)2𝜉+𝑓⁽¹⁾₂₀𝑤¹₁₁(0)+𝑤¹₂₀(0)2+𝑓⁽¹⁾₀₂𝑤¹₁₁(0)𝑤𝜉+¹²⁰(0)2𝜉𝑓,1₍₂₎₁₁₀𝑒^{−𝑖𝜔1𝜏1𝑗}𝑤²₁₁(−1)+𝑒^{2𝑖𝜔1𝜏1j}𝑤²₂₀(−1)2+𝑤¹₁₁(−1)𝜉+𝑒^{2𝑖𝜔1𝜏1𝑗}𝑤¹₂₀(−1)2𝜉+𝑓⁽²⁾₁₀₁𝑒^{−𝑖𝜔1𝜏1𝑗}𝑤²₁₁(0)+𝑒^{𝑖𝜔1𝜏1𝑗}𝑤²₂₀(0)2+𝑤¹₁₁()𝑤−1𝜉+¹²⁰(−1)2𝜉+𝑓⁽²⁾₀₁₁𝑒^{−𝑖𝜔1𝜏1𝑗}𝑤²₁₁(0)𝜉+𝑒^{𝑖𝜔1𝜏1𝑗}𝑤²₂₀(0)2𝜉+𝑤²₁₁()𝑤−1𝜉+²²⁰(−1)2𝜉+12𝑓⁽²⁾₂₀₀2𝑒^{−𝑖𝜔1𝜏1𝑗}𝑤¹₁₁(−1)+𝑒^{𝑖𝜔1𝜏1𝑗}𝑤¹₂₀()+1−12𝑓⁽²⁾₀₂₀2𝑒^{−𝑖𝜔1𝜏1𝑗}𝑤²₁₁(−1)𝜉+𝑒^{𝑖𝜔1𝜏1𝑗}𝑤²₂₀()𝜉𝑧−1,1²𝑧+⋯. (4.26)

Noting that Ψ₁(0)−𝑖Ψ₂(0)=(2(1−𝑖𝜔₁)/(1+𝜔²₁)(1+𝜉𝜁))(1,𝜁), therefore, 𝑔₂₀=𝜏¹_𝑗1−𝑖𝜔₁1+𝜔²₁(1+𝜉𝜁)𝜉𝑓⁽¹⁾₁₁+12𝑓⁽¹⁾₂₀+12𝜉²𝑓⁽¹⁾₀₂+𝑒^{−2𝑖𝜔1𝜏1𝑗}𝜉𝑓⁽²⁾₁₁₀+𝑒^{𝑖𝜔1𝜏1𝑗}𝜉𝑓⁽²⁾₁₀₁+𝑒^{𝑖𝜔1𝜏1𝑗}𝜉²𝑓⁽²⁾₀₁₁+12𝑓⁽²⁾₂₀₀+12𝜉²𝑓⁽²⁾₀₂₀𝜁,𝑔₁₁=𝜏¹_𝑗1−𝑖𝜔₁1+𝜔²₁(1+𝜉𝜁)𝜉+𝜉𝑓⁽¹⁾₁₁+𝑓⁽¹⁾₂₀+𝜉𝜉𝑓⁽¹⁾₀₂+𝜉+𝜉𝑓⁽²⁾₁₁₀+𝑒^{−𝑖𝜔1𝜏1𝑗}𝜉𝑓⁽²⁾₁₀₁+𝜉𝑓⁽²⁾₀₁₁+𝑒^{𝑖𝜔1𝜏1𝑗}𝜉𝑓⁽²⁾₁₀₁+𝜉𝑓⁽²⁾₀₁₁+𝑓⁽²⁾₂₀₀+𝜉𝜉𝑓⁽²⁾₀₂₀𝜁,𝑔₀₂=𝑔₂₀𝑔₂₁=2𝜏¹_𝑗1−𝑖𝜔₁1+𝜔²₁)×𝑓(1+𝜉𝜁₍₁₎₁₁𝑤²₁₁(0)+𝑤²₂₀(0)2+𝑤¹₁₁(0)𝑤𝜉+¹²⁰(0)2𝜉+𝑓⁽¹⁾₂₀𝑤¹₁₁(0)+𝑤¹₂₀(0)2+𝑓⁽¹⁾₀₂𝑤¹₁₁(0)𝑤𝜉+¹²⁰(0)2𝜉+𝑓,1₍₂₎₁₁₀𝑒^{−𝑖𝜔1𝜏1𝑗}𝑤²₁₁(−1)+𝑒^{2𝑖𝜔1𝜏1𝑗}𝑤²₂₀(−1)2+𝑤¹₁₁(−1)𝜉+𝑒^{2𝑖𝜔1𝜏1𝑗}𝑤¹₂₀(−1)2𝜉+𝑓⁽²⁾₁₀₁𝑒^{−𝑖𝜔1𝜏1𝑗}𝑤²₁₁(0)+𝑒^{𝑖𝜔1𝜏1𝑗}𝑤²₂₀(0)2+𝑤¹₁₁()𝑤−1𝜉+¹²⁰(−1)2𝜉+𝑓⁽²⁾₀₁₁𝑒^{−𝑖𝜔1𝜏1𝑗}𝑤²₁₁(0)𝜉+𝑒^{𝑖𝜔1𝜏1𝑗}𝑤²₂₀(0)2𝜉+𝑤²₁₁()𝑤−1𝜉+²²⁰(−1)2𝜉+12𝑓⁽²⁾₂₀₀2𝑒^{−𝑖𝜔1𝜏1𝑗}𝑤¹₁₁(−1)+𝑒^{𝑖𝜔1𝜏1𝑗}𝑤¹₂₀()+1−12𝑓⁽²⁾₀₂₀2𝑒^{−𝑖𝜔1𝜏1𝑗}𝑤²₁₁(−1)𝜉+𝑒^{𝑖𝜔1𝜏1𝑗}𝑤²₂₀(−1)𝜉,1𝜁. (4.27) Since 𝑤₂₀(𝜃) and 𝑤₁₁(𝜃) for (𝜃∈[−1,0]) appear in 𝑔₂₁, we still need them.

It follows easily from (4.22) that 𝑤̇(𝑧,𝑧)=𝑤₂₀𝑧𝑧̇+𝑤₁₁𝑧𝑧̇+𝑧𝑧̇+𝑤₀₂𝑧̇𝐴𝑧+⋯,_𝜏1𝑗𝑤=𝐴_𝜏¹_𝑗𝑤₂₀𝑧²2+𝐴_𝜏¹_𝑗𝑤₁₁𝑧𝑧+𝐴_𝜏¹_𝑗𝑤₀₂𝑧²2+⋯. (4.28)(4.29) According to [22] we can know, 𝑤̇=𝐴_𝜏¹_𝑗𝑤+𝐻(𝑧,𝑧), (4.30) where 𝐻(𝑧,𝑧)=𝐻₂₀𝑧²2+𝐻₁₁𝑧𝑧+𝐻₀₂𝑧²2+⋯=𝑋₀𝐹𝑈^∗_𝑡,0−ΦΨ,𝑋₀𝐹𝑈^∗_𝑡,0,𝑓₁⋅𝑓₁ (4.31) and 𝐻_𝑖𝑗∈𝒫_𝑄𝒞,𝑖+𝑗=2.

Thus, by using the chain rule (𝑤̇=𝜕𝑤𝑧,𝑧)̇(𝜕𝑧𝑧+𝜕𝑤𝑧,𝑧)𝜕𝑧𝑧̇. (4.32)

From (4.23) and (4.30), we can obtain 2𝑖𝜔₁𝜏¹_𝑗−𝐴_𝜏¹_𝑗𝑤₂₀=𝐻₂₀,−𝐴_𝜏¹_𝑗𝑤₁₁=𝐻₁₁,−2𝑖𝜔₁𝜏¹_𝑗−𝐴_𝜏¹_𝑗𝑤₀₂=𝐻₀₂. (4.33) Noticing that 𝐴_𝜏¹_𝑗 has only two eigenvalues ±𝑖𝜔₁, therefore, (4.33) has the unique solution 𝑤_𝑖𝑗(𝑖+𝑗=2) in 𝒫_𝑄𝒞 and 𝑤₂₀=2𝑖𝜔₁𝜏¹_𝑗−𝐴_𝜏¹_𝑗⁻¹𝐻₂₀,𝑤₁₁=−𝐴_𝜏⁻¹_1𝑗𝐻₁₁,𝑤₀₂=−2𝑖𝜔₁𝜏¹_𝑗−𝐴_𝜏¹_𝑗⁻¹𝐻₀₂. (4.34) Note that for −1≤𝜃<0, 𝐻(𝑧,𝑧)=−Φ(𝜃)Ψ(0)𝐹𝑈_𝑡,0,𝑓₁⋅𝑓₁𝑝=−¹(𝜃)+𝑝₂(𝜃)2,𝑝₁(𝜃)−𝑝₂(𝜃)Ψ2𝑖¹(0)Ψ₂(0)𝐹𝑈_𝑡,0,𝑓₁⋅𝑓₁1=−2𝑝₁(𝜃)Ψ₁(0)−𝑖Ψ₂(0)+𝑝₂(𝜃)Ψ₁(0)+𝑖Ψ₂(0)𝐹𝑈_𝑡,0,𝑓₁⋅𝑓₁1=−4𝑔₂₀𝑝₁(𝜃)+𝑔₀₂𝑝₂(𝜃)𝑧²⋅𝑓₁−12𝑔₁₁𝑝₁(𝜃)+𝑔₁₁𝑝₂(𝜃)𝑧𝑧⋅𝑓₁. (4.35) So, for −1≤𝜃<0, 𝐻₂₀(𝜃)1=−2𝑔₂₀𝑝₁(𝜃)+𝑔₀₂𝑝₂(𝜃)⋅𝑓₁,𝐻₁₁(𝜃)1=−2𝑔₁₁𝑝₁(𝜃)+𝑔₁₁𝑝₂(𝜃)𝑧𝑧⋅𝑓₁,𝐻₂₀(0)=𝜏¹_𝑗2𝜉𝑓⁽¹⁾₁₁+12𝑓⁽¹⁾₂₀+12𝜉²𝑓⁽¹⁾₀₂𝑒^{−2𝑖𝜔1𝜏1𝑗}𝜉𝑓⁽²⁾₁₁₀+𝑒^{𝑖𝜔1𝜏1𝑗}𝜉𝑓⁽²⁾₁₀₁+𝑒^{𝑖𝜔1𝜏1𝑗}𝜉²𝑓⁽²⁾₀₁₁+12𝑓⁽²⁾₂₀₀+12𝜉²𝑓⁽²⁾₀₂₀−12𝑔₂₀𝑝₁(0)+𝑔₀₂𝑝₂(0)⋅𝑓₁,𝐻₁₁(0)=𝜏¹_𝑗2𝜉+𝜉𝑓⁽¹⁾₁₁+𝑓⁽¹⁾₂₀+𝜉𝜉𝑓⁽¹⁾₀₂𝜉+𝜉𝑓⁽²⁾₁₁₀+𝑒^{−𝑖𝜔1𝜏1𝑗}𝜉𝑓⁽²⁾₁₀₁+𝜉𝑓⁽²⁾₀₁₁+𝑒^{𝑖𝜔1𝜏1𝑗}𝜉𝑓⁽²⁾₁₀₁+𝜉𝑓⁽²⁾₀₁₁+𝑓⁽²⁾₂₀₀+𝜉𝜉𝑓⁽²⁾₀₂₀−12𝑔₁₁𝑝₁(0)+𝑔₁₁𝑝₂(0)⋅𝑓₁. (4.36) By the definition of 𝐴_𝜏¹_𝑗, for −1≤𝜃<0, we have 𝑤̇₂₀(𝜃)=2𝑖𝜔₁𝑤₂₀(𝜃)+12𝑔₂₀𝑝₁(𝜃)+𝑔₀₂𝑝₂(𝜃)⋅𝑓₁,−1≤𝜃<0. (4.37) Note that 𝑝₁(𝜃)=𝑝₁(0)𝑒^{𝑖𝜔1𝜃},−1≤𝜃≤0; hence 𝑤₂₀(𝜃)=𝑖𝑔₂₀𝜔₁𝜏¹_𝑗𝑝₁(𝜃)+𝑖𝑔₀₂3𝜔₁𝜏¹_𝑗𝑝₂(𝜃)⋅𝑓₁+𝑒^{2𝑖𝜔1𝜃}𝐸₁𝑤,−1≤𝜃<0,₁₁(𝜃)=𝑔₁₁𝑝₁(𝜃)+𝑔₁₁𝑝₂(𝜃)⋅𝑓₁+𝐸₂𝐸,−1≤𝜃<0,₁=𝑤₂₀(0)−𝑖𝑔₂₀𝜔₁𝜏¹_𝑗𝑝₁(0)+𝑖𝑔₀₂3𝜔₁𝜏¹_𝑗𝑝₂(0)⋅𝑓₁,𝐸₂=𝑤₁₁(0)−𝑖𝑔₂₀𝜔₁𝜏¹_𝑗𝑝₁(0)+𝑖𝑔₀₂𝜔₁𝜏¹_𝑗𝑝₂(0)⋅𝑓₁. (4.38)

Using the definition of 𝐴_𝜏¹_𝑗 again and combining (4.29) and (4.33), we get 2𝑖𝜔₁𝜏¹_𝑗𝑖𝑔₂₀𝜔₁𝜏¹_𝑗𝑝₁(0)⋅𝑓₁+𝑖𝑔₀₂3𝜔₁𝜏¹_𝑗𝑝₂(0)⋅𝑓₁+𝐸₁−𝜏¹_𝑗𝐴_𝜏¹_𝑗𝑖𝑔₂₀𝜔₁𝑝₁(0)⋅𝑓₁+𝑖𝑔₀₂3𝜔₁𝑝₂(0)⋅𝑓₁+𝐸₁−𝐿𝜏¹_𝑗𝑖𝑔₂₀2𝜔₁𝑝₁(𝜃)⋅𝑓₁+𝑖𝑔₀₂6𝜔₁𝑝₂(𝜃)⋅𝑓₁+𝐸₁𝑒^{2𝑖𝜔1𝜃}=𝜏¹_𝑗2𝜉𝑓⁽¹⁾₁₁+12𝑓⁽¹⁾₂₀+12𝜉²𝑓⁽¹⁾₀₂𝑒^{−2𝑖𝜔1𝜏1𝑗}𝜉𝑓⁽²⁾₁₁₀+𝑒^{𝑖𝜔1𝜏1𝑗}𝜉𝑓⁽²⁾₁₀₁+𝑒^{𝑖𝜔1𝜏1𝑗}𝜉²𝑓⁽²⁾₀₁₁+12𝑓⁽²⁾₂₀₀+12𝜉²𝑓⁽²⁾₀₂₀−12𝑔₂₀𝑝₁(0)+𝑔₀₂𝑝₂(0)⋅𝑓₁. (4.39) As 𝜏¹_𝑗𝐴_𝜏¹_𝑗𝑝₁(0)⋅𝑓₁+𝐿𝜏¹_𝑗𝑝₁(𝜃)⋅𝑓₁=𝑖𝜔₁𝜏¹_𝑗𝑝₁(0)⋅𝑓₁,𝜏¹_𝑗𝐴_𝜏¹_𝑗𝑝₂(0)⋅𝑓₁+𝐿𝜏¹_𝑗𝑝₂(𝜃)⋅𝑓₁=−𝑖𝜔₁𝜏¹_𝑗𝑝₂(0)⋅𝑓₁, (4.40) then 2𝑖𝜔₁𝜏¹_𝑗𝐸₁−𝜏¹_𝑗𝐴_𝜏¹_𝑗−𝐿𝜏¹_𝑗𝐸₁𝑒^{2𝑖𝜔1𝜏1𝑗𝜃}=𝜏¹_𝑗2𝜉𝑓⁽¹⁾₁₁+12𝑓⁽¹⁾₂₀+12𝜉²𝑓⁽¹⁾₀₂𝑒^{−2𝑖𝜔1𝜏1𝑗}𝜉𝑓⁽²⁾₁₁₀+𝑒^{𝑖𝜔1𝜏1𝑗}𝜉𝑓⁽²⁾₁₀₁+𝑒^{𝑖𝜔1𝜏1𝑗}𝜉²𝑓⁽²⁾₀₁₁+12𝑓⁽²⁾₂₀₀+12𝜉²𝑓⁽²⁾₀₂₀. (4.41) From the above expression, we can see easily that 𝐸₁=12−2𝑖𝜔₁+𝛼₁₁+𝑑₁𝛼₁₂𝛼₂₁−2𝑖𝜔₁+𝛼₂₂+𝑑₂⁻¹×𝜉𝑓⁽¹⁾₁₁+12𝑓⁽¹⁾₂₀+12𝜉²𝑓⁽¹⁾₀₂𝑒^{−2𝑖𝜔1𝜏1𝑗}𝜉𝑓⁽²⁾₁₁₀+𝑒^{𝑖𝜔1𝜏1𝑗}𝜉𝑓⁽²⁾₁₀₁+𝑒^{𝑖𝜔1𝜏1𝑗}𝜉²𝑓⁽²⁾₀₁₁+12𝑓⁽²⁾₂₀₀+12𝜉²𝑓⁽²⁾₀₂₀. (4.42) Similarly 𝐸₂=12𝛼₁₁+𝑑₁𝛼₁₂𝛼₂₁𝛼₂₂+𝑑₂⁻¹×𝜉+𝜉𝑓⁽¹⁾₁₁+𝑓⁽¹⁾₂₀+𝜉𝜉𝑓⁽¹⁾₀₂𝜉+𝜉𝑓⁽²⁾₁₁₀+𝑒^{−𝑖𝜔1𝜏1𝑗}𝜉𝑓⁽²⁾₁₀₁+𝜉𝑓⁽²⁾₀₁₁+𝑒^{𝑖𝜔1𝜏1𝑗}𝜉𝑓⁽²⁾₁₀₁+𝜉𝑓⁽²⁾₀₁₁+𝑓⁽²⁾₂₀₀+𝜉𝜉𝑓⁽²⁾₀₂₀. (4.43)

Thus, 𝑔₂₁ can be determined by the parameters and delay; we get 𝑐₁(0)=𝑖2𝜔₁𝑔₁₁𝑔₂₀−2𝑔₁₁²−𝑔²₀₂3+𝑔₂₁2. (4.44)

Then, we can compute the following values: 𝜎₂=Re𝑐₁(0)Re𝜆^′𝜏¹_𝑗,𝜖₂=2Re𝑐₁(0),𝑇₂=Im𝑐₁(0)+𝜎₂Im𝜆^′𝜏¹_𝑗𝑤₁. (4.45) 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 when 𝜏∈[0,𝜏⁰_𝑗). As the delay 𝜏 crosses through each 𝜏⁰_𝑗, there exist a sequence of critical values 𝜏⁰_𝑗(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).

Acknowledgment

This work is supported by the Fundamental Research Fund of Henan University (B2011088).