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

and Applied Analysis 3 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 ∈W2,2 0, π : du dx dv dx 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) Λ 2 ( b c1δ/β ) , v∗ δ β x∗, Λ ( a1 − b − c1δ/β δ/β )2 4a1(b 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 1 i!j! f 1 ij u i t, x v t, x , ∂v t, x ∂t d2Δv t, x α21u t − τ, x α22v t − τ, x ∑ i j l≥2 1 i!j!l! f 2 ijl u t − τ, x v t − τ, x v t, x , 2.2 4 Abstract and Applied Analysis


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: 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: where δ s/r, β sh/α, a 1 a/K, c 1 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: 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; d 1 > 0, d 2 > 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 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.

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 E 1 1, 0 and a unique positive equilibrium E * u * , v * , where

2.1
Obviously, E 1 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 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 Abstract and Applied Analysis where

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 u 1 t u t, · , u 2 t v t, · , U t u 1 t , u 2 t T ; therefore, system 2.2 can be rewritten as an abstract form in the phase space C C −τ, 0 , X : where y ∈ dom Δ \ {0} and dom Δ ⊂ X.

Abstract and Applied Analysis 5
It is well known that the linear operator Δ on 0, π with homogeneous Neumann boundary conditions has the eigenvalues −k 2 k ∈ N 0 {0, 1, 2, . . .} , and the corresponding eigenfunctions are and thus any element y in X can be expanded a Fourier series in the form In addition, some easy computations can show that From 2.9 and 2.11 , 2.7 is equivalent to Thus λ is a characteristic root of 2.7 if and only if for k ∈ N 0 , λ satisfies 2.14 6 Abstract and Applied Analysis When τ 0, 2.13 reduces to the following quadratic equation with respect to λ: 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 α 11 < 0. Thus, we can have the following conclusions. 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 ∈ N 0 , ω satisfies the following equation: Thus Letting ω 2 z, then 2.18 can be written as Equation 2.19 with k 0 has only one positive real root: where ω 0 √ z 0 . Thus Clearly, if A 2 k − 2B k − α 2 22 > 0 and B 2 k − C 2 k > 0, there are no ω * such that 2.13 with k ≥ 1 has purely imaginary roots ±ω * i.
By computing, we have

2.25
In addition, according to α 11 < 0, we have B k ≥ 0. It is clear that Therefore, 2.13 with k ≥ 1 has no purely imaginary roots when the conditions H and α 11 < 0 hold. Thus the proof of Theorem 2.2 is accomplished.

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 τ 0 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 H 1 : d 1 d 2 d 1 α 22 − d 2 α 11 > α 11 α 22 − α 12 α 21 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 If the condition H is not satisfied, and From the discussion in Section 2, we know that there exists k 0 > 0, k 0 ∈ N such that 2.13 with k ≥ 1 has only characteristic roots with negative real parts when k > k 0 24 . In addition, from 2.17 , we have

3.3
Abstract and Applied Analysis 9 Thus In particular, it is easy to know from H 1 that B k > C k when α 11 Therefore, we can obtain the following.
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 τ 1 j are spatially nonhomogeneous. Therefore, we have the following conclusion. In general, we have the following.
and H 2 holds, then 2.13 with k k 0 has purely imaginary roots iω k 0 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 k 0 j , where ω k 0 and τ k 0 j are defined by 3.2 and 3.4 with k k 0 , k 0 ∈ 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.

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 τ 1 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 where f 1 , f 2 are defined by 2.2 . Letting τ τ 1 j α, j ∈ N 0 , then, 4.1 can be written in abstract form in C C −1, 0 : X as Linearizing 4.2 at 0, 0 leads to the following linear equation: Let Λ 1 {−iω 1 , iω 1 }; consider the following FDE on C −1, 0 , X : that is, Obviously, L τ 1 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 Thus, we can choose 4.8 then 4.7 is satisfied.

4.11
where η θ η θ, τ 1 j and A * are the formal adjoint of A τ 1 j . It is easy to see from Section 2 that A τ 1 j has a pair of simple purely imaginary eigenvalues ±iω 1 and they are also eigenvalues of A * since A τ 1 j and A * are adjoint operators. Let P and P * be the center spaces, that is, the generalized eigenspaces, of A τ 1 j and A * associated with Λ 1 , respectively. Then P * is the adjoint space of P and dim P dim P * 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 thatΦ ΦB, where B is the

4.15
Then A τ 1 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: