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

and Applied Analysis 3 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: 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∗ . 2.2 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 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 , 2.3 where a1 sP10 u∗, v∗ , a2 sP01 u∗, v∗ , b1 rP10 u∗, v∗ , b2 rP01 u∗, v∗ , and


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: where u and v denote the population densities of prey and predator species at time t and space x, respectively; the positive constants d 1 and d 2 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 R N 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 d 1 d 2 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 u 2 v/ 1 u 2 , 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 Section 2, 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.

Analysis of the Characteristic Equations
In this section, by choosing the delay τ 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 E 1 0, 0 and E 2 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 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: where where where φ φ 1 , φ 2 T ∈ .
The linearization of 2.5 is given by and its characteristic equation is λy − DΔy − L e λ y 0, 2.8 where y ∈ dom Δ and y / 0, dom Δ ⊂ X.
From the properties of the Laplacian operator defined on the bounded domain, the operator Δ on X has the eigenvalues −k 2 with the relative eigenfunctions where 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
Some simple computations show that From 2.10 -2.11 , 2.8 is equivalent to u * , then we conclude that the characteristic equation 2.8 is equivalent to the sequence of the characteristic equations: Obviously, for all k ∈ N 0 , λ 0 is not a root of 2.13 .Equation 2.13 with τ 0 is equivalent to the following quadratic equations: Let λ 1 and λ 2 be the two roots of 2.14 .Then, for all k ∈ N 0 ,

2.15
Therefore, we have the following Lemma.
Assume that 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.Let λ iω ω > 0 be a root of the characteristic equation 2.13 .Then ω satisfies the following equation for some k ∈ N 0 :

2.16
Separating the real and imaginary parts of 2.16 leads to

2.18
Let z ω 2 , then 2.18 can be rewritten into the following form:

6
Abstract and Applied Analysis By (A3) and (A4), for all k ∈ N 0 , we have 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: It is easy to get that when 21 Equation 2.13 with k 0 has simple imaginary roots ±iω 0 , and Re dλ/dτ τ τ 0 j > 0, where λ τ is the root of 2.13 with k 0 satisfying λ τ 0 j iω 0 , and We have the following result.
Proof.Let λ iω 1 ω 1 > 0 be a root of 2.13 with k ≥ 1.By the same way in Theorem 2.2, we can obtain Abstract and Applied Analysis 7 Set z ω 2 1 , then Let z 1 and z 2 be the roots of 2.24 with k ≥ 1.We know that if z 1 z 2 < 0 and z 1 z 2 > 0, then 2.13 with k ≥ 1 has no purely imaginary roots.By B2 , it follows that, for ∀k ≥ 1,

2.25
Therefore, 2.13 with k ≥ 1 have no purely imaginary roots.Summarizing the above results and combining Theorem 2.3, 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: 0 , then the equilibrium point E u * , v * is unstable; III τ τ 0 j 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 Theorem 2.2, let λ iω ω > 0 be a root of the characteristic equation 2.13 , then ω satisfies the following equation: for k ∈ N 0 .Now, we make the following assumptions.For a certain k 0 {1, 2, . ..},

Abstract and Applied Analysis
Under the assumptions C1 and C2 , 2.26 with k k 0 has only a positive solution ω k 0 , then 2.26 can be transformed into the following equation:
In addition, similar to the proof of Theorem 2.2, we have F ω k 0 , Abstract and Applied Analysis E ω k 0 .

2.32
From the above analysis, we have the following Theorem.
Let λ τ α τ iβ τ be the root of 2.13 near τ τ k 0 j satisfying where ω k 0 and τ k 0 j are given by 2.27 and 2.32 , respectively.Then we have the following transversality condition.

Re
, then the equilibrium point E u * , v * is asymptotically stable; II If τ > τ k 0 0 , then the equilibrium point E u * , v * is unstable; III τ τ k 0 j j 0, 1, 2, . . .are Hopf bifurcation values of system 1.1 , and these Hopf bifurcations are all spatially inhomogeneous.

The Effect of Diffusion on Hopf Bifurcations
In the previous section, we have studied the Hopf bifurcations from the positive constant steady-state E u * , v * of 1.1 when τ crosses through the critical value τ k j k 0, k 0 ; 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 Theorems 2.4 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: 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 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.

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 τ k j by τ.Let and drop the tilde for the sake of simplicity.Then system 1.1 can be written as where 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 τ, the characteristic equation of 4.5 has a pair of simple purely imaginary eigenvalues 0 {iω k τ, −iω k τ}, k 0, k 0 .
Consider the ordinary functional differential equation By the Riesz representation theorem, there exists a 2 × 2 matrix function η θ, τ −1 ≤ θ ≤ 0 , whose entry is of bounded variation such that In fact, we can choose

4.8
Let A τ denote the infinitesimal generators of the semigroup induced by the solutions of 4.6 and A * be the formal adjoint of A τ under the bilinear pairing Then A τ and A * are a pair of adjoint operators.
From the discussion in Section 2, we know that A τ has a pair of simple purely imaginary eigenvalues ±iω k τ, and they are also eigenvalues of A * .Let P and Q be the center subspaces, that is, the generalized eigenspace of A τ and A * associated with 0 , respectively.Then Q is the adjoint space of P and dim P dim Q 2, see 17 .Direct computations give the following results.

4.10
Then, is a basis of P with 0 , and for θ ∈ −1, 0 , and Abstract and Applied Analysis then Ψ construct a new basis for Q and Ψ, Φ I 2 , see 17 .In addition, f k β 1 k , β 2 k , where Then the center subspace of linear equation 4.5 is given by P CN , where and P CN ⊕ P s , and P s denotes the complement subspace of P CN in .Let A τ be the infinitesimal generator induced by the solution of 4.5 .Then 4.2 can be rewritten as the abstract form

4.19
Using the decomposition P CN ⊕ P s and 4.17 , the solution of 4.2 can be written as where Ψ, U t , f k , and h x 1 , x 2 , μ ∈ P s , h 0, 0, 0 0, Dh 0, 0, 0 0. In particular, the solution of 4.2 on the center manifold is given by Equation 4.21 can be transformed into Abstract and Applied Analysis

4.33
Abstract and Applied Analysis Thus, from 4.24 , 4.27 -4.33 , we can obtain that

4.34
Noticing that A τ has only two eigenvalues ±iω k τ; therefore, 4.34 has a unique solution W ij in Q given by

4.38
Abstract and Applied Analysis 19

4.39
By the definition of A τ , we have from 4.34 ,

4.42
Using the definition of A τ , and combining 4.34 and 4.41 , we have

4.43
Notice that

4.45
From the above expression, we obtain that
iω k τ − sP 11 C − 1 k 2 − 1 u * a 1 u * e −2iω k τ a 2 −b 1 2iω k d 2 k 2 − b 2 d cos ω k τ − sP 11 C C − sP 20 − sP 02 CC rP 11 C CrP 20 rP 02 CC W 20 θ and W 11 θ have been expressed by the parameters of the system 1.1 .And, hence, g 21 can be expressed also.Thus, we can compute the following values: 2 determines the direction of Hopf bifurcation, β 2 determines the stability of bifurcating periodic solution, and T 2 determines the period of the bifurcating periodic solution.Hence, we have the following result.The signs of μ 2 , β 2 , T 2 determine the properties of Hopf bifurcation described in Theorems 2.4 and 2.7.If μ 2 > 0 μ 2 < 0 , then the Hopf bifurcation is supercritical (subcritical), and the bifurcating periodic solutions exist (nonexist) for τ > τ k τ < τ k .If β 2 < 0 β 2 > 0 , then the bifurcating periodic solutions are stable (unstable).If T 2 > 0 T 2 < 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 of E 1 and E 2 contain the diffusion coefficient.According to the definition of g 21 , the values of g 21 have related on E 1 and E 2 .Therefore, the signs of β 2 and μ 2 which determine the stability and direction of spatially inhomogeneous periodic solutions strictly depend on the diffusion coefficient of d 1 and d 2 .