Existence and Stability of Periodic Solutions for a Delayed Prey–predator Model with Diffusion Effects

Existence and stability of spatially periodic solutions for a delay prey– predator diffusion system are concerned in this work. We obtain that the system can generate the spatially nonhomogeneous periodic solutions when the diffusive rates are suitably small. This result demonstrates that the diffusion plays an important role in deriving the complex spatiotemporal dynamics. Meanwhile, the stability of the spatially periodic solutions is also studied. Finally, in order to verify our theoretical results, some numerical simulations are also included.


Introduction
In recent years, the interactions between two species have attracted much attention due to their theoretical and practical significance since the pioneering theoretical works by Lotka [22] and Volterra [28], see [4,8,10,19,30,32].It is well known that the interactions between two species have mainly three kinds of fundamental forms such as competition, cooperation and prey-predation in population biology.Among these interactions, extreme attention has been payed to the prey-predation mechanism because it possesses a very significant function as a kind of restriction factor in the process of evolvement of biology [6,9,17,23,25].Understanding the dynamics of predator-prey models will be very helpful for investigating multiple species interactions.In [1], Beretta and Kuang have explored the dynamics of the following delayed Leslie-Gower model.
where u(t), v(t) are the population densities of the prey and the predator, respectively; r 1 > 0, r 2 > 0 denote the intrinsic growth rates of the prey and the predator, respectively.K > 0 is the carrying capacity of the prey and ru takes on the role of a prey-dependent carrying capacity for the predator.The parameter r > 0 is a measure of the quality of the prey as food for the predator.They presented some results on the boundedness of solutions permanence, global stability of the boundary equilibrium and local stability results of the positive equilibrium.Following this work, Song et al. [27] considered the properties of the local Hopf bifurcation and the global continuation of the local Hopf bifurcation for model (1.1).
In fact, the distribution of species is generally spatially inhomogeneous and therefore the species always tend to migrate toward regions of lower population density to improve the possibility of survival [29].Therefore, spatial diffusion should be considered in modelling biological interactions, see [2,3,12,13,16,20,21,26,31].Thus, the dynamics behavior of two species to model (1.1) should be described by the following model with Neumann boundary conditions with the inner product •, • .The main goals of the present paper are to consider the existence and stability of spatially periodic solutions of system (1.2).By regarding the time delay τ as the bifurcation parameter and analyzing the associated characteristic equation, we find that an increase of τ can lead to the occurrence of spatially nonhomogeneous periodic solutions at (u * , v * ).Moreover, the stability of the spatially nonhomogeneous periodic solutions is studied.
The remaining parts of this paper are organized as follows.In Section 2, the existence of spatially nonhomogeneous periodic solutions is investigated.In Section 3, we derive conditions for determining the stability of the spatially nonhomogeneous periodic solutions on the center manifold.Finally, some conclusions and numerical simulations are presented in Section 4. Throughout the paper, we denote by N the set of all positive integers, and N 0 = N ∪ {0}.

Existence of spatially periodic solutions
In this section, we focus on investigating the local stability and the existence of spatially periodic solutions of the positive constant steady-state of system (1.2).It is easy to see that system (1.2) has two feasible boundary equilibria (0, 0), (K, 0) and a unique positive constant steady-state E * (u * , v * ), where , for convenience, we still use u and v to denote u and v. Then system (1.2) can be transformed into the following reaction-diffusion system when Ω is restricted to the one-dimensional spatial domain (0, π): (2.1) Thus, the positive constant steady state E * (u * , v * ) of system (1.2) is transformed into the zero steady state of system (2.1).By virtue of the Taylor expansions, system (2.1) can be rewritten as the following system where •), and U(t) = (u 1 (t), u 2 (t)) T .According to [11,12], then system (2.2) can be rewritten as a delay differential equation in the phase space where 3) at (0, 0) gives the linear equation The characteristic equation for the linearized equation (2.4) is where y ∈ dom(∆)\{0} and dom(∆) ⊂ X.
It is well known that the linear operator ∆ on (0, π) with homogeneous Neumann boundary conditions has the eigenvalues −k 2 (k ∈ N 0 ) and the corresponding eigenfunctions are k=0 construct an orthogonal basis of the Banach space X (see [12]).Therefore k } and thus any element y in X can be expanded a Fourier series in the form In addition, some easy computations can show that where ϕ = (ϕ 1 , ϕ 2 ) T ∈ C. From (2.6) and (2.7), (2.5) is equivalent to Hence, we conclude that the characteristic equation (2.4) is equivalent to the sequence of the characteristic equations It is obvious that equation (2.8) has no zero roots since β 11 < 0, When τ = 0, (2.8) reduces to the following quadratic equation with respect to λ where Consequently, all roots of equations (2.9) have negative real parts.Therefore, the positive steady state E(u * , v * ) of system (1.2) is locally asymptotically stable in the absence of delay. When (2.10) then (2.8) with k = 1 has purely imaginary roots ±iω 1 , where Proof.Assuming iω (ω > 0) is a solution of (2.8) with k ≥ 1, then substituting iω into equation (2.8) and separating the real and imaginary parts, one can get that ) where From (2.12) and (2.13), it is easy to see that By computing, we have ) ≥ 0, then (2.14) with k = 1 has at least one positive root ω 1 .From (2.10), (2.11), (H) and (2.14), we have That is, it has ω 1 such that (2.8) with k = 1 has purely imaginary eigenvalues ±iω 1 .Thus the proof is complete.
Proof.Taking the derivative for equation (2.8) with respect to τ at τ j , we have (2.17) From (2.17), we get sign Re dλ dτ Thus the proof is complete.
Therefore, we have the following conclusions, Theorem 2.3.Suppose that the conditions in Lemma 2.1 are satisfied.Let τ j be defined as in (2.16).

Stability of spatially periodic solutions
In the previous section, we have obtained the existence of spatially periodic solutions of system (1.2) when the parameter τ crosses through the critical value τ j (j = 0, 1, 2, . . .).In this section, we shall study the stability of periodic solutions by applying the normal form theory of partial functional differential equations developed by [15,29].
where η (θ) = η θ, τ j and A * is the formal adjoint of A τ j .Obviously, the characteristic equation of the linear operator A τ j is (2.8) with k = 1.So, it is easy to see from Section 2 that A(τ j ) has a pair of simple purely imaginary eigenvalues ±iω 1 and they are also eigenvalues of A * since A τ j and A * are adjoint operators.Let P and P * be the center spaces, that is, the generalized eigenspaces, of A τ 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 [11,27], 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 2 Then the center space of linear equation (3.3) is given by P CN C, where and C = P CN C ⊕ P Q C, here P Q C denotes the complementary subspace of P CN C in C.

Conclusions and numerical simulations
In this paper, by studying the existence and stability of spatially periodic solutions for a delay Leslie-Gower diffusion system, we obtain that the system can generate the spatially nonhomogeneous periodic solutions when the diffusive rates are suitably small.We discover that system (1.2) have more abundant dynamic behavior than system (1.1), this indicates that diffusion plays a fundamental role in classifying the rich dynamics.In addition, we considered the stability of periodic solutions by applying the normal form theory of partial functional differential equations.
To illustrate the analytical results obtained, we give some numerical simulations and consider the following case of model ( 1 When τ passes through the critical value τ 0 , E * = (0.3333, 0.6667) loss its stability, a family of periodic solution bifurcates from equilibrium E * = (0.3333, 0.6667) which is depicted by the numerical simulation in Figs.11-12.From Theorem 3.1 and system (4.1),we can compute c 1 (0) = −2.0788e+ 0.03 − 6.9946e + 0.02i by the software package Matlab R2009b, and 2 = 2 Re(c 1 (0)) < 0. Therefore, the bifurcated periodic solutions are orbitally asymptotically stable on the center manifold.These properties are depicted by the numerical simulation in Figs.11-12.

Figures
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