TRAVELING WAVE SOLUTIONS OF A REACTION-DIFFUSION PREDATOR-PREY MODEL

. This paper is concerned with the dynamics of traveling wave solutions for a reaction-diﬀusion predator-prey model with a nonlocal delay. By using Schauder’s ﬁxed point theorem, we establish the existence result of a traveling wave solution connecting two steady states by constructing a pair of upper-lower solutions which are easy to construct in practice. We also investigate the asymptotic behavior of traveling wave solutions by employing the standard asymptotic theory.


1.
Introduction. Predator-prey model is an important tool that helps us to understand the ecological and biological systems surrounding us, and it is also one of the basic models between different species in nature [5]. Recently, traveling wave solutions of delayed reaction-diffusion equations have been widely studied due to the significant applications in mathematical theory and other practical fields ( see, for example, Liang and Zhao [8], Ma [11], Schaaf [12], Huang and Zou [6], Lin et al. [9], Britton [1]).
In the traditional Lotka-Volterra type, we can see that the spatial content of environment has always been ignored by people. Traditionally, these models have been worked out in connection with the time evolution of uniform population distribution in the habitat and are strictly dominated by ordinary differential equation. Whereas, as argued in, the species taken into consideration in many ecological systems many disperse both in space and time. The spatial dispersal or diffusion results from the resource limitation which brings the trends of some species towards regions of lower population density. In recent years, great important has been attached on the influence of dispersion of a population in a bordered area, and in this case a system of reaction-diffusion equation are applied to describe the governing equations concerning the population density. The problem that interest both an ecological system and the mathematically research is to determine the condition under which the time-dependent solution converges to a positive steady-state solution, and the role of the influence of diffusion and time delays. It is argued that any delays should be spatially inhomogeneous in more realistic ecological models. That is to say that the delays influence not only temporal variables but also spatial variables. This is because the fact that any given individual in former times may not necessarily has been at the same spatial. Such delays are called nonlocal. Lately, the influence of nonlocal delays on the dynamics of ecological models has been taken into consideration ( see [1,14,2,17,16,15,7,4,19,13,18]).
Gan et al. [4] discussed the following three-species food-chain model with spatial diffusion and time delays      ∂u ∂t = D 1 ∂ 2 u ∂x 2 + u(t, x)(r 1 − a 11 u(t, x) − a 12 v(t, x)), ∂v ∂t = D 2 with initial conditions where u(x, t), v(t, x) and w(t, x) represent the densities of the prey, predator and top predator at time t and location x, respectively. The parameter a 11 , a 21 , a 12 , a 22 , a 23 , a 32 , a 33 , τ , D i , r i (i = 1, 2, 3) are positive constants. The existence of traveling wave solutions for the system (1) with initial conditions was obtained by using Schaudar's fixed point theorem, the cross iteration method and constructing a pair of upper-lower solutions [4]. Shang, Du and Lin [13] considered a n-dimension diffusive system with delays where I = {i = 1, . . . , n}, J = {i = 1, . . . , n − 1}, u i (x, t) (i ∈ J) and u n represent the densities of the prey and predator populations at location x and time t, respectively. The parameters d i (i ∈ J) and d n are the diffusion rates of the prey and predator populations, respectively. The existence of traveling wave solutions for the n-dimensional delayed reaction-diffusion system was established by using Schauder's fixed point theorem and constructing a pair of upper-lower solutions [13]. Zhang and Xu [18] were concerned with the following reaction-diffusion predatorprey model stage structure and nonlocal delay , v(x, θ) = ψ(x, θ) ≥ 0. By applying the cross iteration method and Schauder's fixed point theorem, the existence result of traveling wave solution for the above system with initial conditions was established in [18].
Lv and Wang [10] were concerned with the following diffusive and time-delayed integro-differential equation where the constants d, a and b are positive, u(x, t) is normalized spatial density of an infections host at time t and at point x, and F (x, y, t, s) is the convolution kernel. The authors [10] obtained the asymptotic behavior and uniqueness of the traveling wave fronts by using the standard asymptotic theory and sliding method. Xu and Ma [15] considered the following reaction-diffusion predator-prey model with a nonlocal delay: for t > 0, x ∈ (0, π), with homogeneous Neumann boundary conditions and initial conditions where is the solution of ∂G ∂t = D ∂ 2 G ∂y 2 , subject to ∂G ∂y = 0 at y = 0, π and G(x, y, 0) = δ(x − y).
And the function f in (2) is called the delay kernel and satisfies f (t) ≥ 0 for all t ≥ 0 together with ∞ 0 f (t) = 1. In system (2), u 1 (x, t) and u 2 (x, t) represent the densities of the prey and predator populations at location x and time t, respectively. The parameters D 1 and D 2 are the diffusion rates of the prey and predator populations, respectively. The parameter r 1 is the intrinsic growth rate of the prey; the parameter r 2 is the death rate of the predator; a 11 is the capture rate of the predator; a 21 is the conversion rate of the predator by consuming prey; a 22 is the intra-specific competition rate of the predator. φ 1 (x, θ)and φ 2 (x, θ) are nonnegative and Hölder continuous and satisfy represents a time delay due to the gestation of the predator. In [15], the global dynamics of problem (2) -(4) was discussed by using upper-lower solutions and monotone iteration technique.
In this paper, we will further study the corresponding spatial-temporal patterns by traveling wave solutions of system (2). In other words, we will discuss the existence of traveling wave solutions for system (2) by using Schauder's fixed point theorem and constructing a pair of upper-lower solutions in ( [4,18]) and study the asymptotic behavior of the traveling wave by applying the standard asymptotic theory( [10]).
The rest of this paper is organized as follows. In Section 2, we introduce some preliminaries and several lemmas which will be essential to our proofs. In Section 3, it is devoted to establishing the existence of traveling wave solution for system (2). In Section 4, the asymptotic behavior of the traveling wave is obtained by employing the standard asymptotic theory.
2. Preliminaries. In order to establish the existence results of traveling wave solution for system (2), we introduce some basic notations and concepts. Letting Then the system with boundary and initial conditions (3), (4) is equivalent to the system for t > 0, x ∈ (0, π), with homogeneous Neumann boundary conditions and initial conditions First, we consider the general delayed reaction-diffusion system which satisfies the hypotheses: (A2) There exist three positive constants L i > 0, (i = 1, 2, 3), such that Then system (9) satisfies the following partial quasi-monotonicity conditions (P QM ), i.e., (P QM ). There exist three positive constants β 1 , β 2 , β 3 > 0 so that

Definition 1.
A traveling wave solution of (9) is a special solution of the form is the wave profile that propagates through the one-dimension spatial domain R at the constant wave speed c > 0.

Definition 2.
A pair of continuous functions ρ = (φ, ϕ, ψ) and ρ = (φ, ϕ, ψ) are called an upper solution and a lower solution for system (9), respectively, if ρ and ρ are twice differentiable almost everywhere in R and essentially bounded on R, and content and We assume that a pair of upper and lower solutions (φ(t), ϕ(t), ψ(t)) and (φ(t), ϕ(t), ψ(t)) are given such that We will look for traveling wave solutions of system (9) in the following profile set Obviously, Γ((φ, ϕ, ψ), (φ, ϕ, ψ)) is non-empty, closed, convex and bound. For the constants β i > 0 in (10), define H : The operators H 1 , H 2 and H 3 admit the following properties: Lemma 1. Assume that (A1) and (10) hold, then Proof. From (10), a direct calculation shows that Similar to reference ( [18]), we have the following Lemma 2 and omit the proof.
From the definitions of H 1 , H 2 and H 3 in (19)-(21), system (11) can be rewritten as We define Thus, a fixed point of the operator F is a solution of (22). On the other hand, a solution of (22) is also a fixed point of the operator F . Obviously, a bounded solution Φ of (22) or a bounded fixed point of the operator F allows bounded first and second derivatives. Corresponding to Lemma 1 and Lemma 2, we have the following results for F .

Lemma 5. ([4]) Assume the (10) holds, then
is compact. Now, we are in a position to state and prove the following theorem.
On substituting u 1 (t, x) = φ(x + ct), u 2 (x, t) = ϕ(x + ct), u 3 (x, t) = ψ(x + ct) and denoting the traveling wave coordinate x + ct still by t, we derive from (2) that We are interested in the possibility of a transition between the equilibria E 0 and E * in the form of a traveling wave solution.
Choosing appropriate constants such that Then we complete the proof.
Remark 2. In [15] and [4], the authors considered the existence of traveling wave solution, but they don't investigate the asymptotic behavior of traveling wave solutions. In this paper, we also consider the asymptotic behavior for system (2).
Remark 3. In theorem 3, in order to facilitate the calculation, we assume that D 1 = D 2 = 1. In fact, D 1 , D 2 can be the positive integers.