Dynamical Behaviors of a Delayed Reaction-diffusion Equation

In this paper, we derive a delayed reaction-diffusion equation to describe a two-species predator-prey system with diffusion terms and stage structure. By coupling the uniformly approximate approach with the method of upper and lower solutions, we prove that the traveling wave fronts exist, which connect the zero solution with the positive steady state. Finally, we draw a conclusion that the existence of traveling wave fronts for the delayed reaction-diffusion equation is an interesting and difficult problem.


INTRODUCTION
Delay ordinary differential equations (also called by retarded functional differential equations) have been extensively studied by many authors, such as [2], [3], [4], [7], [10] and so on.Recently, a two-species predator-prey system described by a delayed ordinary differential equation was considered in [9], where the delay means the stage for the prey population.We remark that the above mentioned models did not consider the effect of diffusion on the stability of the equilibrium and traveling wave fronts.However, the specie's diffusion is a natural tendency to move into areas of smaller population density.So we follow the normal technique to handle with the diffusion (see [3], [5], [6] and [12]) to give the following delayed reaction-diffusion equations , ( , ), ( , ) u x t u x t x -space and t -time, respectively; represents the density of predator population at ( , ) v x t x -space and t -time;

u D u u x t u x t e u x t t u D u u x t a u x t v x t e u x t hu x t t
is the transformation coefficient of mature predator population; 1 a represents the transformation of immature to mature; is the birth rate of the immature prey population; is the harvesting effort of the prey species; 0 h ≥ 0 β > represents the death and overcrowding rate of the mature prey population.And the constants And the constant is positive, and the initial functions are continuous in Ω and 2 ( , ) In this paper, we aim to study the dynamical behaviors of the system (1).Note that and of the system (1) are independent of , so we obtain the dynamical behaviors of the system (1) by studying the following system 2 ( , ) where and ( , ), ( , ) x t x t ϕ ϕ of the system (1), respectively.For this single specie model of [10], S.A. Gourley and Y. Kuang pointed out that the existence of wave front solutions is an open question.Motivated by the results of [10], we study the existence of traveling wave fronts of the two-species delayed system (2).The key idea is to couple the uniformly approximated approach introduced by J. Canosa in [1] with the method of upper and lower solutions.The method to construct the upper and lower solutions of the system (2) is derived from the idea of [11].The remaining parts of the paper are organized as follows.In section 2, we study the locally asymptotical stability of the constant equilibrium and the existence of traveling wave fronts of the system (2).Finally, we draw a conclusion.

DYNAMICAL BEHAVIORS OF THE SYTEM (2)
It is easy to check that the system (2) has only three nonnegative constant solutions: and the positive equilibrium β β + Using the linearization technique ( [8] or [12]) and omitting the detailed derivation ( [12]), we have Next, we study the existence of traveling wave solution for the infinite spatial .To seek a pair of traveling wave fronts of the system (2), we set ( , x ∈ −∞ +∞)  ( ( ), ( )) ( , ) Proof.To prove the theorem, we need to check that a quasi-monotone condition (see [7] or [11]) holds and show that there exists a pair of upper and lower solutions ( ) ( ( ), ( ) where I is a identity matrix, 2 2 × Next, we show that there exists a pair of upper and lower solutions.To do that, we introduce the following set where And it is easy to check that 1 2 ( ) .( ) ∈Γ Next, we check that ψ is a pair of upper solutions to (5).To do that, we have two cases Case i: 0 η > .Then, we have From ( 8) and ( 9), for the case η τ > it follows that and for the case 0 η τ < ≤ , we get Case ii: 0.
η < Using ( 8) and ( 9), we have From Case i and Case ii, we know that ψ is an upper solution to (5).From (13), we get Using ( 13) and ( 14), for η τ ≥ we have for 0 η τ < < we obtain from [11] we know that there exists at least one solution in the set Γ .The proof of the theorem is completed.

CONCLUSION
In our work, we prove the existence of traveling wave fronts for the two-species model for large values of the wave speed .The system ( 2) is a new model and the method to prove the existence of traveling wave fronts is also novel, and it is effective to deal with the case of large wave speeds, which is deserved future study.
open and bounded with smooth boundary ∂Ω , / n ∂ ∂ is differentiation in the direction of the outward unit normal, represent the immature and mature prey population densities at1 2