Positive Periodic Solution for a Two-Species Predator-Prey System

A two-species predator-prey system with time delay in a two-patch environment is investigated. By using a continuation theorem based on coincidence degree theory, we obtain some sufficient conditions for the existence of periodic solution for the system.


Introduction
Dynamical systems generated by predator-prey models have long been the topic of research interest of many biomathematical scholars, and there have been vast studies to investigate the dynamics of predator-prey models, see e.g., Refs.[1]- [12] and references therein.In 1975, Beddington [13] and DeAngelis [14] proposed the predator-prey system with the Beddington-DeAngelis functional response as follows.
In the last years, some experts have studied the system [15]- [21].Recently, Li and Takeuchi [22] proposed the following model with both Beddington-DeAngelis functional response and density dependent predator and discussed the dynamic behaviors of the model.In this paper, we consider the following nonautonomous two-species predator-prey system with diffusion and time delays.
where ( ) i x t represents the prey population in the ith patch ( ) , and ( ) x t represents the predator population.
( ) i D t denotes the dispersal rate of the prey in the ith patch ( ) . We always make the following fundamental assumptions for system (1.3): τ is positive constant and ( ) ( ) m t re positive continuous ω-periodic functions.The main purpose of this paper is, by using the coincidence degree theory to derive the sufficient conditions for the existence of periodic solution of (1.3).

Preliminaries
The method to be used in this paper involves the applications of the continuation theorem of coincidence degree.we shall use some concepts and results from the book by Gaines and Mawhin [23].
Let X, Z be real Banach spaces, : be a linear mapping, and  ) be an open bounded set, L be a Fredholm mapping of index zero and N be L-compact on Ω .Assume 1) for each ( ) 2) for each , 0; has at least one solution in DomL Ω  .Throughout this paper, we adopt the notations ( ) where ( ) f t is an ω -periodic continuous function.
Then system (1.3) has at least one positive ω-periodic solution.
Proof.Let ( ) ( ) , , u u u u = , ( ) 3) can be rewritten as follows: where all function are defined as ones in system (1.3).It is easy to know that if (3.1) has one ω -periodic solu- tion ( ) ( ) ( ) ( ) T e ,e ,e x t x t x t * * * is a positive ω -periodic solution of system (1.3)Therefore, to complete the proof , it suffices to show that system (3.1) has one ω -periodic solution. Take , then X and Z are Banach space with the norm ⋅ .
Therefore, L is a Fredholm map- ping of index zero.Through an easy computation we find that the inverse p K of p L has the form . Clearly, QN and tinuous.By using Arzela-Ascoli theorem, it is not difficult to prove that ( ) ( ) Suppose that u X ∈ is a solution of (3.2) for an appropriate ( ) Multiplying the first equation of (3.2) by This yields

(
u 14), (3.15) to (3.6) and (3.7), which leads to is a constant vector in R 3 and u ρ = .It follows from the definition of ρ that ( ) 0 QN u ≠ , so the condition (2) in Lemma 2.1 is satisfied.In order to verify (3) in Lemma 2.1, we define


By now we have proved the condition (3) in Lemma 2.1.This completes the proof of Theorem 3.1.

the mapping N will be called L-compact on Ω if ( ) QN Ω is bounded and
and Im L is closed in Z.If L is a Fredholm mapping of index zero and there exist continuous projectors