ABSTRACT

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.

Keywords:Predator-Prey System, Diffusion, Periodic Solution, Coincidence Degree

1. 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.

(1.1)

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

(1.2)

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.

(1.3)

where represents the prey population in the ith patch, and represents the predator population. denotes the dispersal rate of the prey in the ith patch. We always make the following fundamental assumptions for system (1.3): is positive constant and, , , , , , , , , , re positive continuous -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).

2. 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 a continuous mapping. The mapping L is called a Fredholm mapping of index zero if

and is closed in Z. If L is a Fredholm mapping of index zero and there exist continuous projectors and such that, , then the restriction L_P of L to is invertible. Denote the inverse of L_P by. If is an open bounded subset of X, the mapping N will be called L-compact on if is bounded and is compact. Since is isomorphic to, there exists isomorphism.

Lemma 2.1 (Continuation theorem [23] ) Let be an open bounded set, L be a Fredholm mapping of index zero and N be L-compact on. Assume 1) for each ,;

2) for each

3)

Then has at least one solution in.

Throughout this paper, we adopt the notations, , where is an -periodic continuous function.

3. Main Result

Theorem 3.1 Assume that 1);

2);

3);

4).

Then system (1.3) has at least one positive -periodic solution.

Proof. Let, , , , then (1.3) can be rewritten as follows:

(3.1)

where all function are defined as ones in system (1.3). It is easy to know that if (3.1) has one -periodic solution, then is a positive -periodic solution of system (1.3) Therefore, to complete the proof , it suffices to show that system (3.1) has one -periodic solution.

Take and, then X and Z are Banach space with the norm.

Set, , , ,. Obviously, ,

is closed in Z and. Therefore, L is a Fredholm mapping of index zero. Through an easy computation we find that the inverse of has the form

,. Clearly, QN and are continuous. By using Arzela-Ascoli theorem, it is not difficult to prove that is compact for any open bounded set. Moreover, is bounded. Therefore, N is L-compact on with any open bounded set.

Corresponding to the operator equation, , we have

. (3.2)

Suppose that is a solution of (3.2) for an appropriate. Integrating (3.2) over the interval leads to

(3.3)

, (3.4)

. (3.5)

From (3.2)-(3.5), we have

(3.6)

(3.7)

(3.8)

Multiplying the first equation of (3.2) by and integrating over gives.

which implies

(3.9)

By using the inequalities

.

It follows from (3.9) that

, (3.10)

This yields

. (3.11)

By using the inequalities

.

It follows from (3.10) that

. (3.12)

Multiplying the second equation of (3.2) by and integrating over, similarly, we can obtain

. (3.13)

Substitute (3.13) to (3.12), which leads to

So, there exist a positive constant such that

. (3.14)

It follows from (3.13) and (3.14) that there exist a positive constant such that

. (3.15)

Substitute (3.14), (3.15) to (3.6) and (3.7), which leads to

, (3.16)

. (3.17)

From (3.3) we have

(3.18)

From (3.4) we have

. (3.19)

It follows from (3.14), (3.15), (3.18) and (3.19) that there exist such that

, (3.20)

, (3.21)

. (3.22)

From (3.16), (3.17) and (3.20)-(3.22) we have

,

.

So, for we have

, (3.23)

. (3.24)

From (3.5) we have

So, there exist such that

. (3.25)

From (3.5) we also have

. (3.26)

It follows from (3.26) that there exist such that

.

So

. (3.27)

It follows from (3.8), (3.25) and (3.27) that for we have

.

So we have

.

Clearly, are independent of. On other hand, we consider the following algebraic equation

(3.28)

Take, where is large enough such that the solution of (3.28) satisfies

.

Let, then satisfies the condition (1) in Lemma 2.1. When, u is a constant vector in R³ and. It follows from the definition of that, so the condition (2) in Lemma 2.1 is satisfied. In order to verify the condition (3) in Lemma 2.1, we define by

where is a parameter. When, u is a constant vector in and. It is easy to obtain that, then. So, is a Homotopy mapping, due to homogoy invariance theorem of topology degree, we have

It is not difficult to see that the following algebraic equation

has a unique solution

Thus

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

References

NOTES

^*Corresponding author.