Abstract

By constructing a suitable Lyapunov functional, the global attractivity of positive periodic solutions for a delayed predator-prey system with diffusion and impulses is studied in this paper. Finally, an example and numerical analysis are given to show the effectiveness of the main results.

1. Introduction

Recently, the existence of positive periodic solution of predator-prey system has attracted more and more attention. By using Mawhin continuation theorem and some analysis techniques, sufficient conditions of the existence of positive periodic solution are derived; see [1–3] and references cited therein.

For example, the authors in [3] proposed the following delayed periodic predator-prey system with impulses and prey diffusion in -patches environments: with initial conditions where denotes the densities of prey species in patch and denotes the density of predator. is the intrinsic growth rates of the prey. is the density-dependent coefficients of the prey species. is the capturing rates of the predator; is the conversion rates of nutrients into the reproduction of predator; . is the death rate of the predator. is dispersal rate of prey species, denotes the delay due to negative feedback of the predator species, and is the time delays due to gestation; that is, mature predators can only contribute to the production of predator biomass; .

By using Mawhin continuation theorem, the authors derived the sufficient conditions of the existence of positive periodic solution. However, in addition to periodic behavior, a hallmark of observed population densities is their permanent and stable behaviors [4–8]. Then, it is necessary to discuss the permanent and stable behaviors of (1) with initial conditions (2). Therefore, in this paper, we are devoted to the study of global attractivity of positive periodic solution for systems (1) and (2).

Throughout this paper, for ,, the following conditions are assumed.,, and are continuous, positive periodic functions with period . is positive constant satisfying and there exists an integer such that

For periodic function , we denote

The organization of this paper is as follows. In Section 2, some definitions and lemmas are introduced. In Section 3, by constructing suitable Lyapunov functional, the sufficient conditions ensuring the global attractivity of periodic solution for system (1) are established. In Section 4, an example and simulations are given to show the validity of the main results. Finally in Section 5, we conclude this paper with a brief discussion.

2. Preliminaries

In this section, some definitions and lemmas are introduced as follows.

Definition 1. System (1) is uniformly persistent if there exists a compact region such that every solution of (1) eventually enters and remains in region .

Definition 2. A bounded positive solution of (1) is globally asymptotically stable if for any other positive bounded solution of (1), the equality holds.
The following lemma ensuring the existence of positive periodic solution for system (1) is from [3].

Lemma 3 (see [3]). System (1) has at least one strictly positive -periodic solution provided thatwhere .

Lemma 4 (see [4]). If , for and , then

The following lemma is from [9]. It will be employed in establishing the asymptotic stability of (1).

Lemma 5. Let be a real number and let be a nonnegative function defined on such that is integrable and uniformly continuous on ; then, .
Under , we consider the nonimpulsive delay differential equation with initial conditions where .

Considering (1) and (8), we have the following lemma which plays key role in the proof of the main results. The proof is similar to that of Theorem 1 in [10], and it is omitted.

Lemma 6. Assume that hold. Then, one has the following.(i)If is a solution of (8) on , then is a solution of (1) on .(ii)If is a solution of (1) on , then is a solution of (1) on .

Lemma 7. Let be a solution of (8); then, there exists , where .

Proof. Define . Calculating the upper right derivative of along the positive solution of (8), then By Lemma 4, for arbitrary small positive constant , there exists such that for . Let ; then, holds for .
In addition, from the second equation of (8), for , ; we have
Similar argument in the proof of Lemma 2.1 of [11] shows that there exists such that for . Take ; then, Lemma 7 follows immediately.

3. Main Results

In this section, by constructing suitable functional, we study the permanence and globally asymptotic stability of the periodic solution of (1).

Theorem 8. Suppose that hold. Further,Then, system (1) is uniformly persistent; that is, there exist and such that where is defined by (10), .

Proof. Define . Suppose that . Calculating the lower right derivative of along the positive solution of (8), then there exist , for any ; we have By Lemma 4, there exists such that Equation (16) implies that there exists a positive integer , for ; we have Noting that is an arbitrary small positive number, we can choose small enough such that Thus, for any , (17) and (18) lead to Let , ; then,
On the other hand, from the th equation of (8) and (20), for , we have According to the assumption , there exists satisfying . Let . If for all , then by (21) it follows that , which leads to a contradiction. Thus, there must exist such that . If for any , then we take ; the conclusions hold. If not, suppose , where ; then, from the above discussion, there exist and such that and for , where . Now suppose that attains its maximum at ; then, . Equation (21) implies that Equation (22) leads to From (21) again, we have Integrating the above inequality from to , we have This contradicts with for . Therefore, holds for any . Let ; then, holds for . In addition, by Lemma 7, . Hence, holds for .This completes the proof.