Abstract and Applied Analysis

Volume 2013 (2013), Article ID 956703, 9 pages

http://dx.doi.org/10.1155/2013/956703

Research Article

Dynamics in a Lotka-Volterra Predator-Prey Model with Time-Varying Delays

Changjin Xu¹ and Yusen Wu²

¹Guizhou Key Laboratory of Economics System Simulation, School of Mathematics and Statistics, Guizhou University of Finance and Economics, Guiyang 550004, China

²School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471023, China

Received 31 May 2013; Revised 18 August 2013; Accepted 18 August 2013

Academic Editor: Mark McKibben

Copyright © 2013 Changjin Xu and Yusen Wu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

A Lotka-Volterra predator-prey model with time-varying delays is investigated. By using the differential inequality theory, some sufficient conditions which ensure the permanence and global asymptotic stability of the system are established. The paper ends with some interesting numerical simulations that illustrate our analytical predictions.

1. Introduction

In 1992, Berryman [1] pointed out that the dynamical relationship between predators and their prey has long been and will continue to be one of the dominant themes in both ecology and mathematical ecology due to its universal existence and importance. Dynamical behavior of predator-prey models has been studied by a lot of papers. It is well known that the investigation on predator-prey models not only focuses on the discussion of stability, periodic oscillatory, bifurcation, and chaos [2–26], but also involves many other dynamical behaviors such as permanence. In many applications, the nature of permanence is of great interest. Recently, Chen [27] investigated the permanence of a discrete -species food-chain system with delays. Fan and Li [28] gave a theoretical study on permanence of a delayed ratio-dependent predator-prey model with Holling type functional response. Chen [29] focused on the permanence and global attractivity of Lotka-Volterra competition system with feedback control. Zhao and Jiang [30] analyzed the permanence and extinction for nonautonomous Lotka-Volterra system. Teng et al. [31] addressed the permanence criteria for delayed discrete nonautonomous-species Kolmogorov systems. For more research on the permanence behavior of predator-prey models, one can see [32–40].

In 2010, Lv et al. [41] investigated the existence and global attractivity of periodic solution to the following Lotka-Volterra predator-prey system:

where denotes the density of prey species at time , and stand for the density of predator species at time , and . Using Krasnoselskii’s fixed point theorem and constructing Lyapunov function, Lv et al. obtained a set of easily verifiable sufficient conditions which guarantee the permanence and global attractivity of system (1).

For the viewpoint of biology, we shall consider (1) together with the initial conditions . The principle object of this paper is to explore the dynamics of system (1), applying the differential inequality theory to study the permanence of system (1). Using the method of Lyapunov function, we investigated the globally asymptotically stability of system (1).

The remainder of the paper is organized as follows: in Section 2, basic definitions and Lemmas are given, and some sufficient conditions for the permanence of the Lotka-Volterra predator-prey model in consideration are established. A series of sufficient conditions for the global stability of the Lotka-Volterra predator-prey model in consideration are included in Section 3. In Section 4, we give an example which shows the feasibility of the main results. Conclusions are presented in Section 5.

2. Permanence

For convenience in the following discussing, we always use the notations:

where is a continuous function. In order to obtain the main result of this paper, we shall first state the definition of permanence and several lemmas which will be useful in the proving the main result.

Definition 1 (see [41]). We say that system (1) is permanence if there are positive constants and such that for each positive solution of system (1) satisfies

Lemma 2 (see [42]). If ,, and , when and , we have

If ,, and , when and , we have

Now we state our permanence result for system (1).

Theorem 3. Let ,,, and be defined by (11), (18), (24), and (30), respectively. Suppose that the following conditions:

(H1), (H2)

hold, and then system (1) is permanent; that is, there exist positive constants which are independent of the solution of system (1), such that, for any positive solution of system (1) with the initial condition , one has

Proof. It is easy to see that system (1) with the initial value condition has positive solution passing through . Let be any positive solution of system (1) with the initial condition . It follows from the first equation of system (1) that

Integrating both sides of (7) from to , we get

which leads to

Substituting (9) into the first equation of system (1), it follows that

It follows from (10) and Lemma 2 that

For any positive constant , it follows from (11) that there exists a such that, for all ,

For , from (12) and the second equation of system (1), we have

Integrating both sides of (13) from to , we get

which leads to

Substituting (15) into the second equation of system (1), it follows that

Thus, as a direct corollary of Lemma 2, according to (16), one has

Setting , it follows that

For , from (12) and the third equation of system (1), we have

Integrating both sides of (19) from to , we get

which leads to

Substituting (21) into the third equation of system (1), it follows that

Thus, as a direct corollary of Lemma 2, according to (22), one has

Setting , it follows that

For , it follows from the first equation of system (1) that

Integrating both sides of (25) from to , one has

which leads to

Substituting (27) into the first equation of system (1), it follows that

According to Lemma 2, it follows from (28) that

Setting in (29), we can get

For , from the second equation of system (1), we have

Integrating both sides of (31) from to leads to

which leads to

Substituting (33) into the second equation of system (1), it follows that

By Lemma 2 and (34), we can get

Setting in the above inequality, it follows that

For , it follows from the third equation of system (1) that

Integrating both sides of (37) from to , we get

Hence

Substituting (39) into the third equation of system (1), we derive

In view of Lemma 2 and (40), one has

Setting in (41) leads to

Equations (11), (18), (24), (30), (36), and (42) show that system (1) is permanent. The proof of Theorem 3 is complete.

3. Global Asymptotically Stability of Positive Solutions

In this section, we formulate the global asymptotically stability of positive solutions of system (1).

Definition 4. A bounded positive solution of system (1) is said to be globally asymptotically stable if, for any other positive bounded solution of system (1), the following equality holds:

Definition 5 (see [24]). Let be a real number and be a nonnegative function defined on such that is integrable on and is uniformly continuous on , then .

Theorem 6. In addition to (H1)-(H2), assume further that

(H3),

where are defined by (48), (49), and (50), respectively. Then system (1) has a unique positive solution which is global attractivity.

Proof. According to the conclusion of Theorem 3, there exists and positive constants such that

Define

Calculating the upper-right derivative of along the solution of (1), it follows for that

It follows that

where is defined by Theorem 3 and

By hypothesis , there exist constants and such that

Integrating both sides of (51) on interval yields

It follows from (51) and (52) that

Since are bounded for , so are uniformly continuous on . By Barbalat’s Lemma [24], we have

By Theorems 7.4 and 8.2 in [43], we know that the positive solution of (1) is uniformly asymptotically stable. The proof of Theorem 6 is complete.

4. Numerical Example

To illustrate the theoretical results, we present some numerical simulations. Let us consider the following discrete system:

Here

All the coefficients ,, are functions with respect to , and it is easy to see that

Then ,,,. Thus it is easy to see that all the conditions of Theorem 6 are satisfied. Thus system (55) is permanent which is shown in Figures 1, 2, and 3.

Figure 1: The dynamical behavior of the first component of the solution .

Figure 2: The dynamical behavior of the second component of the solution .

Figure 3: The dynamical behavior of the third component of the solution .

5. Conclusions

In this paper, we have investigated the dynamical behavior of a Lotka-Volterra predator-prey model with time-varying delays. Sufficient conditions which ensure the permanence of the system are derived. Moreover, we also deal with the global stability of the system. It is shown that delay has influence on the permanence and the global stability of system. Thus delay is an important factor to decide the permanence and global stability of the system. Numerical simulations show the feasibility of our main results.

Acknowledgments

This work is supported by National Natural Science Foundation of China (no. 11261010 and no. 11101126), Soft Science and Technology Program of Guizhou Province (no. 2011LKC2030), Natural Science and Technology Foundation of Guizhou Province (J[2012]2100), Governor Foundation of Guizhou Province ([2012]53), and Doctoral Foundation of Guizhou University of Finance and Economics (2010).

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