Journal of Applied Mathematics

Volume 2012 (2012), Article ID 358918, 7 pages

doi:10.1155/2012/358918

Research Article

Uniformly Strong Persistence for a Delayed Predator-Prey Model

Changjin Xu,¹ Yuanfu Shao,² and Peiluan Li³

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

²Department of Mathematics and Physics, Guilin University of Technology, Guilin 541004, China

³Department of Mathematics and Statistics, Henan University of Science and Technology, Luoyang 471003, China

Received 6 July 2012; Accepted 18 September 2012

Academic Editor: Wan-Tong Li

Copyright © 2012 Changjin Xu et al. 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.

An asymptotically periodic predator-prey model with time delay is investigated. Some sufficient conditions for the uniformly strong persistence of the system are obtained. Our result is an important complementarity to the earlier results.

1. Introduction

The dynamical behavior including boundedness, stability, permanence, and existence of periodic solutions of predator-prey systems has attracted a great deal of attention and many excellent results have already been derived. For example, Gyllenberg et al. [1] studied limit cycles of a competitor-competitor-mutualist Lotka-Volterra model. Mukherjee [2] made a discussion on the uniform persistence in a generalized prey-predator system with parasitic infection. Aggelis et al. [3] considered the coexistence of both prey and predator populations of a prey-predator model. Agiza et al. [4] investigated the chaotic phenomena of a discrete prey-predator model with Holling type II. Sen et al. [5] analyzed the bifurcation behavior of a ratio-dependent prey-predator model with the Allee effect. Zhang and Luo [6] gave a theoretical study on the existence of multiple positive periodic solutions for a delayed predator-prey system with stage structure for the predator. Nindjin and Aziz-Alaoui [7] focused on the persistence and global stability in a delayed Leslie-Gower-type three species food chain. Ko and Ryu [8] discussed the coexistence states of a nonlinear Lotka-Volterra-type predator-prey model with cross-diffusion. Fazly and Hesaaraki [9] dealt with periodic solutions of a predator-prey system with monotone functional responses. One can see [10–19] and so forth for more related studies. However, the research work on asymptotically periodic predator-prey model is very few at present.

The so-called asymptotically periodic function is that a function can be expressed by the form , where is a periodic function and satisfies .

In 2006, Kar and Batabyal [20] investigated the stability and bifurcation of the following predator-prey model with time delay

with initial conditions , where denotes the densities of prey; and denote the densities of two predators, respectively, at time ; and denote the intraspecific competition coefficients of the predators; and denote the conversion of biomass constant; and are the death rate of first and second predator species, respectively; is the maximum values of per capita reduction rate of due to and is the maximum values of per capita reduction rate of due to ; and are half saturation constants. is time delay in the prey species. All the parameters are positive constants. For details, one can see [20].

It will be pointed out that all biological and environment parameters in model (1.1) are constants in time. However, any biological or environmental parameters are naturally subject to fluctuation in time. Thus the effects of a periodically varying environment are important for evolutionary theory as the selective forces on systems in a fluctuating environment differ from those in a stable environment. Therefore, the assumptions of periodicity of the parameters are a way of incorporating the periodicity the environment (such as seasonal effects of weather, food supplies, and mating habits). Inspired by above considerations and considering the asymptotically periodic function, in this paper, we will modify system (1.1) as follows:

with initial conditions .

The principle object of this paper is to explore the uniformly strong persistence of system (1.2). There are very few papers which deal with this topic, see [10, 21].

In order to obtain our results, we always assume that system (1.2) satisfies , , , ,, , , are continuous, nonnegative periodic functions; ,  , , , , , , are continuous, nonnegative asymptotically items of asymptotically periodic functions.

2. Uniformly Strong Persistence

In this section, we will present some result about the uniformly strong persistence of system (1.2). For convenience and simplicity in the following discussion, we introduce the notations, definition, and Lemmas. Let

In view of the definitions of lower limit and upper limit, it follows that for any , there exists such that

Definition 2.1. The system (1.2) is said to be strong persistence, if every solution of system (1.2) satisfied

Lemma 2.2. Both the positive and nonnegative cones of are invariant with respect to system (1.2).

It follows from Lemma 2.2 that any solution of system (1.2) with a nonnegative initial condition remains nonnegative.

Lemma 2.3 (see [10]). If , and , where is a positive constant, when and , we have

In the following, we will be ready to state our result.

Theorem 2.4. Let , , , and be defined by (2.7), (2.10), (2.13), and (2.16), respectively. Assume that conditions and

(H2), (H3)

hold, then system (1.2) is uniformly strong persistence.

Proof. It follows from (2.2) that for any , there exists such that for ,

Substitute (2.5) into the first equation of system (1.2), then we have

By Lemma 2.3, we get

Then for any , there exists such that

Similarly, from (2.2) and the second equation of system (1.2), we obtain that for any , there exists such that

In view of Lemma 2.3, we derive

Then for any , there exists such that

From (2.2) and the third equation of system (1.2), we obtain that for any , there exists such that

In view of Lemma 2.3, we derive

Then for any , there exists such that

According (2.8), (2.11), (2.14) and the first equation of system (1.2), we obtain that for any , there exists such that

Using Lemma 2.3 again, we have

Thus for any , there exists such that

According (2.8), (2.11), (2.14) and the second equation of system (1.2), we obtain that for any , there exists such that

Using Lemma 2.3 again, we have

Thus for any , there exists such that

According (2.8), (2.11), (2.14) and the third equation of system (1.2), we obtain that for any , there exists such that

Using Lemma 2.3 again, we have

Thus the proof of Theorem 2.4 is complete.

Acknowledgments

This work is supported by National Natural Science Foundation of China (no. 11261010 and no. 11161015), 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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