Abstract

A class of predator-prey system with distributed delays and competition term is considered. By considering the time delay as bifurcation parameter, we analyze the stability and the Hopf bifurcation of the predator-prey system. According to the theorem of Hopf bifurcation, some sufficient conditions are obtained for the local stability of the positive equilibrium point.

1. Introduction

In general, delay differential equations display more complicated dynamics than ordinary differential equations as a time delay could bring a switch in the stability of equilibria and induce various oscillations and periodic solutions. Neural Networks with time delay or distributed delays have been investigated by some researchers [1–4]. Different from [1–4], distributed delays are introduced into a Lotka-Volterra predator-prey system. Lotka-Volterra predator-prey system is always the hot question in mathematical ecology, and many scholars have made deep and broad study in this area [5–10]. May [5] first proposed a model describing a predator-prey system with delay: where , denote the densities of prey and predator at time , respectively, denotes growth time of prey, , denote the growth rates of prey and predator at time , respectively, denote the inner impact factors in two populations, respectively, and denote the mutual impact factors in two populations, respectively.

Recently, Song and Wei [9] developed a Logistic predator-prey model with discrete and distributed delays: where , and are positive constants and is the kernel function satisfying and .

In nature, living space and resources are limited. Therefore it is advisable to study a predator-prey model with competition term. The competition term describes the competitive interactions between the two species. Associated with these works in [5, 9, 10], in this paper, a predator-prey model with distributed delays and competition term is considered:

Obviously, system (3) improves and contains most of existing models. For (3), we define the kernel function

Denote Then we have

Let , , and ; by (6),

Denote , , , and ; from (7), it follows that

2. Main Results

To ensure system (8) with positive equilibrium points, let us suppose system (8) with conditionsG₁:,where .

Clearly, is a positive equilibrium point of system (8) if and only if satisfies

Obviously, under the assumption , system (8) has unique positive equilibrium points , where

Let , , and ; then

Linearizing system (11) at , then the characteristic equation of (12) is where

By (13), we can get

Clearly, is a solution of (15) if and only if satisfies Substituting , into (16), then that is, thus

Let so Combined with , we get where

Let

AssumeG₂:system (22) has at least one positive real solution.

On the other hand, thus where , . Define

When , we can obtain By Hurwitz criterion, it follows that the solutions of (28) are all of negative real parts when , so the positive equilibrium point is locally asymptotically stable.

In the following, we assumeG₃:.

From (13), therefore and then Substituting , into (31), we get

Let Combined with then we can derive the following theorem.