Periodic Solution of Prey‐Predator Model with Beddington‐DeAngelis Functional Response and Impulsive State Feedback Control

Chunjin Wei,Lansun Chen

doi:10.1155/2012/607105

Abstract

A prey‐predator model with Beddington‐DeAngelis functional response and impulsive state feedback control is investigated. We obtain the sufficient conditions of the global asymptotical stability of the system without impulsive effects. By using the geometry theory of semicontinuous dynamic system and the method of successor function, we obtain the system with impulsive effects that has an order one periodic solution, and sufficient conditions for existence and stability of order one periodic solution are also obtained. Finally, numerical simulations are performed to illustrate our main results.

Highlights

The study of the dynamics of prey-predator system is one of the dominant subjects in both ecology and mathematical ecology due to the fact that predator-prey interaction is the fundamental structure in population dynamics
It is well known that Beddington-DeAngelis functional response which was introduced by Beddington and DeAngelis et al 1, 2 can avoid some of the singular behavior of ratiodependent models at low densities and provide better description of predator feeding over a range of prey-predator abundances
Impulsive differential equations have been widely used in various fields of applied sciences, for example, physics, ecology, and pest control

Summary

Introduction

The study of the dynamics of prey-predator system is one of the dominant subjects in both ecology and mathematical ecology due to the fact that predator-prey interaction is the fundamental structure in population dynamics. Many scholars have carried out the study of prey-predator system with various functional responses, such as Monod-type and Hollingtype. In this paper, we investigate prey-predator system with Beddington-DeAngelis functional response. The functional response in system 1.1 is similar to the well-known Holling type II with an extra term γy in the denominator which models the mutual interference among predators. It has some of the same qualitative behaviours as the classical ratio-dependent model i.e., α 0 , but is free from the singular behaviors of ratio-dependent model at low densities which is, the source of controversy 12–14.

Definitions and Lemmas

Existence of Order One Periodic Solution

D A dy dt

Numerical Analysis and Discussion