Existence of Periodic Solutions in a Discrete Predator‐Prey System withBeddington‐DeAngelis Functional Responses

Changjin Xu,Maoxin Liao

doi:10.1155/2011/970763

Abstract

A discrete predator‐prey model with Holling II and Beddington‐DeAngelis functional responses is investigated. With the aid of differential equations with piecewise constant arguments, a discrete version of continuous nonautonomous delayed predator‐prey model with Beddington‐DeAngelis functional responses is proposed. By using Gaines and Mawhin′s continuation theorem of coincidence degree theory, sufficient conditions for the existence of positive solutions of the model are established.

Highlights

In population dynamics, the functional response refers to the number of prey eaten per predator per unit time as a function of prey density
Where x t stands for the density of the prey, y t and z t are the densities of the predators, respectively, and r, a, A, d, e, D, E, K, b, and B are positive constants
Assuming that one predator consumes prey according to the Holling II functional response and the other predator consumes prey according to the Beddington-DeAngelis functional response, Cantrell et al 3 proposed the revised version of system 1.1 as follows: dx dt rx t

Summary

Research Article

A discrete predator-prey model with Holling II and Beddington-DeAngelis functional responses is investigated. With the aid of differential equations with piecewise constant arguments, a discrete version of continuous nonautonomous delayed predator-prey model with BeddingtonDeAngelis functional responses is proposed. By using Gaines and Mawhin’s continuation theorem of coincidence degree theory, sufficient conditions for the existence of positive solutions of the model are established

Introduction

Ex t Bx t Cz t

Then it is obvious that