Oscillatory dynamics in a discrete predator-prey model with distributed delays.

Changjin Xu,Ying Guo,Lilin Chen,Peiluan Li,Pan-Ping Liu

doi:10.1371/journal.pone.0208322

Abstract

This work aims to discuss a predator-prey system with distributed delay. Various conditions are presented to ensure the existence and global asymptotic stability of positive periodic solution of the involved model. The method is based on coincidence degree theory and the idea of Lyapunov function. At last, simulation results are presented to show the correctness of theoretical findings.

Highlights

It is well known that the qualitative analysis of predator-prey models is an interesting mathematical problem and has received great attention from both theoretical and mathematical biologists [1,2,3,4,5]
Liu [8] dealt with the impulsive periodic oscillation of a predator-prey model with Hassell-Varley-Holling functional response
The principle aim of this paper is to propose a discrete version of system (1) and analyze the effect of the periodicity of the ecological and environmental parameters on the dynamics of discrete time predator-prey model

Summary

Introduction

It is well known that the qualitative analysis of predator-prey models is an interesting mathematical problem and has received great attention from both theoretical and mathematical biologists [1,2,3,4,5]. Liu and Yan [7] considered positive periodic solutions for a neutral delay ratiodependent predator-prey model with a Holling type II functional response. Liu [8] dealt with the impulsive periodic oscillation of a predator-prey model with Hassell-Varley-Holling functional response. We proposed the following predator-prey model with Holling. The principle aim of this paper is to propose a discrete version of system (1) and analyze the effect of the periodicity of the ecological and environmental parameters on the dynamics of discrete time predator-prey model. If L is a Fredholm mapping of index zero and there exist continuous projectors P: X ! L is a bounded linear operator and KerL 1⁄4 lca; ImL 1⁄4 l0a and dimKerL 1⁄4 2 1⁄4 codim ImL; L is a Fredholm mapping of index zero.

X oÀ 1

À s2k3ju2ðkÞj þ s2 ð1 þ mx1 Þ2 ju1ðkÞj

Conclusions