Stability in a Simple Food Chain System with Michaelis-Menten Functional Response and Nonlocal Delays

Wenzhen Gan,Zhigui Lin,Qunying Zhang,Canrong Tian

doi:10.1155/2013/936952

Abstract

This paper is concerned with the asymptotical behavior of solutions to the reaction-diffusion system under homogeneous Neumann boundary condition. By taking food ingestion and species' moving into account, the model is further coupled with Michaelis-Menten type functional response and nonlocal delay. Sufficient conditions are derived for the global stability of the positive steady state and the semitrivial steady state of the proposed problem by using the Lyapunov functional. Our results show that intraspecific competition benefits the coexistence of prey and predator. Furthermore, the introduction of Michaelis-Menten type functional response positively affects the coexistence of prey and predator, and the nonlocal delay is harmless for stabilities of all nonnegative steady states of the system. Numerical simulations are carried out to illustrate the main results.

Highlights

The overall behavior of ecological systems continues to be of great interest to both applied mathematicians and ecologists
Motivated by the work above, we are concerned with the following food chain model with Michaelis-Menten type functional response:
We show the existence and uniqueness of positive steady state

Summary

Introduction

The overall behavior of ecological systems continues to be of great interest to both applied mathematicians and ecologists. The predator-prey system for three species with Michaelis-Menten type functional response was studied by many authors [1,2,3,4]. The asymptotic behavior of solutions of the reaction-diffusion equations with nonlocal delay was investigated in [11, 12] by using an iterative technique and in [13,14,15] by the Lyapunov functional. The stability and Hopf bifurcation were discussed in [16] for a diffusive logistic population model with nonlocal delay effect. Motivated by the work above, we are concerned with the following food chain model with Michaelis-Menten type functional response:. A brief discussion is given to conclude this work

Preliminary Results

Global Stability

Numerical Illustrations

Discussion