Positive Steady States of a Strongly Coupled Predator-Prey System with Holling-(n+1) Functional Response

Xiao-Zhou Feng,Zhi-Guo Wang

doi:10.1155/2013/851028

Abstract

This paper discusses a predator-prey system with Holling-(n+1) functional response and the fractional type nonlinear diffusion term in a bounded domain under homogeneous Neumann boundary condition. The existence and nonexistence results concerning nonconstant positive steady states of the system were obtained. In particular, we prove that the positive constant solution(u~,v~)is asymptotically stable when the parameterksatisfies some conditions.

Highlights

We are interested in the positive steady states of the strongly coupled predator-prey system with Holling(n + 1) functional response
Paper [4] considers the positive steady states for a prey-predator model with some nonlinear diffusion terms, and the sufficient conditions for the existence of positive steady state solutions were obtained by bifurcation theory
The main work of this paper is to study the effects of the fractional type nonlinear diffusion pressures on the existence of nonconstant positive steady states of (1)

Summary

Introduction

We are interested in the positive steady states of the strongly coupled predator-prey system with Holling(n + 1) functional response. Paper [4] considers the positive steady states for a prey-predator model with some nonlinear diffusion terms, and the sufficient conditions for the existence of positive steady state solutions were obtained by bifurcation theory. We point out that most efforts have concentrated on the Lotka-Volterra competition system which was proposed first by Shigesada et al [5] Since their pioneering work, many authors have studied population models with cross-diffusion terms from various mathematical viewpoints, for example, the global existence of time-depending solutions [6,7,8,9,10,11], the stability analysis for steady states [12,13,14], and the steady state problems [15,16,17,18,19,20,21]. We prove that the positive constant solution (ũ, ̃V) is asymptotically stable for different ranges of parameters

Upper and Lower Bounds for Positive Solutions

A Result on Degree Theory

Existence of Nonconstant Positive Solutions