Existence, Multiplicity, and Stability of Positive Solutions of a Predator-Prey Model with Dinosaur Functional Response

Xiaozhou Feng,Kenan Shi,Jianhui Tian,Tongqian Zhang

doi:10.1155/2017/4912032

Abstract

We investigate the property of positive solutions of a predator-prey model with Dinosaur functional response under Dirichlet boundary conditions. Firstly, using the comparison principle and fixed point index theory, the sufficient conditions and necessary conditions on coexistence of positive solutions of a predator-prey model with Dinosaur functional response are established. Secondly, by virtue of bifurcation theory, perturbation theory of eigenvalues, and the fixed point index theory, we establish the bifurcation of positive solutions of the model and obtain the stability and multiplicity of the positive solution under certain conditions. Furthermore, the local uniqueness result is studied whenbanddare small enough. Finally, we investigate the multiplicity, uniqueness, and stability of positive solutions whenk>0is sufficiently large.

Highlights

The dynamic relationship between predator and their prey is one of dominant themes in ecology and mathematical ecology
We investigate the property of positive solutions of a predator-prey model with Dinosaur functional response under Dirichlet boundary conditions
By virtue of bifurcation theory, perturbation theory of eigenvalues, and the fixed point index theory, we establish the bifurcation of positive solutions of the model and obtain the stability and multiplicity of the positive solution under certain conditions

Summary

Introduction

The dynamic relationship between predator and their prey is one of dominant themes in ecology and mathematical ecology. The research on existence and uniqueness of the limit cycle of a predator-prey model with Ivlev response can be found in [22, 23]. The permanence and existence and stability of positive periodic solutions of the model were studied in [24,25,26]. Some dynamical behavior analysis of the Ivlev response predator-prey systems was established in [17,18,19,20,21]. This paper mainly aims at establishing the existence, bifurcation, and multiplicity of positive solutions on the corresponding elliptic equations to system (1). We investigate the multiplicity, uniqueness, and stability of positive solutions when k > 0 is sufficiently large

Preliminaries

Existence of Positive Solutions

Bifurcation and Multiplicity

Stability and Multiplicity of Positive Solutions

Conclusion