Uniqueness and Multiplicity of a Prey‐Predator Model with Predator Saturation and Competition

Xiaozhou Feng,Lifeng Li

doi:10.1155/2012/627419

Abstract

We investigate positive solutions of a prey‐predator model with predator saturation and competition under homogeneous Dirichlet boundary conditions. First, the existence of positive solutions and some sufficient and necessary conditions is established by using the standard fixed point index theory in cones. Second, the changes of solution branches, multiplicity, uniqueness, and stability of positive solutions are obtained by virtue of bifurcation theory, perturbation theory of eigenvalues, and the fixed point index theory. Finally, the exact number and type of positive solutions are proved when k or m converges to infinity.

Highlights

Considering the destabilizing force of predator saturation and the stabilizing force of competition for prey, Bazykin 1 proposed the function response f u, v 1/ 1 mu 1 kv in the prey-predator model instead of the classical Holling-type II functional response
We are concerned with the positive solution of the boundary value problem of the following elliptic system corresponding to the system 1.1 :
It has been conjectured that there is at most one positive solution, but this was shown only for the case the space dimension n is one, see 9

Summary

Introduction

Considering the destabilizing force of predator saturation and the stabilizing force of competition for prey, Bazykin 1 proposed the function response f u, v 1/ 1 mu 1 kv in the prey-predator model instead of the classical Holling-type II functional response. In 12 , Blat and Brown studied the existence of positive solutions to 1.2 by making use of both local and global bifurcation theories. The case when m goes to infinity was extensively studied by Du and Lou in 13, 14, 18 They gave a good understanding of the existence, stability, and number of positive solutions for large m. More works can refer to 19 , Wang studies the existence, multiplicity, and stability of positive solutions of 1.2. This paper is organized as follows: in Section 2, we give sufficient and necessary conditions for the existence of coexistence states of 1.2 by using index theory.

Existence and Nonexistence of Coexistence States

Global Bifurcation and Stability of Positive Solution