The Problem Of A Three-Species Predator-Prey Model With Common Prey

In this paper,we study a three-species predator-prey model with common prey under the homogeneous Neumann boundary condition.In the classical predator-prey model,the carrying capacity is usually assumed as a constant.In this paper,the carrying capacity of the predator and the prey are variables changing with the amount of common prey which consumed by both predator and prey.At first,based on the study of [18],we prove the existence and uniqueness of the solution by the methods of the upper and lower solutions and the semigroup theory.Secondly,we consider the constant equilibria of nondimensionalized system.We give some sufficient conditions for locally and globally asymptotic stability of constant equilibria by eigenvalue analysis and iteration technique.At last,we consider the steady state system.When the parameters satisfy some conditions,we obtain the nonexistence of nonconstant positive solution by energy estimation.In Chapter 1,we summarize the background and the development of the problem we considered,and briefly introduce the contents of this paper.In Chapter 2,we prove the existence and uniqueness of the solution.In chapter 3,we study the existence and stability of constant equilibria for the parabolic system.In Chapter 4,we prove the nonexistence of nonconstant positive solutions for steady state system under some conditions.