Qualitative Analysis Of A Kind Of Predator - Prey Model

Population ecology is one of the important branches of biomathermatics. The main purpose of population ecology is to discuss the internal transformation rules among populations, the interaction relationship between the population and other population, and the interaction relationship between population and environment. By discussing these contents, we can understand the primary cause of the population to decline and extinct, and control population growing more reasonably. The rela-tionship between predator and prey is one of the principal concern for researchers. In this thesis, by using the knowledge of nonlinear functional analysis, bifurcation the-ory, and reaction diffusion equations, we discuss a predator-prey model and obtain some conditions that can ensure the predator and prey coexist.We consider a predator-prey model with Beddington-DeAngelis functional re-sponse In chapter 1, under the Dirichlet boundary conditions, the coexistence of the popu-lation about this model is discussed. In chapter 2, without harvesting and under the Neumann boundary conditions, the coexistence of the population about this model is investigated.In chapter 1, under the Dirichlet boundary conditions, the influence of the growth rate of predator to the positive solutions of the system is discussed. First, by using the nonlinear functional analysis and local bifurcation theorem, regarding the growth rate of the predator as the bifurcation parameter, we discuss the existence of the local bifurcation solutions, which emanate from the semi-trivial solution. Then the asymptotic stability of the local bifurcation solution is considered by the related knowledge of bifurcation theory. The second, by using the Gauss-Green formula, Poincare inequality and the upper and lower solution method, we obtain a priori estimates of the positive equilibrium solution to this system. Combing the priori estimates with global bifurcation theory, we extend the local branch to the global branch and get the sufficient condition which ensure the existence of the non-constant positive solutions to this system.In chapter 2, we discuss the large time behavior and the asymptotic stability of the model incorporating the Neumann boundary conditions and without harvesting. In the first part, large time behaviors consisting of global attractor and persistence property are discussed. Firstly, by discussing the global attractor, one can know that every nonnegative solution of this system lies in a certain bounded region, that is to say this solution exists globally. Secondly, the persistence property is investigated and the sufficient condition for the existence of the positive solutions to this system is obtained, in other word, one gets the conditions which can make sure the popu-lation coexist. In the second part, the asymptotic stability of the positive constant solution is considered. First of all, we give the conditions, which can make sure this system has the unique positive constant solution. Under this assumption, we stud-ied the local asymptotic stability of this solution by using the related knowledge of the reaction diffusion equations. Next, by constructing the Lyapunov function, we discuss the global asymptotic stability of the positive constant solution and obtain the conditions which can make sure the non-constant solution do not exist.