Qualitative Analysis Of A Class Of Improved Predator-prey Models

In the thesis, by using the theory and method of the nonlinear analysis and partial differential equations, we deal with a modified Leslie-Gower predator-prey model with Beddington-DeAngelis functional response under homogeneous Neuman-n boundary condition Firstly, we discuss the large time behavior of solutions?the local and global asymp-totic stability of the semi-trivial solution and the positive constant equilibrium for the parabolic system. Secondly, we discuss the existence of the non-constant pos-itive steady states as well as the local and global bifurcation solutions emanating from the positive constant equilibrium for the corresponding equilibrium system.The main contents in the thesis are as follows:In chapter 1, we briefly introduce the research background and development status of predator-prey models, and also present the model to study in the thesis.In chapter 2, we consider the large time behavior and stability of parabolic sys-tem. Firstly, the large time behavior which includes global attractor and persistence is studied by the comparison principle of parabolic equation. Secondly, the sufficient conditions for the local and global asymptotic stability of the semi-trivial solution are given by the spectrum analysis and comparison principle of the parabolic equa-tions. Finally, the local asymptotic stability of the positive constant equilibrium is discussed by the linearization method; The global asymptotic stability of the positive constant equilibrium is discussed by the monotone iteration principle and constructing Lyapunov function method.In chapter 3, we consider the existence of the solutions of equilibrium system. Firstly, a priori estimate for the positive steady states is obtained by the Maximum Principle and the Harnack Inequality. Secondly, on the basis of a priori estimate, the non-existence and existence of non-constant positive steady states are given by employing the energy integral method and the Leray-Schauder degree argument. Finally, regarding the diffusion coefficient as a bifurcation parameter, the local and global bifurcation emanating from the positive constant equilibrium are proved by the topological degree theory.