Equilibrium Analysis Of Two Types Of Predator-prey Models With Diffusion

In recent years, strong actual demand have prompted scientists to propose a number of mathematical models, and a large number of mathematical models can be grouped into so-called reaction-diffusion equation. By establishing mathematical models, people can describe the real world, thus making it a powerful tool to protect the natural environment. Those models with Leslie type and strong Allee effect are very important mathematical models in the field of population dynamics, thus more and more scholars have concerned about those models and conducted extensive research.This thesis systematically investigates the dynamical properties of two reaction-diffusion predator-prey systems, mainly using the knowledge and theories of the reaction-diffusion equations and the corresponding equilibrium equations to obtain some conditions that can ensure the predator and prey coexist. The knowledge used here includes the maximum principle, upper and lower solution methods, Schauder fixed points theorem, bifurcation theory and stability theory.The main content and structure of this thesis are as follows:Chapter 1 introduces the biological background and presents research situation of predator-prey models, and some preliminary knowledge which are very useful in this thesis are given.Chapter 2 investigates the existence and stability of coexistence solutions of the predator-prey model of modified Leslie type with the generalized Holling type III functional response under Dirichlet boundary conditionFirstly, by the maximum principle, a priori estimate of negative solutions is given; Secondly, the local existence of positive solution is given by regarding c as a bi-furcation parameter and making use of maximum principle and bifurcation theory, and further analysis of the global structure of local bifurcation solution is given; Finally, the stability of positive solution is discussed by the stability theory of linear operator.In chapter 3, we study the equilibrium system of the predator-prey model with strong Allee effect under homogeneous Neumann boundary conditionFirstly, the stability of constant solution (u, v) is discussed by the stability theory of linear operator; Secondly, taking the diffusion coefficient d as the parameter, the local and global bifurcation from the positive constant solution (u*, v*) are studied by the theorems of bifurcation in one dimension. The sufficient conditions for existence of bifurcation solutions of the system and its trend are obtained.