Positive Solutions Of Three Types Of Reaction-Diffusion Equations

By using the theories in the nonlinear functional analysis and the reaction-diffusion equation, including the implicit function theorem, bifurcation the-ory, topological degree theory, upper and lower solution method, compari-son principle and regularization theory of elliptic and parabolic equations, stability theory and numerical simulation technology based on platform of MATLAB, this article will study a predator-prey model with C-M function-al response under homogeneous Dirichlet boundary conditions, the Lengyel-Epstein reaction-diffusion model under homogeneous Neumann boundary con-ditions and a chemical reaction-diffusion model with Degn-Harrison reaction scheme.In Chapter 1, we first present the research background and present situa-tion of the Lotka-Volterra prey predator model, the Lengyel-Epstein reaction-diffusion system as well as a chemical reaction-diffusion model with Degn-Harrison reaction scheme. Next, the main works of this paper are introduced.In Chapter 2, we are concerned with positive solutions of a predator-prey model with Crowley-Martin functional response under homogeneous Dirichlet boundary conditions. First, we prove the existence and reveal the structure of the positive solutions by using bifurcation theory. Then, we investigate the uniqueness and stability of the positive solutions for a large key parameter. In addition, we derive some sufficient conditions for the uniqueness of the positive solutions by using some specific inequalities. Moreover, we discuss the extinction and persistence results of time-dependent positive solutions to the system. Finally, we present some numerical simulations to supplement the analytic results in one dimension.In Chapter 3, we continue to study the positive solutions of a predator-prey model with Crowley-Martin functional response under homogeneous Dirich-let boundary conditions. First of all, we state some known results and give the existence of positive solutions. Furthermore, the effect of large a is extensively studied. By analyzing the asymptotic behaviors of positive solutions when a goes to oo, we derive a complete understanding of the multiplicity, uniqueness and stability of positive solutions.In Chapter 4, we continue the mathematical study started in [78,79] on the analytic aspects of the Lengyel-Epstein reaction-diffusion system. First, we further analyze the fundamental properties of nonconstant positive solutions. On the other hand, we continue to consider the effect of the diffusion coefficient d. We obtain another nonexistence result for the case of large d by the implicit function theory, and investigate the direction of bifurcation solutions from (u*, v*). These results promote the Turing patterns arising from the Lengyel-Epstein reaction-diffusion system.In Chapter 5, we consider a chemical reaction-diffusion model with Degn-Harrison reaction scheme under homogeneous Neumann boundary conditions. The existence of Hopf bifurcation to ordinary differential equation (ODE) and partial differential equation (PDE) models are derived, respectively. Further-more, by using of the center manifold theory and the normal form method, we establish the bifurcation direction and stability of periodic solutions.