Qualitative Analysis Of Two Kinds Of Reaction Diffusion Model

Partial differential equation is an important branch of mathematics. It has im-portant practical background and theoretical value in the modern science and tech-nology. Many problems of biology, physics, chemistry, economics and engineering are solved by establishing the mathematical model and applying the mathematics the-ory and method of reaction diffusion equation. Also because of this, many scholars pay attention to the research of reaction diffusion equation. Many mathematicians have made research and exploration on this topic and obtained a lot of meaningful results, which promote the development of modern science and technology. This thesis mainly involves a tumor-virus model with diffusion and a ratio-dependent prey-predator model with diffusion.In the first chapter, we introduce the related biological background and devel-opment situation of tumor-virus models and ratio-dependent prey-predator models. We also introduce some related research works and results.In the second chapter, a model for tumor-virus dynamics with homogeneous Neumann boundary conditions is concerned. We first give a priori estimate for the positive solution and analyze the stability of the positive constant solution. When the positive constant solution is unstable, the system may have non-constant positive solution. In order to explain the existence of non-constant positive solutions, we mainly discuss the effects of the rate of killing the infected cells on the bifurcation solution emanating from the positive constant solution by taking the killing rate as the bifurcation parameter. In the high-dimensional case, we find all the bifurcation points and give a precise description for the structure of positive solutions near the bifurcation points by using the local bifurcation theory. In the case of one dimension, we prolong the bifurcation curves by global bifurcation theory.In the third chapter, we investigate a prey-predator model with diffusion and ratio-dependent functional response subject to the homogeneous Neumann bound-ary condition. First, we analysis stability of the positive constant solution of the ODE system. Then, we consider the Hopf bifurcation phenomenon of the model. Taking the growth rate of the prey as the bifurcation parameter, we perform a Hopf bifurcation analysis of the positive equilibrium for both the ODE system and PDE system and find all the bifurcation points by applying the Poincare-Andronov-Hopf bifurcation theorem. What’s more, we discuss the conditions for determining the bifurcation direction and the stability of the bifurcating periodic solution.