Qualitative Analysis Of Two Types Of Reaction Diffusion Models

In this paper, we mainly discuss two kinds of reaction diffusion models from biology. By means of nonlinear partial differential equations and nonlinear analysis of knowledge, we mainly study the existence, stability, uniqueness and asymptotic behavior of positive solutions to the steady state problems.This paper includes two parts. In the first part, the following reaction diffu-sion systems of predator-prey model with Holling type III functional response is considered. By applying the bifurcation theory, we show the existence of coexistence solution.In the second part, the following system with saturating terms and toxic sub-stances is discussed. We study the bifurcation of equilibrium solutions and the stability of bifurcation solutions.The main contents of this paper are as follows:In chapter 1, the biological background, the research status of the Lotka-Volterra prey predator model are introduced. At the same time, we give the main results of this thesis, we list some basic theories and classical results of reaction diffusion systems, such as the eigenvalue problems, the uniqueness of some classical problems, bifurcation theory and so on. These are the basic parts that will be very important in the following contents.In chapter 2, A class of predator-prey model with Holling type III functional response is discussed. First, we consider the stability of the trivial and semi-trivial solutions, using the theory of linear operators and the method of spectral analysis. In order to research the following context, a priori estimate of non-negative solutions is given by the maximum principle. At the same time, it shows that range of birth rate of a and b when the coexistence solution exists. Then, by means of the bifurcation theory, the positive steady-state solutions bifurcating from the semi-trivial solution branch{(b, ?a,0):a> ?1} are given by taking the birth rate of one species b as the bifurcation parameter. Furthermore, the local bifurcation branch can be extended to the global one. In order to better understand the nature of the bifurcation solutions, the stability of the bifurcation solutions are investigated.In chapter 3, we discuss the uniqueness of the positive solutions and the disap-pearance and asymptotic behavior of positive solutions.In chapter 4, We study the model with saturating terms and toxic substances. We consider the stability of the trivial and semi-trivial solutions. After that, the existence of the positive constant equilibrium is taken into account. The stability of the bifurcation solutions are obtained by using perturbation the theory of linear operators and stability theory of bifurcation solutions.