The Coexistence States Of Several Reaction Diffusion Systems

The theoretical system of reaction diffusion systems is produced by the research on the reciprocities of interacting populations within a population dynamic sys- tem. With the investigation going deeply, the reaction diffusion equations are widely used to investigate the population dynamic systems with diffusion. The applications of reaction diffusion equations play an active role in physics, chemistry, medicine, the protection of creatures and plants and in the comprehensive control and exploitation of environment.In the light of the current researches and applications of the theoretics of reaction diffusion systems, based on some pioneering works, for a two substances autocatalytic re-action model, a two species predator-prey model with non-monotonic functional response and diffusion, a three species periodic cooperating model and a periodic competition model with diffusion, we apply the methods of nonlinear analysis and the tools of nonlinear par-tial differential equations(mainly the theory of second order elliptic and parabolic partial differential equations) to study the dynamical behavior of these models(including the ex-istence, nonexistence, stability of the steady-state solutions and the asymptotic behavior of these models, etc.), some valuable results are obtained.The structure and contents of this paper are as follows:In chapter 1, we list some basic theory and classical results of reaction diffusion systems, such as the maximum principle, the sub- and super-solutions method and the eigenvalue problems of second order elliptic and parabolic partial differential equations, the fixed point index theory, bifurcation theory and stability theory. These are the basic parts that will be very useful in the forthcoming contents.In chapter 2, a two substances autocatalytic reaction model with homogeneous Neu-mann boundary conditions is considered. We first discuss a few characteristics of the positive solutions by integration and some known inequalities, and by the boundedness of the solution, it is shown that there is no positive solution when the diffusion rates are too large. Secondly, taking the constant appeared in the system and the diffusion rates as the bifurcation parameter respectively, we analysis the bifurcation solution which em-anates from the constant solution. The stability of the positive constant solution and the bifurcation solution are investigated. In succession, using the fixed point index theory, the existence of non-constant positive solutions is exploded. Finally, we discuss the global ex-istence of the positive solutions, it shows that, in one dimensional, the bifurcation solution which emanates from the constant solution must extend to infinity.In chapter 3, a two-species predator-prey model with non-monotonic functional re-sponse and homogeneous Dirichlet boundary conditions is studied. We consider the sta-bility of the trivial and semi-trivial solutions. Taking the birth rate of one species as the bifurcation parameter, we discuss the existence, uniqueness and stability of the bifurca-tion solution which emanates from the semi-trivial solution. Furthermore, we take the birth rates of two species simultaneously as the bifurcation parameters, using Lyapunov-Schmidt methods, the existence, uniqueness and stability of the bifurcation solution which emanates from the trivial solution(i. e., emanates from double eigenvalues) are also inves-tigated. And the existence of the coexistence states is discussed by using degree theory and the fixed point index theory in the remainder section.In chapter 4, a three-species time-periodic cooperating model with homogeneous Dirichlet boundary conditions is discussed. Using sub-and super-solution method, we give several sufficient conditions for the existence of positive solution. Some a prior estimates are established. For a concrete case, combining algebra with functional analysis, we obtain some sufficient and necessary conditions for the existence of positive solution. Different from many other time-periodic models, the system given in this chapter is coefficient variable. It shows that the discussion on existence of positive solution of three-species time-periodic cooperating model is more complicated than that of in two-species model.In chapter 5, the asymptotic behavior of the coexistence states of a three-species time-periodic competition model with homogeneous Neumann boundary conditions is dis-cussed. The main purpose in this part is to seek for an asymptotic behavior for the coexistence states, that is, one species dies out, and the other two coexist. First, for the positive solutions, we give some estimates by several coefficient functions. A few sufficient conditions of nonexistence for the strict positive solution are obtained. Secondly, applying the sufficient conditions of nonexistence, the asymptotic behavior of the coexistence states is considered. In precisely, one species dies out, and the other two coexist. Similar to chapter 4, the system in this chapter is also coefficient variable. We still find that the dis-cussion on existence of positive solution of three-species time-periodic competition model is more complicated than that of in two-species model.