Non-Constant Positive Solutions Of Two Reaction-Diffusion Equations

In this paper, positive steady-states of two reaction-diffusion equations will be considered. The paper is mainly divided into five chapters.The first section is the introduction of the whole paper. We talk about the background of these problems, main problems and applied methods.The second section is composed of some basic definitions and theorems. They are the foundations and tools of the later work, and we will use them without proof.Next section of this paper mainly investigates the non-constant positive steadystate of the simple Schnakenberg model:We obtain the non-existence of nontrivial solutions through energy method and Implicit Function theorem, and we prove the global existence of non-constant positive solutions by degree theory. At last we give some theorems about bifurcation.The fourth section mainly investigates the competitive model with spatial diffusion and stage-structure:We obtain the stability of the positive constant solution of the competitive model. Moreover, the results of the non-existence of non-constant positive solutions are given by using two different ways which means some certain conditions under which the pattern formation does not occur.At last, we summarize the results of the whole paper.