Analysis On A Generalized Sel'kov-schnakenberg Reaction-diffusion System

Reaction-diffusion equation has important applications in mathematical biology.Generally,it has an important significance for the stability of the constant steady state solution and the existence or nonexistence of nonconstant steady state solutions to the reaction-diffusion system.Especially,it has important application value to understand Turing pattern formation.In recent years,a great deal of research have been devoted to the study of Sel'kov-Schnakenberg model.The results revealed that each parameter regions have different effect on the stability or instability of the constant steady state solution and the existence or nonexistence of nonconstant steady state solutions.In this thesis,we analyse a generalized Sel'kov-Schnakenberg reaction-diffusion system.In addition,using a priori estimates,energy estimates and topological degree and so on to analyze the stability of constant steady state solution and the existence,nonexistence of nonconstant steady state solutions.Our main contents are included in the following seven chapters.Chapter 1,we introduce the background of Sel'kov-Schnakenberg reaction-diffusion system,the previous works and our main results.Chapter 2,we analyze the linearized stability,especially,Turing stability of Sel'kov-Schnakenberg system at the constant steady state solution.Generally,it is hard to study the global stability of the constant steady state solu-tion.In chapter 3,we consider the global stability of the constant steady state solution when p=1 and p=2.Chapter 4,we state a priori estimates for the positive steady state solution to Sel'kov-Schnakenberg system.This will make necessary preparation for the investi-gation of the existence and nonexistence of nonconstant steady state solutions.Chapter 5 and Chapter 6 are devoted to the investigation of the existence and nonexistence of nonconstant steady state solutions.We will show that how the diffusion coefficient d2 and the index p effect the existence and nonexistence of nonconstant steady state solutions.In fact,we have proved that Sel'kov-Schnakenberg reaction-diffusion system admits no nonconstant steady state solution provided that d2 is large enough and 0<p?1,and it has nonconstant steady state solution if d2 is large enough and p>1.This implies,when d2 is large enough,the index p=1 is the critical value of yielding the nonconstant steady state solution,especially,Turing patterns.Thus,our main results essentially improve the previous works.Chapter 7,a brief discussion completes this thesis.