Qualitative Analysis Of A Class Of Templater Models

In recent years,it has become one of the hot topics to study models in the field of biochemistry.The Templator model studied in this paper is a chemical model based on monomer-dimer reaction system with self complementary template.In this paper,some properties of positive solutions for Templator model are discussed by means of the theory of nonlinear elliptic equations.The specific discussion is divided into the following parts:In Chapter 1,the current research status and significance of the Templator system are put forward.In Chapter 2,the existence and nonexistence of positive solutions of Templator system under homogeneous Neumann boundary conditions are discussed.Firstly,priori estimates of the positive solution are given by means of the maximum principle and Harnack inequality.Then the nonexistence of the nonconstant positive solution is proved by the Young inequality with ? and Poincare inequality.In addition,the sufficient conditions for the existence of the nonconstant positive solution are obtained by means of the degree theory.At last,the local bifurcation and global bifurcation of the system are discussed based on the topological degree theory.In Chapter 3,firstly,the stability for the positive constant equilibrium state of the local system is discussed by using the spectral analysis method.Moreover,we give conditions of the parameters in the reaction-diffusion system should satisfy when the steady state is stable and when the system is Turing unstable.At last,the theoretical results of the stability are verified by numerical simulations.