Two Types Of Three Molecules Autocatalytic Model Differences And Stability

In order to understand the chemical process and the mechanism of life phe-nomenon, the mathematical models are used to describe and study the reaction process, which is an important mean to investigate the biochemical process and has great significance for recognizing the life phenomenon. Almost all biochemical reac-tions involve autocatalytic reaction, among which glycolysis model and Schnaken-berg model are two kinds of important three molecular autocatalytic models. These biochemical reaction phenomena are closely related to our actual production and life, such as the improvement of productive fermentation technology, the biological control of mosquitoes and the storage of fruits and vegetables. Making clear the dy-namical property of biochemical reaction can accurately grasp the catalytic reaction condition to give full play to the role of catalyst, explain some phenomena of the corresponding areas, and predict development law of research object, which plays an important guiding role in the process of production and living and promotes the development of the nonlinear science.Based on the research status of glycolysis model and Schnakenberg model, we study the dynamic behaviors of these two kinds of models in this dissertation, such as the existence, multiplicity and stability of stationary structure, and the existence and stability of spatio-temporal structure. The tools used here include maximum principle, degree theory, bifurcation theory, stability theory, Lyapunov-Schmidt re-duction method and singularity theory. The main contents and results are as follows:In chapter1, the background and research status of three molecular autocat-alytic model are presented, and then the basic theory and knowledge, such as max-imum principle, bifurcation theory, stability theory, Lyapunov-Schmidt reduction method and singularity theory, are given in this chapter.In chapter2, the steady state structure and spatio-temporal structure of gly-colysis model with homogeneous Neumann boundary conditions are investigated. A necessary condition for the existence of non-constant positive steady-state solutions is obtained by using the striking theory of "diffusion-driven instability" proposed by Turing. The degree theory is combined with a priori estimates to give a sufficient condition for the existence of non-constant positive steady-state solutions, which gives a more general result. Furthermore, taking diffusion coefficient d1as bifur-cation parameter, the local structure and global structure of non-constant steady state solution arising from simple bifurcation are analyzed. In particular, Lyapunov-Schmidt technique and singularity theory are applied to study the double bifurca-tion, and the multiplicity, bifurcation direction and stability of simple bifurcation solutions, the application of which is an innovative research for double bifurcation and also breaks through the limitation of stability theory. Then, viewing input δ as bifurcation parameter, the bifurcation direction and stability of the spatial ho-mogeneous periodic solution are demonstrated by Hopf bifurcation theory, however, those of the spatial inhomogeneous periodic solution are the further work.In chapter3, the stationary bifurcation and stability of glycolysis model with fixed boundary conditions are studied. Still taking d1as bifurcation parameter, the simple bifurcation and double bifurcation are obtained in detail, and the stability of the bifurcation solutions are analyzed by stability theory. Especially, the ODE method is our new attempt to deal with the inversion of linear operator. The results obtained in this chapter complement qualitative analysis of glycolysis model, and are different from those of glycolysis model with homogeneous Neumann bound-ary conditions, such as the solution form and equivalence problem of the reduced equation. The Hopf bifurcation of the model with fixed boundary conditions is still blank, and there is the preliminary research thinking for the further reasoning.In chapter4, the stationary structure and dynamical stability of Schnakenberg model are analyzed. The key point of this chapter is the steady state structure, thus the further discussions are based on γ∈(0,2(1/2)-1]. However, the classical simple bifurcation theory is not valid for every bifurcation point of this model, then Lyapunov-Schmidt method and singularity theory are the main tools for simple bifurcation, double bifurcation and stability of the bifurcation solutions, which make up the singularity of simple bifurcation theory and also obtain finely detailed and complete results, such as the existence, multiplicity and stability of the non-constant steady state solutions. The numerical simulation results show that there are the spatial inhomogeneous periodic solutions for γ∈(2(1/2)-1,1), which provide numerical basis for the further Hopf bifurcation.