Qualitative Analysis Of Two Kinds Of Autocatalysis Reaction-diffusion Models

Autocatalysis reaction is a common reaction in chemical reaction.Study on the dynamics of the autocatalysis model can help us to understand the action mechanism and evolution law of the reaction process,and to accurately predict the development trend of the reactants.In this paper,we mainly investigates the dynamics of two kinds of autocatalytic reaction models with delay and saturation effect,including the stability of equilibrium,Turing instability,the existence of Hopf bifurcation,the existence of non-constant solution,the existence as well as the properties of steady-state bifurcation.The main work is as follows:In chapter 1.Firstly,the research background and work of reactiondiffusion equation,autocatalysis,delay differential equation and saturation effect are introduced.Secondly,the significance of parameters in Brusselator model and autocatalysis model and the research status at home and abroad are given.Finally,the main contents of this paper are pointed out.In chapter 2.The Brusselator model with delay is studied under homogeneous Neumann boundary conditions.The stability of equilibrium,the Turing instability caused by diffusion and the existence of Hopf bifurcation are analyzed by using linear stability theory and Hopf bifurcation theory.For ordinary differential equation and partial differential equation,taking delay τ as bifurcation parameter,the stability of equilibrium and existence of Hopf bifurcation are studied by utilizing distribution of eigenvalues.The results show that the stable equilibrium point will become unstable after adding delay,and delay will destroy the stability of the equilibrium of the model,and induce the occurrence of periodic solutions.The influence of diffusion coefficient on the stability of equilibrium is analyzed in the diffusion system.The conditions of equilibrium stability and Turing instability are given.The results show that Turing instability will occur in the system when d1/d2 is properly small.Based on the basis of theoretical analysis,numerical simulation is carried out to verify the theoretical analysis,which shows the rich dynamic behavior of system.In chapter 3.An arbitrary-order autocatalysis reaction-diffusion model with saturation effect is established.We mainly discuss the stability of equilibrium,Hopf bifurcation,Turing instability,existence and nonexistence of non-constant positive solutions,local steady-state bifurcation and global steady-state bifurcation under homogeneous Neumann boundary conditions.Firstly,taking the saturation coefficient k as bifurcation parameter,the stability of positive equilibrium,the existence of Hopf bifurcation are obtained through linear analysis,and the direction and stability of bifurcation are discussed.At the same time,a sufficient condition for Turing instability caused by diffusion is established.Secondly,a priori estimates and correlation properties of positive solutions of the system are given based on the maximum principle,Poincare inequality and H(?)lder inequality.Then,the effects of diffusion and saturation on the nonexistence of non-constant positive solutions are fully considered.The results show that there is no non-constant positive solution when d1 is small,d2 large or k large.The existence of non-constant positive solution of the model is proved by using Leray-Schauder topological degree.Finally,taking the diffusion coefficient d1 as the bifurcation parameter,we obtain the local steadystate bifurcations at single eigenvalue and double eigenvalues,respectively.The classical Crandall-Rabinowitz theorem is used to prove the steady-state bifurcation at single eigenvalue,and the techniques of space decomposition and implicit function theorem are used to deal with the steady-state bifurcation at double eigenvalues.Meanwhile,some conclusions of the local steady-state bifurcation are generalized to the global steady-state bifurcation,and a formula for determining the direction of steady-state bifurcation is given.Moreover,numerical simulation is carried out to verify the theoretical analysis.