Bifurcation Analysis Of A Kind Of Autocatalysis Reaction Diffusion Model

Autocatalytic reaction diffusion is a common phenomenon in the field of biochemical reaction.Its reaction mechanism is complex and changeable,but they still have certain rules to find.With the development of Mathematical Sciences,people begin to use differential equations to establish various mathematical models to describe the process of reaction diffusion.The mathematical model we call the phenomenon of autocatalytic reaction diffusion is the autocatalytic model.Through the analysis of the theoretical analysis of autocatalysis model,the dynamic properties of the model are clear,and it is theoretically explained that these phenomena have an important guiding role in practical production.In this paper,we mainly use the nonlinear analysis and the central manifold theorem of partial differential equations,the canonical theory,the Hopf bifurcation,the local steady bifurcation and the global steady bifurcation theory,to study the dynamic behavior of a class of autocatalytic reaction diffusion models with high order in the homogeneous Neumann boundary condition.First,we use the maximum principle,Harnck inequality,Holder inequality and Poincare inequality to give a priori estimation and correlation properties of positive solutions of the autocatalytic model.Secondly,using the central manifold theorem and the canonical theory,the existence and stability of the Hopf bifurcation of the autocatalytic model and the diffusion system are studied for the bifurcation parameters.The results show that the different values of parameters only determine the direction and stability of the Hopf bifurcation of the ordinary differential system,and the Hopf bifurcation of the diffusion system is unconditionally stable.And Matlab software is used to carry out numerical simulation to confirm the conclusion.Finally,with the diffusion coefficient as the branch parameter,the partial steady state bifurcation of the single eigenvalue is obtained by using the Crandall Rabinowitz partial branch theory.For the case of double eigenvalues,the existence and stability of stable branches are proved by means of spatial decomposition and implicit function theorem.At the same time,by using the global steady state bifurcation theory,some conclusions of the local steady state bifurcation are generalized to the global steady state bifurcation.Finally,we calculate the direction and stability of the steady state bifurcation.In many fields,such as physics,chemistry and biology,are involved in the process of autocatalytic reaction,and the autocatalytic model is the mathematical description of the catalytic reaction process.Through the theoretical analysis of the autocatalytic model,the autocatalytic reaction process can be scientifically intervened,thus making the reaction process in the most favorable direction and utilizing the resources.Production efficiency and ecological protection are of great value.