Bifurcation Analysis Of Two Saturated Reversible Biochemical Reaction Models

The study of biochemical reaction is an important means to study the inter-nal mechanism of organisms.in this paper,we mainly study two kinds of rever-sible biochemical reaction models with different saturation rates under the Neu-mann boundary condition.One is the multiple-saturation reversible biochemical reaction model,the other is the reversible biochemical reaction model with Mic-halis saturation.For the multiple-saturation reversible biochemical reaction model,the stabi-lity,Turing instability,Hopf bifurcation and steady-state bifurcation are studied by using stability theory and bifurcation theorem.Firstly,the reversible coeff-icient c is used as the parameter in the ordinary differential system,and the local stability of the unique positive equilibrium point is studied by analyzing the eigenvalue problem in detail.Then,the effect of diffusion coefficient on the stability of the model is studied,and the stability conditions of the positive equilibrium and the Turing instability of the diffusion system are analyzed.Secondly,Hopf bifurcation of ordinary differential system and diffusion system are given by appling the normal form theory and the center manifold theorem respectively.When the reversible coefficient c varies around the constant c0,the system may produce Hopf bifurcation.In addition,the existence of local steady-state bifurcation is discussed.Appling Crandall-Rabinowitz local bifurca-tion theory,the bifurcation structure at simple eigenvalues is established.Spatial decomposition technique and implicit function theorem are used to deal with double eigenvalues.Finally,numerical simulations are carried out on the basis of theoretical analysis,and while verifying the supplementary theoretical results,it also intuitively demonstrates the rich dynamic behavior of the system.For the reversible biochemical reaction model with Michalis saturation,the type and stability of the positive equilibrium,Hopf bifurcation of spatial homo-geneity and spatial inhomogeneity,Turing instability and steady-state bifurcation are discussed.Firstly,the existence of unique positive equilibrium is found by solving algebraic equations,and the types and stability of unique positive equili-brium are discussed.Secondly,the existence,stability and direction of Hopf bifurcation of the ordinary differential systems are given by using reversible coefficients c as parameter.Then,the existence of spatial homogeneous Hopf bifurcation and spatial inhomogeneous Hopf bifurcation in reaction-diffusion system are analyzed,and the existence conditions and bifurcation points deter-mined by diffusion coefficient are obtained.The bifurcation theory is used to prove that the spatial homogeneous Hopf bifurcation is stable and the direction is subcritical.In addition,Turing instability caused by diffusion is established,and some sufficient conditions for the existence of Turing instability are given.Moreover,the existence of local steady-state bifurcation in one-dimensional space is discussed.The steady-state bifurcation at single eigenvalue and double eigenvalues are studied respectively.Finally,the results are verified by numerical simulation.