Turing Instability And Hopf Bifurcation In A Diffusive Bimolecular Model With Saturation Law

In this paper, the stability and Hopf bifurcation of the constant steady state solution of a bimolecular reaction-diffusion model with saturation law are studied. This paper is organized as follows:In Chapter One, the study background, current situation of the model, main study contents, and some preliminary knowledge involved in the paper are given.In Chapter Two, the stability of the unique positive equilibrium and Hopf bifurcation of the local system corresponding to the diffusive system are considered by considering (m, a) as parameters.In Chapter Three, the local stability of the diffusive system is analyzed first; then, in the stability domain of the local system, by analyzing the effects of spatial domain size and variations of diffusive coefficients on the local system stability, the conditions of Turing instability in diffusive system are deduced; further, the corrections of the theorem is verified by giving the corresponding numerical simulations.In Chapter Four, in the instability domain of the local system, by choosing a,m,a as bifurcation parameters, spatial homogeneous and non-homogeneous Hopf bifurcation values of the diffusive system are found. Furthermore, when choosing a as the bifurcation parameter, the stability of spatial homogeneous bifurcation periodic solutions as well as bifurcation direction of the diffusive system are found by applying reduction method of the center manifold and the Hopf bifurcation theorem. Relative numerical simulations are also provided.