Bifurcation And Stability Of A Bacteria Colony Model With Density-suppressed Motility

This dissertation is a study on the local bifurcation,global bifurcation,the exis-tence conditions of nonconstant steady-state solution,metastability of two or multi-step solutions and pattern propagation way in a large domain for a class of bacterial colony reaction-diffusion models with density-suppressed motility.In the first chapter,we introduce the research background and significance of the model,the mathematical notations and preliminary knowledges used throughout this thesis as well as our main work.In the second chapter,we study the model with the logistic growth term.The growth coefficient a is regarded as a bifurcation parameter.We discuss the existence and structure of nonconstant steady states by using Rabinowitz local and global theorems.Then the formulas of non-constant steady states are obtained by applying the asymptotic analysis.Based on this,we establish the sufficient conditions and necessary conditions for the sta-bility of nonconstant steady states through dealing with the principal eigenvalue problem of its linearized system.Numerical simulation demonstrates our theoretical resultsIn the third chapter,we consider the model without the growth term(i.e,cells have no proliferation ability).First,we obtain the unstable mode interval.Then,the existence of nonconstant steady states is proved by applying phase plane analysis.Next,the expression of the principal eigenvalue of the multi-step solution is deterived by using the stability analysis,which implies the metastability of multi-step solutions.Finally,the analytial results are clarified by numerical simulationIn the fourth chapter,when the motility function is taken as(?)we deal with the propagation way of pattern invading a large domain.With the help of a weakly nonlinear analysis,we derive the equation of Ginzburg-Landau type for the amplitude of pattern.Then numerical simulation is carried out to show our theoretical results.In the fifth chapter,we present a conclusion and put forward some problems which can be further investigated.