Nonlinear Instability And Asymptotic Stability Of Volume-Filling Chemotaxis Models

In this thesis,the dynamic behavior of solutions for volume-filling chemotaxis models with bistable source term or logistic growth are studied.This thesis is composed of three parts including five chapters.In the first part,we mainly introduce the backgrounds and research advance of the volume-filling chemotaxis-diffusion-growth models.In the second part,by using the energy estimates and the bootstrap arguments,the nonlinear instability of constant equilibrium and the early spatiotemporal pattern formation in the volume-filling chemotaxis model with bistable source term are discussed.In the third part,we deals with the asymptotic behavior of positive equilibrium to the three kinds of chemotaxis-diffusion-growth models by applying the energy estimates,the comparative method and the heat operator semigroups theory.First,the local asymptotic stability of the positive and linearly stable constant equilibrium in Keller-Segel chemotaxis model with bistable source term is analyzed.Then it is studied the local asymptotic stability of the positive constant equilibrium of linear stability for the volume-filling chemotaxis model with bistable source term.Finally,we prove the global asymptotic stability of the unique positive equilibrium to the volume-filling chemotaxis model with logistic growth.Our results show that if the ratio between the self-restraint rate of species and chemotactic sensitivities is sufficiently large,and time is relatively long,then the densities of population and chemoattractant all tend to homogeneous state.