The Pattern Dynamics Of Two Kinds Of Classical Mathematical Models

In this dissertation,we study the pattern dynamics of two kinds of classical mathematical models,one of which is the diffusion-chemotaxis model with volume-filling effect and no cell proliferation.Another one is the bacteria colony model with cell growth term and densitysuppressed motility.The main contents are as follows: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 the main work.In the second chapter,we study the stability and metastability of stationary patterns in a diffusion-chemotaxis model without cell proliferation.We first establish the interval of unstable wave modes of the homogeneous steady state and show the chemotactic flux is the key mechanism for pattern formation.Then we treat the chemotaxis coefficient as a bifurcation parameter to obtain the asymptotic expressions of steady states.Based on this,we derive the sufficient conditions for the stability of the patten solution and show the metastability of two or more step patterns.All the analytical results are demonstrated by numerical simulations.In the third chapter,we study the pattern formation process of the density-suppressed motility model with cell proliferation by combining theoretical analysis with numerical analysis.Firstly,the stability analysis shows that the growth rate of cells plays an important role in the process of pattern formation.Then,for the supercritical case,the cubic and quantic StuartLandau equations describing the amplitude evolution of the most unstable mode are obtained by using multi-scale weak nonlinear analysis and adjoint theory,and then the asymptotic expressions of the pattern solution are obtained.Next,we study the selection law of the principle wave mode of the stationary pattern under different conditions for the diffusion rate of the substance AHL and the motion value of cells,and further deduce the ordinary differential equation system which characterizes the nonlinear interaction between two unstable modes,so as to explain the influence of the initial value on the pattern shape.In the fourth chapter,we deduced the quintic Stuart-Landau equation for the subcritical case,and then the asymptotic expression of the fifth order of the pattern solution is obtained.Also we analyzed the existence of the hysteresis cycle.It is shown that when the bifurcation parameter does not lie in the range of the bifurcation,there is a large amplitude bifurcation solution.In the fifth chapter,we present the conclusion and put forward some problems which can be further investigated.