Patterns For A Chemotaxis-Reaction-Diffusion System In A Two Dimensional Domain

This dissertation is devoted to study the pattern formation of a chemotaxis-reaction-diffusion system with the volume-filling effect in a two dimensional domain.The main contents are as follows:In the first chapter,we briefly introduce the background and the significance of the studied problems,the preliminaries and the mathematical notations used throughout this paper.In the second chapter,we prove that chemotaxis is the key mechanism for pattern formation by applying the globally asymptotic stability analysis,and further determine the unstable mode band and the conditions.In the third chapter,we focus on the single eigenvalue case.For both supercritical and subcritical phenomena,the Stuart-Landau equations for the amplitude are obtained by applying a weakly nonlinear analysis with multiple scales and Fredholm theorem.Then we establish the asymptotic expression of the pattern for each case.The analytical results are corroborated by comparing the figures from directly integrating the full underlying chemotaxis system and asymptotic expression,respectively.In the fourth chapter,we concentrate on the double eigenvalue case.By applying a weakly multiple scales nonlinear analysis and Fredholm theorem we derive the Stuart-Landau equations of the amplitudes for both the noresonance and resonance phenomena.Moreover,the asymptotic expressions of the patterns are obtained for all cases.The theoretical results show excellent qualitative and good quantitative agreement with the simulations of the full chemotaxis system.The fifth chapter summaries our work for the thesis,and points out the future research questions.