Pattern Formation For A Volume-filling Chemotaxis Model With Logistic Growth

This paper is devoted to investigate the pattern formation of a volume-filling chemotaxis model with logistic cell growth.The main contents are as follows:In the first chapter,we briefly introduce the development background and current research of the research system.In the second chapter,We first apply the local stability analysis to establish sufficient conditions of destabilization for uniform steady-state solution.Then,by weakly nonlinear analysis with multi-scales,we study the process of pattern formation near the bifurcation point.For the single unstable mode case,we derive the Stuart-Landau equations governing the evolution of the amplitude,and thus the asymptotic expressions of patterns are obtained in both supercritical case and subcritical case.While for the case of multiple unstable modes,we also derive competitive amplitude equations to study the competitive behavior between two unstable modes through the phase plane analysis.In particular,we find that the initial data play a dominant role in the competition.All the theoretical and numerical results are in excellently qualitative agreement and better quantitative agreement than that in [23].Moreover,in the subcritical case,we confirm the existence of stationary patterns with larger amplitudes when the bifurcation parameter is less then the bifurcation point,which gives an positive answer to the open problem proposed in [24].In the third chapter,By deriving the real Ginzburg-Landau equation governing the evolution of pattern amplitude we show that the pattern invades the whole domain in the form of a traveling wave,which implies the existence of a wavefront solution connecting a uniform steady state and a stationary pattern in the considered model with a large chemotactic parameter.The numerical results are in good agreement with the theoretical results.This work is an significant supplement of the related literature.