Pattern formation for a volume-filling chemotaxis model with logistic growth

This paper is devoted to investigating the pattern formation of a volume-filling chemotaxis model with logistic cell growth. We first apply the local stability analysis to establish sufficient conditions of destabilization for the uniform steady-state solution. Then, the weakly nonlinear analysis with multi-scales is used to deal with the emerging process of patterns near the bifurcation point. For the single unstable mode case, we derive the Stuart-Landau equations describing 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 coupled 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 [10]. Moreover, in the subcritical case, we confirm the existence of stationary patterns with larger amplitudes when the bifurcation parameter is less than the first bifurcation point, which gives an positive answer to the open problem proposed in [9].