Chemotaxis-driven pattern formation for a reaction–diffusion–chemotaxis model with volume-filling effect

Abstract

In this paper we analytically and numerically investigate the emerging process of pattern formation for a reaction–diffusion–chemotaxis model with volume-filling effect. We first apply globally asymptotic stability analysis to show that the chemotactic flux is the key mechanism for pattern formation. Then, by weakly nonlinear analysis with multiple scales and the adjoint system theory, we derive the cubic and the quintic Stuart–Landau equations to describe the evolution of the amplitude of the most unstable mode, and thus the analytical approximate solutions of the patterns are obtained. Next, we present the selection law of principal wave mode of the emerging pattern by considering the competition of the growing modes, and for this we deduce the change rule of the most unstable mode and the coupled ordinary differential equations that indicates the significant nonlinear interaction of two competing modes. Finally, in the subcritical case we clarify that there exists the phenomenon of hysteresis, which implies the existence of large amplitude pattern for the bifurcation parameter values smaller than the first bifurcation point. Therefore, we answer the open problems proposed in the known references and improve some of results obtained there. All the theoretical results are tested against the numerical results showing excellent qualitative and good quantitative agreement.