Pattern formation for a two-dimensional reaction-diffusion model with chemotaxis

Abstract

This paper is devoted to study the formation of stationary patterns for a chemotaxis model with nonlinear diffusion and volume-filling effect over a bounded rectangular domain. By using linear stability analysis around the homogeneous steady states we establish conditions for the existence of unstable mode bands that lead to the formation of spatial patterns. We derive the Stuart-Landau equations for the pattern amplitudes by means of weakly nonlinear multiple scales analysis and Fredholm theory. In particular, we find asymptotic expressions for a wide range of patterns sustained by the system. These patterns include mixed-mode, square, hexagonal, and roll stationary configurations. Our analytical results are corroborated by direct simulations of the underlying chemotaxis system.