Turing instability and traveling fronts for a nonlinear reaction–diffusion system with cross-diffusion

In this work we investigate the phenomena of pattern formation and wave propagation for a reaction–diffusion system with nonlinear diffusion. We show how cross-diffusion destabilizes uniform equilibrium and is responsible for the initiation of spatial patterns. Near marginal stability, through a weakly nonlinear analysis, we are able to predict the shape and the amplitude of the pattern. For the amplitude, in the supercritical and in the subcritical case, we derive the cubic and the quintic Stuart–Landau equation respectively. When the size of the spatial domain is large, and the initial perturbation is localized, the pattern is formed sequentially and invades the whole domain as a traveling wavefront. In this case the amplitude of the pattern is modulated in space and the corresponding evolution is governed by the Ginzburg–Landau equation.