Cross-diffusion driven instability for a nonlinear reaction-diffusion system

In this work we investigate the possibility of the pattern formation for a system of two coupled reaction-diffusion equations. The nonlinear diffusion terms has been introduced to describe the tendency of two competing species to diffuse faster (than predicted by the usual linear diffusion) toward lower densities areas. The reaction terms are chosen of the Lotka-Volterra type in the competitive interaction case. The system is supplemented with the initial conditions and no-flux boundary conditions. Through a linear stability analysis we find the conditions which allow a homogeneous steady state (stable for the kinetics) to become unstable through a Turing mechanism. In particular, we show how cross-diffusion effects, together with the usual scale effect, are the responsible of the initiation of spatial patterns. Through a weakly nonlinear analysis we are able to predict the shape and the amplitude of the pattern near the marginal stability. In the case of 1--D spatial domain, we find the Stuart-Landau equation for the amplitude which correctly describes the pattern close to the bifurcation value. When the domain size is large, the pattern is formed sequentially and travelling wavefronts are the precursors to patterning. In this case the amplitude of the pattern is modulated in space and its corresponding evolution equation is the Ginzburg-Landau equation.