Front propagation and segregation in a reaction–diffusion model with cross-diffusion

A study of front propagation and segregation in a system of reaction–diffusion equations with cross-diffusion is presented. The reaction models predator–prey dynamics involving two fields. The diffusive part is nonlinear in the sense that the diffusion coefficient, instead of being a constant as in the well-studied case, depends on one of the fields. A key element of the model is a cross-diffusion term according to which the flux of one of the fields is driven by gradients of the other field. The original motivation of the model was the study of the turbulence–shear flow interaction in plasmas. The model also bears some similarities with models used in the study of spatial segregation of interacting biological species. The system has three nontrivial fixed points, and a study of traveling fronts solutions joining these states is presented. Depending on the stability properties of the fixed points, the fronts are uniform or have spatial structure. In the latter case, a cross-diffusion-driven pattern-forming (k≢0) instability leads to segregation in the wake of the front. The segregated state consists of layered structures. A Ginzburg–Landau amplitude equation is used to describe the dynamics near marginal stability.