Cross-diffusion Induced Instability and Stability in Reaction-diffusion Systems

In a reaction-diffusion system, diffusion can induce the instability of a uniform equilibrium which is stable with respect to a constant perturbation, as shown by Turing in 1950s. We show that cross-diffusion can destabilize a uniform equilibrium which is stable for the kinetic and self-diffusion reaction systems; on the other hand, cross-diffusion can also stabilize a uniform equilibrium which is stable for the kinetic system but unstable for the selfdiffusion reaction system. Application is given to predator-prey system with preytaxis and vegetation pattern formation in a water-limited ecosystem.