CROSS-DIFFUSION INDUCED INSTABILITY AND STABILITY IN REACTION-DIFFUSION SYSTEMS

Abstract

In a reaction-difiusion system, difiusion 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-difiusion can destabilize a uniform equilibrium which is stable for the kinetic and self-difiusion reaction systems; on the other hand, cross-difiusion can also stabilize a uniform equilibrium which is stable for the kinetic system but unstable for the selfdifiusion reaction system. Application is given to predator-prey system with preytaxis and vegetation pattern formation in a water-limited ecosystem.