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Volume 1, Number 1, 2011, Pages 95-120                                                                DOI:10.11948/2011007
Cross-diffusion induced instability and stability in reaction-diffusion systems
Junping Shi,Zhifu Xie,Kristina Little
      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 self-diffusion reaction system. Application is given to predator-prey system with preytaxis and vegetation pattern formation in a water-limited ecosystem.
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