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

Junping Shi; Zhifu Xie; Kristina Little

doi:10.11948/2011007

2011 Volume 1 Issue 1

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Junping Shi, Zhifu Xie, Kristina Little. CROSS-DIFFUSION INDUCED INSTABILITY AND STABILITY IN REACTION-DIFFUSION SYSTEMS[J]. Journal of Applied Analysis & Computation, 2011, 1(1): 95-119. doi: 10.11948/2011007

Citation:

Junping Shi, Zhifu Xie, Kristina Little. CROSS-DIFFUSION INDUCED INSTABILITY AND STABILITY IN REACTION-DIFFUSION SYSTEMS[J]. Journal of Applied Analysis & Computation, 2011, 1(1): 95-119. doi: 10.11948/2011007

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

1 Department of Mathematics, College of William and Mary, Williamsburg, Virginia, 23187-8795, USA;
2 Y. Y. Tseng Functional Analysis Research Center, Harbin Normal University, Harbin, Heilongjiang, 150025, P. R. China;
3 Department of Mathematics and Computer Science, Virginia State University, Petersburg, VA 23806, USA

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Abstract

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.
- Reaction-difiusion systems /
- instability /
- cross-difiusion /
- pattern formation
MSC: 92C15;35K57;37L15;92D40

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References

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