American Institute of Mathematical Sciences

April  1996, 2(2): 255-270. doi: 10.3934/dcds.1996.2.255

Pattern formation in reaction-diffusion systems with $D_2$-symmetric kinetics

 1 Department of Mathematics, University of North Carolina at Wilmington, Wilmington, NC 28403, United States 2 MCNC - North Carolina Supercomputing Center, 3021 Cornwallis Road, Research Triangle Park, NC 27709, United States

Received  August 1995 Revised  January 1996 Published  February 1996

We study the bifurcations of stationary solutions in a class of coupled reaction-diffusion systems on 1-dimensional space where the steady-state system is $D_2$-symmetric and reversible with respect to two involutions.
Because the stationary patterns of such reaction-diffusion systems are the symmetric cycles of its steady-state system, we investigate the bifurcations of manifolds of symmetric cycles near equilibria in general $D_2$-symmetric reversible systems. This is done through an analysis of the bifurcation regimes at strong resonances using 1-dimensional universal unfoldings of $D_2$-symmetric reversible normal forms. We prove there are two disjoint manifolds at "odd" resonance and four disjoint manifolds at "even" resonance. The number of these disjoint manifolds, in turn, determines the number of different types of stationary patterns.
Applications of our analysis to the study of pattern formation in reaction-diffusion systems are illustrated with a predator-prey model arising from mathematical ecology. Numerical results are obtained as a verification of our analysis.
Citation: Joseph G. Yan, Dong-Ming Hwang. Pattern formation in reaction-diffusion systems with $D_2$-symmetric kinetics. Discrete & Continuous Dynamical Systems - A, 1996, 2 (2) : 255-270. doi: 10.3934/dcds.1996.2.255
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