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

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.