Simple and superlattice Turing patterns in reaction-diffusion systems: Bifurcation, bistability, and parameter collapse

We use equivariant bifurcation theory to investigate pattern selection at the onset of a Turing instability in a general two-component reaction-diffusion system. The analysis is restricted to patterns that periodically tile the plane in either a square or hexagonal fashion. Both simple periodic patterns (stripes, squares, hexagons, and rhombs) and "superlattice" patterns are considered. The latter correspond to patterns that have structure on two disparate length scales; the short length scale is dictated by the critical wave number from linear theory, while the periodicity of the pattern is on a larger scale. Analytic expressions for the coefficients of the leading nonlinear terms in the bifurcation equations are computed from the general reaction-diffusion system using perturbation theory. We show that no matter how complicated the reaction kinetics might be, the nonlinear reaction terms enter the analysis through just four parameters. Moreover, for hexagonal problems, all patterns bifurcate unstably unless a particular degeneracy condition is satisfied, and at this degeneracy we find that the number of effective system parameters drops to two, allowing a complete characterization of the possible bifurcation results at this degeneracy. For example, we find that rhombs, squares and superlattice patterns always bifurcate unstably. We apply these general results to some specific model equations, including the Lengyel-Epstein CIMA model, to investigate the relative stability of patterns as a function of system parameters, and to numerically test the analytical predictions.