Interface proliferation and the growth of labyrinths in a reaction-diffusion system

Abstract

In the bistable regime of the FitzHugh-Nagumo model of reaction-diffusion systems, spatially homogeneous patterns may be nonlinearly unstable to the formation of compact "localized states." The formation of space-filling patterns from instabilities of such structures is studied in the context of a nonlocal contour dynamics model for the evolution of boundaries between high and low concentrations of the activator. An earlier heuristic derivation [D. M. Petrich and R. E. Goldstein, Phys. Rev. Lett. 72, 1120 (1994)] is made more systematic by an asymptotic analysis appropriate to the limits of fast inhibition, sharp activator interfaces, and small asymmetry in the bistable minima. The resulting contour dynamics is temporally local, with the normal component of the velocity involving a local contribution linear in the interface curvature and a nonlocal component having the form of a screened Biot-Savart interaction. The amplitude of the nonlocal interaction is set by the activator-inhibitor coupling and controls the "lateral inhibition" responsible for the destabilization of localized structures such as spots and stripes, and the repulsion of nearby interfaces in the later stages of those instabilities. The phenomenology of pattern formation exhibited by the contour dynamics is consistent with that seen by Lee, McCormick, Ouyang, and Swinney [Science 261, 192 (1993)] in experiments on the iodide-ferrocyanide-sulfite reaction in a gel reactor. Extensive numerical studies of the underlying partial differential equations are presented and compared in detail with the contour dynamics. The similarity of these phenomena (and their mathematical description) with those observed in amphiphilic monolayers, type I superconductors in the intermediate state, and magnetic fluids in Hele-Shaw geometry is emphasized.