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, 1120Lett. 72, (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 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 are emphasized.

Spatially Localized Structures in Diblock Copolymer Mixtures

“…For pure diblock copolymers, on the other hand, the region between micelletype structures must be filled with a homogeneous mixture, which may or may not be unstable. This paper's results have some qualitative similarity to a variety of systems that arise in the reaction-diffusion literature [13,21,27,45]. Pattern formation in variational models similar to ours has been studied by Muratov [26].…”

Bumps and rings in a two-dimensional neural field: splitting and rotational instabilities

“…This feature is reminiscent of spot splitting in the Gray-Scott reaction-diffusion system [33]. We also see the emergence of labyrinthine patterns from a bump that is unstable to mode 3 perturbations (figure 5), again a feature that has been observed in reaction-diffusion systems [34].…”

“…This technique is valid for arbitrary non-rotationally symmetric solutions, irrespective of their detailed shape. A potential tool to study the evolution of intricate structures such as the labyrinthine patterns seen here, is to formulate a description of the dynamics of the interface (where u(r, t) = h), along the lines described for reaction diffusion equations in [34].…”

Order parameter equations for front transitions: Planar and circular fronts

“…The two solutions agree within an accuracy of approximately 1% for the amplitude and 2% for the phase. In addition to the oscillatory instability spot solutions may also be unstable to transverse perturbations [35][36][37]. Numerical solutions of the fully twodimensional model (27) show that for the parameters of fronts leading to a labyrinthine pattern.…”