Self-replicating spots in reaction-diffusion systems

Abstract

In this article we discuss the self-replicating spot, a particlelike phenomenon that occurs in reaction-diffusion ~RD! systems @1#. The spots consist of localized regions in which the concentrations of the reactants differ from the surrounding concentration fields. They grow, reaching a critical size at which time they divide in two. The two resulting spots again grow and divide. This process, which is visually similar to cell division, continues indefinitely. The long-time behavior depends on the precise values of the external control parameters, but typically consists of a chaotic ‘‘soup’’ in which many spots compete for resources as illustrated in Fig. 1. Those spots that find adequate resources continue to grow and divide. Those that are unable to find adequate resources decay into the background. The spots observed in @1# were found during an attempt to model labyrinthine patterns in the ferrocyanide-iodate-sulfate ~FIS! reaction @2#. Since then, replicating spot patterns have been observed both numerically and experimentally in the FIS reaction @3,4#. There are obvious differences in the Gaspar-Showalter model of the FIS reaction @5# and of the models discussed by us and others. This fact suggests that replication is a generic feature characterizing a broad class of reaction-diffusion systems. In @6#, we presented some arguments in support of this proposition. These arguments included both a heuristic description of the process of replication and demonstrations of analytic features common to several related model RD systems. It turns out that replication is more general than our analysis accounts for. Nevertheless, we think it worthwhile to spell out the details of the theory presented in @6#. We remark here that various aspects of the replication phenomenon have been discussed by other authors @7,8#. Kerner and Osipov have a large body of work on largeamplitude dissipative structures including an analysis of the static division of one-dimensional pulses as the system size is changed. Gurevich and Mints have a body of work on replication of thermal hot spots in composite superconductors. In the article by Petrov, Scott, and Showalter replication