Pattern formation in the 1-D Gray-Scott model

Abstract

In this work, we analyze a pair of one-dimensional coupled reaction-diusion equations known as the Gray{Scott model, in which self-replicating patterns have been observed. We focus on stationary and traveling patterns, and begin by deriving the asymptotic scaling of the parameters and variables necessary for the analysis of these patterns. Single{pulse and multiple{pulse stationary waves are shown to exist in the appropriately{scaled equations on the innite line. A (single) pulse is a narrow interval in which the concentration U of one chemical is small, while that of the second, V , is large, and outside of which the concentration U tends (slowly) to the homogeneous steady state U 1, while V is everywhere close to V 0. In addition, we establish the existence of a plethora of periodic steady states consisting of periodic arrays of pulses interspersed by intervals in which the concentration V is exponentially small and U varies slowly. These periodic states are spatially inhomogeneous steady patterns whose length scales are determined exclusively by the reactions of the chemicals and their diusions, and not by other mechanisms such as boundary conditions. A complete bifurcation study of these solutions is presented. We also establish the non-existence of traveling solitary pulses in this system. This non-existence result re ects the system's degeneracy and indicates that some event, for example pulse-splitting, `must' occur when a pair of pulses moving apart from each other (as has been observed in simulations): these pulses evolve towards the non-existent traveling solitary pulses. The main mathematical techniques employed in this analysis of the stationary and traveling patterns are geometric singular perturbation theory and adiabatic Melnikov theory. Finally, the theoretical results are compared to those obtained from direct numerical simulation of the coupled partial dierential equations on a `very large' domain, using a moving grid code. It has been checked that the boundaries do not in uence the dynamics. A subset of the family of stationary single pulses appears to be stable. This subset determines the boundary of a region in parameter space in which the self-replicating process takes place. In that region, we observe that the core of a time-dependent self-replicating pattern turns out to be precisely a stationary periodic pulse-pattern of the type that we construct. Moreover, the simulations reveal some other essential components of the pulse-splitting process and provide an important guide to further analysis.