PATTERN-FORMATION IN THE ONE-DIMENSIONAL GRAY-SCOTT MODEL

In this work, we analyse a pair of one-dimensional coupled reaction-di ffusion equations known as the Gray-Scott model, in which self-replica ting patterns have been observed. We focus on stationary and travellin g patterns, and begin by deriving the asymptotic scaling of the parame ters and variables necessary for the analysis of these patterns. Singl e-pulse and multipulse stationary waves are shown to exist in the appr opriately scaled equations on the infinite 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 conc entration CI tends (slowly) to the homogeneous steady state U = 1, whi le V is everywhere close to V = 0. In addition, we establish the exist ence of a plethora of periodic steady states consisting of periodic ar rays 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 determ ined exclusively by the reactions of the chemicals and their diffusion s, 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 travelling solitary pulses in this system. This n on-existence result reflects the system's degeneracy and indicates tha t some event, for example pulse splitting, 'must' occur when two pulse s move apart from each other (as has been observed in simulations): th ese pulses evolve towards the non-existent travelling solitary pulses. The main mathematical techniques employed in this analysis of the sta tionary and travelling patterns are geometric singular perturbation th eory and adiabatic Melnikov theory. Finally, the theoretical results a re compared to those obtained from direct numerical simulation of the coupled partial differential equations on a 'very large' domain, using a moving grid code. It has been checked that the boundaries do not in fluence the dynamics. A subset of the family of stationary single puls es appears to be stable. This subset determines the boundary of a regi on in parameter space in which the self-replicating process takes plac e. In that region, we observe that the core of a time-dependent self-r eplicating pattern turns out to be precisely a stationary periodic pul se pattern of the type that we construct. Moreover, the simulations re veal some other essential components of the pulse-splitting process an d provide an important guide to further analysis.