Stable localized moving patterns in the 2-D Gray-Scott model

Abstract

I show stable, localized, single and multi-spot patterns of three classes - stationary, moving, and rotating - that exist within a limited range of parameter values in the two-dimensional Gray-Scott reaction-diffusion model with ${\sigma} = D_u/D_v = 2$. These patterns exist in domains of any size, and appear to derive their stability from a constructive reinforcement effect of the standing waves that surround any feature. There are several common elements - including a spot that behaves as a quasiparticle, a U-shaped stripe, and a ring or annulus, or a portion thereof - which combine to form a great variety of stable structures. These patterns interact with each other in a variety of ways. There are similarities to other reaction-diffusion systems and to physical experiments; I offer several suggestions for further research.