Stable pattern selection through invasion fronts in closed two-species reaction-diffusion systems

We study pattern formation in a two-species reaction-diffusion system with a conserved quantity. Such systems arise in the study of closed chemical reactors and recurrent precipitation. We compare pattern forming aspects in these systems to Turing-pattern forming systems. We show that in a zero-diffusion limit, these systems possess stable periodic patterns. We also exhibit a wavenumber-selection mechanism in this limit: While spatially random initial conditions give patterns on arbitrarily fine scales, localized initial conditions evolve into a coherent pattern with a finite wavenumber that is formed in the wake of invasion fronts. We compare our theoretical results with numerical simulations and point to an interesting front instability in a small mass fraction regime.