Pattern Formation on a Periodic Rectangle and the Green’s Function Potential

Abstract

Abstract. Many reaction-diffusion (RD) systems exhibit spot patterns. When the domain is a periodic rectangle, many such patterns can be constructed explicitly using regular integer lattices, and moreover numerical simulations show that they are prevalent even when starting with random initial conditions. For a wide class of RD systems which includes the Schnakenberg model, these equilibria are local minima of the corresponding Green’s function potential. We use integer lattice enumeration and Floquet theory to classify all stable lattices on a square up to [Formula: see text] spots. We then investigate stability boundaries and bifurcations as the aspect ratio of the rectangle is varied. For a rectangle with a small aspect ratio, we derive explicit thresholds for stability of a pattern consisting of two interlaced stripes. For certain aspect ratios and [Formula: see text], there exists a perfect hexagonal lattice without any defects. We determine stability boundaries of such lattices as the aspect ratio changes. We also explore bifurcations and transitions that result when the lattice is deformed beyond its stable regime. The results are illustrated and confirmed using the Schnakenberg RD system.