Spatially localized structures in the Gray–Scott model

Abstract

Spatially localized structures in the one-dimensional Gray–Scott reaction–diffusion model are studied using a combination of numerical continuation techniques and weakly nonlinear theory, focusing on the regime in which the activator and substrate diffusivities are different but comparable. Localized states arise in three different ways: in a subcritical Turing instability present in this regime, and from folds in the branch of spatially periodic Turing states. They also arise from the fold of spatially uniform states. These three solution branches interconnect in complex ways. We use numerical continuation techniques to explore their global behaviour within a formulation of the model that has been used to describe dryland vegetation patterns on a flat terrain. This article is part of the theme issue ‘Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 2)’.