The Stability and Dynamics of Localized Spot Patterns for a Bulk-Membrane Coupled Brusselator Model

We consider a bulk-membrane-coupled partial differential equation in which a single diffusion equation posed within the unit ball is coupled to a two-component reaction diffusion equation posed on the bounding unit sphere through a linear Robin boundary condition. Specifically, within the bulk we consider a process of linear diffusion with point-source generation for a bulk-bound activator. On the bounding surface we consider the classical two-component Brusselator model where the feed term is replaced by the restriction of the bulk-bound activator to the membrane. By considering the singularly perturbed limit of a small diffusivity ratio between the membrane-bound activator and inhibitor species, we use formal asymptotic expansions to construct strongly localized quasi-equilibrium spot solutions and study their linear stability. Our analysis reveals that bulk-membrane-coupling can restrict the existence of localized spot solutions through a recirculation mechanism. In addition we derive stability thresholds that illustrate the effect of coupling on both competition and splitting instabilities. Finally, we use higher-order matched asymptotic expansions to derive a system of differential algebraic equations that describe the slow motion of spots. The potential for new coupling induced dynamical behaviour is illustrated by considering examples of one-, two-, and three-spot solutions.