Localized Spot Patterns on the Sphere for Reaction-Diffusion Systems: Theory and Open Problems

Abstract

A new class of point-interaction problem characterizing the time evolution of spatially localized spots for reaction-diffusion (RD) systems on the surface of the sphere is introduced and studied. This problem consists of a differential algebraic system (DAE) of ODEs for the locations of a collection of spots on the sphere, and is derived from an asymptotic analysis in the large diffusivity ratio limit of certain singularly perturbed two-component RD systems. In Trinh and Ward (The dynamics of localized spot patterns for reaction-diffusion systems on the sphere. Nonlinearity Nonlinearity 29 (3), 766–806 (2016)), this DAE system was derived for the Brusselator and Schnakenberg RD systems, and herein we extend this previous analysis to the Gray-Scott RD model. Results and open problems pertaining to the determination of equilibria of this DAE system, and its relation to elliptic Fekete point sets , are highlighted. The potential of deriving similar DAE systems for more complicated modeling scenarios is discussed.