The dynamics of localized spot patterns for reaction-diffusion systems on the sphere

TL;DR

This paper derives and analyzes a differential algebraic equation describing the slow dynamics of localized spot patterns in reaction-diffusion systems on a sphere, revealing complex bifurcations, instabilities, and connections to vortex and Fekete point problems.

Contribution

It introduces a novel DAE framework for understanding spot pattern dynamics on a sphere, including bifurcation analysis and instability conditions, for the Brusselator model.

Findings

Derived the DAE system for spot dynamics on the sphere.

Identified bifurcation structures and imperfection sensitivity.

Established conditions for spot instability and stability basins.

Abstract

In the singularly perturbed limit corresponding to a large diffusivity ratio between two components in a reaction-diffusion (RD) system, quasi-equilibrium spot patterns are often admitted, producing a solution that concentrates at a discrete set of points in the domain. In this paper, we derive and study the differential algebraic equation (DAE) that characterizes the slow dynamics for such spot patterns for the Brusselator RD model on the surface of a sphere. Asymptotic and numerical solutions are presented for the system governing the spot strengths, and we describe the complex bifurcation structure and demonstrate the occurrence of imperfection sensitivity due to higher order effects. Localized spot patterns can undergo a fast time instability and we derive the conditions for this phenomena, which depend on the spatial configuration of the spots and the parameters in the system. In the absence of these instabilities, our numerical solutions of the DAE system for $N = 2$ to $N = 8$ spots suggest a large basin of attraction to a small set of possible steady-state configurations. We discuss the connections between our results and the study of point vortices on the sphere, as well as the problem of determining a set of elliptic Fekete points, which correspond to globally minimizing the discrete logarithmic energy for $N$ points on the sphere.