Dynamics of N-spot rings with oscillatory tails in a three-component reaction-diffusion system

In two-dimensional space, we investigate the slow dynamics of multiple localized spots with oscillatory tails in a specific three-component reaction-diffusion system, whose key feature is that the spots attract or repel each other alternatively according to their mutual distances, leading to rather complex patterns. One fundamental pattern is the ring pattern, consisting of $N$ equally distributed spots on a circle with a certain radius. Depending on the parameters of the system, stationary or moving (i.e., traveling and rotating) $N$-spot rings can be observed. In order to understand the emergence of these patterns, we describe the dynamics of $N$ spots by a set of reduced ordinary differential equations (ODEs) encoding the information of each spot's location and velocity. On the basis of the reduced system, we analytically study the existence and stability of stationary and moving $N$-spot ring solutions, similar to the collective motion of self-propelled particles. Numerical simulations are provided to verify our results with fixed parameters.