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 attract or repel each other alternatively according to their mutual distances, leading rather complex patterns. One fundamental pattern ring pattern, consisting $N$ equally distributed on circle certain radius. Depending parameters stationary moving (i.e., traveling and rotating) $N$-spot rings can be observed. order understand emergence these patterns, describe by set reduced ODEs encoding information spot's location velocity. On basis analytically study existence stability solutions, which keep most essential features collective motion self-propelled particles. Numerical simulations both PDEs are provided verify our results, comparison between them implies several future challenges including new effects higher terms.