Various localized patterns have been observed both experimentally and numerically in many reaction-diffusion systems, including stable spots, traveling spots, breathing spot and splitting spot etc. A well-known example is the ferrocyanide-iodate-sulphite reaction that has reproduced similar spot-replication behavior shown by computing Gray-Scott model. Understanding these phenomena could have potential applications in chemical reactions, biological morphogenesis and medical research.

In this talk, I concentrate on analyzing dynamics and instabilities of spike patterns (1D) and spot patterns (2D) in Gray-Scott model. In a specific parameter regime of 1D problem, oscillatory profile and drift instabilities are analyzed through a Stefan type problem. In 2D model, we study the mechanism of spot-replication, for which the splitting criterion is identified. A DAE system is derived to describe the spot dynamics. Competition instability and oscillatory profile instability are also investigated. On top of these results, phase diagrams are plotted in different parameter spaces. The theories are illustrated for infinite domain, unit circle and square domain, and are compared with full numerical simulations.