Localized Patterns in Reaction-Diffusion Systems

A new chemical pattern is discussed, which is a propagationless solitary island in an infinite medium. We demonstrate analytically its existence and stability for a certain simple model. The localization turns out to be a consequence of the rapid diffusion of an inhibiting substance occurring in a potentially excitable system. In order to extract the important features of the localized pattern, the method of singular perturbation is employed, with the following results: (1) A stable motionless solitary pattern can exist either for a monostable or bistable system. (2) Under suitable conditions such a pattern undergoes the Hopf bifurcation, leading to a “breathing motion” of the activated droplet. The analysis is restricted to the one-dimensional case throughout.