General theory of instabilities for patterns with sharp interfaces in reaction-diffusion systems

An asymptotic method for finding instabilities of arbitrary d-dimensional large-amplitude patterns in a wide class of reaction-diffusion systems is presented. The complete stability analysis of two- and three-dimensional localized patterns is carried out. It is shown that in the considered class of systems the criteria for different types of instabilities are universal. The specific nonlinearities enter the criteria only via three numerical constants of order 1. The analysis performed explains the self-organization scenarios observed in the recent experiments and numerical simulations of some concrete reaction-diffusion systems.