Asymmetric Spotty Patterns for the Gray-Scott Model in R^2

Abstract

In this paper, we rigorously prove the existence and stability of asymmetric spotty patterns for the Gray-Scott model in a bounded two dimensional domain. We show that given any two positive integers k_1,\,k_2, there are asymmetric solutions with k_1 large spots (type A) and k_2 small spots (type B). We also give conditions for their location and calculate their heights. Most of these asymmetric solutions are shown to be unstable. However, in a narrow range of parameters, asymmetric solutions may be stable.