Existence and stability of multiple-spot solutions for the Gray–Scott model in R2

In this paper, we rigorously prove the existence and stability of multiple-spot patterns for the Gray–Scott system in a two-dimensional domain which are far from spatial homogeneity. The Green’s function and its derivatives together with two nonlocal eigenvalue problems both play a major role in the analysis. We establish a threshold behavior for stability: if a certain inequality for the parameters holds then we get stability, otherwise we get instability of multiple-spot solutions. The exact asymptotics of the critical thresholds are obtained.