The Stability and Dynamics of Localized Spot Patterns in the Two-Dimensional Gray-Scott Model

The dynamics and stability of multi-spot patterns to the Gray-Scott (GS) reaction-diffusion model in a two-dimensional domain is studied in the singularly perturbed limit of small diffusivity ε of one of the two solution components. A hybrid asymptotic-numerical approach based on combining the method of matched asymptotic expansions with the detailed numerical study of certain eigenvalue problems is used to predict the dynamical behavior and instability mechanisms of multi-spot quasi-equilibrium patterns for the GS model in the limit ε → 0. For ε → 0, a quasi-equilibrium k-spot pattern is constructed by representing each localized spot as a logarithmic singularity of unknown strength Sj , for j = 1, . . . , k, at an unknown spot location xj ∈ Ω, for j = 1, . . . , k. A formal asymptotic analysis is then used to derive a differential algebraic ODE system for the collective coordinates Sj and xj , for j = 1, . . . , k, which characterizes the slow dynamics of a spot pattern. Instabilities of the multi-spot pattern due to the three distinct mechanisms of spot self-replication, spot oscillation, and spot annihilation, are studied by first deriving certain associated eigenvalue problems by using singular perturbation techniques. From a numerical computation of the spectrum of these eigenvalue problems, phase diagrams in the GS parameter space corresponding to the onset of spot instabilities are obtained for various simple spatial configurations of multi-spot patterns. In addition, it is shown that there is a wide parameter range where a spot instability can be triggered only as a result of the intrinsic slow motion of the collection of spots. The construction of the quasi-equilibrium multi-spot patterns and the numerical study of the spectrum of the eigenvalue problems relies on certain detailed properties of the reduced-wave Green’s function. The hybrid asymptotic-numerical results for spot dynamics and spot instabilities are validated from full numerical results computed from the GS model for various spatial configurations of spots.