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

Abstract

The dynamics and stability of multispot patterns to the Gray–Scott (GS) reaction-diffusion model in a two-dimensional domain is studied in the singularly perturbed limit of small diffusivity $\varepsilon$ 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 multispot quasi-equilibrium patterns for the GS model in the limit $\varepsilon\to 0$. For $\varepsilon\to 0$, a quasi-equilibrium k-spot pattern is constructed by representing each localized spot as a logarithmic singularity of unknown strength $S_j$ for $j=1,\ldots,k$ at an unknown spot location ${\mathbf x}_j\in \Omega$ for $j=1,\ldots,k$. A formal asymptotic analysis is then used to derive a differential algebraic ODE system for the collective coordinates $S_j$ and ${\mathbf x}_j$ for $j=1,\ldots,k$, which characterizes the slo...