Spot Self-Replication and Dynamics for the Schnakenburg Model in a Two-Dimensional Domain

The dynamical behavior of multi-spot solutions in a two-dimensional domain Ω is analyzed for the two-component Schnakenburg reaction-diffusion model in the singularly perturbed limit of small diffusivity ε for one of the two components. In the limit ε → 0, a quasi-equilibrium spot pattern in the region away from the spots is constructed by representing each localized spot as a logarithmic singularity of unknown strength Sj for j = 1, . . . , K at unknown spot locations xj ∈ Ω for j = 1, . . . , K. A formal asymptotic analysis, which has the effect of summing infinite logarithmic series in powers of −1/ log ε, is then used to derive an ODE differential algebraic system (DAE) for the collective coordinates Sj and xj for j = 1, . . . , K, which characterizes the slow dynamics of a spot pattern. This DAE system involves the Neumann Green’s function for the Laplacian. By numerically examining the stability thresholds for a single spot solution, a specific criterion in terms of the source strengths Sj , for j = 1, . . . , K, is then formulated to theoretically predict the initiation of a spot-splitting event. The analytical theory is illustrated for spot patterns in the unit disk and the unit square, and is compared with full numerical results computed directly from the Schnakenburg model.