Pacific Institute for the Mathematical Sciences - PIMS

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 Si for i = 1,…,k at unknown spot locations xi ∈ Ω for i = 1,…,k. A formal asymptotic analysis is then used to derive a differential algebraic ODE system for the collective coordinates Si and xi for i = 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 representing 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.

This is joint work with Wan Chen (Oxford Center for Collaborative Applied Mathematics).

Location: LSK 301

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