  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 $\epsilon$ 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 $\epsilon\to 0$. A differential algebraic ODE system for the
collective coordinates $S_j$ and ${\mathbf x}_j$ for $j=1,...,k$ is derived,
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 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.
