Title: Stable spike clusters for the precursor Gierer-Meinhardt system in R2,
Abstract: We consider the Gierer-Meinhardt system with small inhibitor diffusivity,
very small activator diffusivity and a precursor inhomogeneity.
For any given positive integer k we construct a spike cluster consisting of
$k$ spikes which all approach the same nondegenerate local minimum point of the
precursor inhomogeneity. We show that this spike cluster can be linearly
stable. In particular, we show the existence of spike clusters for spikes
located at the vertices of a polygon with or without centre. Further, the
cluster without centre is stable for up to three spikes, whereas the cluster
with centre is stable for up to six spikes.
The main idea underpinning these stable spike clusters is the following: due
to the small inhibitor diffusivity the interaction between spikes is repulsive,
and the spikes are attracted towards the local minimum point of the precursor
inhomogeneity. Combining these two effects can lead to an equilibrium of spike
positions within the cluster such that the cluster is linearly stable.