  For the Schnakenberg model, we consider a highly symmetric configuration of N
spikes whose locations are located at the vertices of a regular N-gon inside
either a unit disk or an annulus. We call such configuration a ring of spikes.
The ring radius is characterized in terms of the modified Green's function. For
a disk, we find that a ring of 9 or more spikes is always unstable with respect
to small eigenvalues. Conversely, a ring of 8 or less spikes is stable inside a
disk provided that the feed-rate $A$ is sufficiently large. More generally, for
sufficiently high feed-rate, a ring of $N$ spikes can be stabilized provided
that the annulus is thin enough. As $A$ is decreased, we show that the ring is
destabilized due to small eigenvalues first, and then due to large eigenvalues,
although both of these thresholds are separated by an asymptotically small
amount. For a ring of 8 spikes inside a disk, the instability appears to be
supercritical, and deforms the ring into a square-like configuration. For less
than 8 spikes, this instability is subcritical and results in spike death.
