  The linear stability of steady-state periodic patterns of localized spots in
$\R^2$ for the two-component Gierer-Meinhardt (GM) and Schnakenburg
reaction-diffusion models is analyzed in the semi-strong interaction limit
corresponding to an asymptotically small diffusion coefficient $\eps^2$ of the
activator concentration. In the limit $\eps\to 0$, localized spots in the
activator are centered at the lattice points of a Bravais lattice with constant
area $|\Omega|$. To leading order in $\nu={-1/\log\eps}$, the linearization of
the steady-state periodic spot pattern has a zero eigenvalue when the inhibitor
diffusivity satisfies $D={D_0/\nu}$, for some $D_0$ independent of the lattice
and the Bloch wavevector $\kb$. From a combination of the method of matched
asymptotic expansions, Floquet-Bloch theory, and the rigorous study of certain
nonlocal eigenvalue problems, an explicit analytical formula for the continuous
band of spectrum that lies within an ${\mathcal O}(\nu)$ neighborhood of the
origin in the spectral plane is derived when $D={D_0/\nu} + D_1$, where
$D_1={\mathcal O}(1)$ is a de-tuning parameter. The periodic pattern is
linearly stable when $D_1$ is chosen small enough so that this continuous band
is in the stable left-half plane $\mbox{Re}(\lambda)<0$ for all $\kb$.
Moreover, for both the Schnakenburg and GM models, our analysis identifies a
model-dependent objective function, involving the regular part of the Bloch
Green's function, that must be maximized in order to determine the specific
periodic arrangement of localized spots that constitutes a linearly stable
steady-state pattern for the largest value of $D$. From a numerical
computation, it is shown within the class of oblique Bravais lattices that a
regular hexagonal lattice arrangement of spots is optimal for maximizing the
stability threshold in $D$.
