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Logarithmic Expansions and the Stability of Periodic Patterns of Localized Spots for Reaction-Diffusion Systems in $\R^2

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 $|Ω|$. To leading order in $ν={-1/\log\eps}$, the linearization of the steady-state periodic spot pattern has a zero eigenvalue when the inhibitor diffusivity satisfies $D={D_0/ν}$, 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 ${\mathcal O}(ν)$ neighborhood of the origin in the spectral plane is derived when $D={D_0/ν} + D_1$, 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 $\mbox{Re}(λ)<0$ 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 $D$.