The Stability and Dynamics of Localized Spot Solutions to Reaction-Diffusion Systems in R2
Michael Ward (UBC)

Abstract:
In the singularly perturbed limit corresponding to an asymptotically large diffusivity ratio of O(ǫ−2)
between two components in a reaction-diffusion (RD) system, many such systems admit quasi-equilibrium
spot patterns, whereby the solution concentrates at a discrete set of points in the domain. For this class
of non-variational problems we discuss two recent results regarding for the stability and dynamics of
localized spot patterns.

Our first problem involves analyzing the linear stability of steady-state periodic patterns of localized
spots in R2 for the two-component Gierer-Meinhardt (GM) and Schnakenburg reaction-diffusion models
where the spots are localized at the lattice points of an arbitrary Bravais lattice of unit cell area. To leading
order in ν = −1/ log ǫ, the linearization of the steady-state periodic spot pattern has a zero eigenvalue
when the inhibitor diffusivity satisfies D = D0/ν, for some D0 independent of the lattice and the Bloch
wavevector k. 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,
involving the regular part of the Bloch Green’s function, is derived for the continuous band of spectrum
that lies within an O(ν) neighborhood of the origin in the spectral plane when D = D0/ν + D1, where
D1 = O(1) is a de-tuning parameter. From a numerical computation, based on an Ewald-type algorithm,
of the regular part of this Bloch Green’s it is shown that a regular hexagonal lattice arrangement of spots
is optimal for maximizing the stability threshold in D.

For our second problem, we derive and then analyze a differential algebraic system of ODEs (DAE)
that characterizes the slow dynamics of an N-spot pattern for the Brusselator model on the surface of a
sphere. Our numerical solution of the DAE system system for N = 2 to N = 10 spots suggest a large
basin of attraction to a small set of possible steady-state configurations. We discuss the connections
between our results for slow spot dynamics and the study of point vortices on the sphere, as well as
the problem of determining a set of elliptic Fekete points, which correspond to globally minimizing the
discrete logarithmic energy for N points on the sphere.

First Part: Joint Work with David Iron and John Rumsey (Dalhousie), and Juncheng
Wei (UBC). Second Part: Joint with Philippe Trinh (Oxford).
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