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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