Through the use of a linearized analysis, Alan Turing in 1952 showed
how a stable spatially non-uniform pattern can develop from small
perturbations of a spatially homogeneous equilibrium state for a coupled
PDE system of reaction-diffusion (RD) equations. Over the past three
decades a comprehensive methodology, based on normal form and centre
manifold theory, has been developed to classify the existence and stability
of small amplitude spatial patterns in RD systems. However, when the ratio
of the diffusion coefficients is very large, certain RD systems allow for
the existence of localized solutions that have a high degree of spatial
heterogeneity. Solutions where the pattern is spatially localized near a
discrete set of points in the domain are called spot patterns. Examples of
this class of solutions include localized patterns of chemical activity
(the Gray-Scott model), hot-spot patterns of urban crime, and localized
patterns in biological morphogenesis. In contrast to the case of spatially
homogeneous solutions, the instabilities and the dynamics of these
localized spot patterns are not nearly as well understood. In this talk, we
begin by giving a brief survey of some problems with localized spot
patterns in \(\mathbf{R}^2\), and we highlight novel dynamical behaviors
that occur for these systems, including spot self-replication and
annihilation. In the second part of the talk, we provide a theoretical
framework to analyze the linear stability of localized spot patterns based
on the spectral analysis of a class of nonlocal eigenvalue problem. For
periodic arrangements of spots in \(\mathbf{R}^2\), it is shown that a
hexagonal lattice of spots provides an optimum stability threshold. Some
open problems and challenges are discussed.
Note: the talk will be accessible to a general
math audience.
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