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Michael Ward

University of British Columbia



Beyond Turing: The Stability and Dynamics of Localized Patterns for Reaction-Diffusion Systems



December 4, 2013
3:00pm    PGH 646



Abstract
 

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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