Topics in Localized Patterns: Hot Spot Patterns of Urban Crime and Localized Patterns on the Sphere

Friday, June 7, 2013 - 9:00am - 9:40am
Keller 3-180
Michael Ward (University of British Columbia)
Two topics in the theory and applications of localized pattern formation are
discussed. 1) Hot-spot patterns of urban crime; 2) Localized patterns for the
Brusselator on the sphere.

We analyze localized patterns of criminal activity for some reaction-diffusion models of urban crime introduced by Short et al. Such patterns, characterized by the concentration of criminal activity in localized spatial regions, are referred to as hot-spot patterns and they occur in a parameter regime far from the Turing point associated with the bifurcation of spatially uniform solutions. Hot-spot patterns are constructed and their stability properties analyzed from a new class of nonlocal eigenvalue problems for a three-component RD system where the effect of police deterrence is included. The stability analysis of hot-spot solutions suggest optimal strategies for police enforcement. Secondly, we show that there is a parameter regime where hot-spot patterns can nucleate from a quiescent background when the inter hot-spot spacing exceeds a threshold. The bifurcation structure and dynamical behavior of hot-spot patterns in this regime is analyzed.

Next, we study the existence and stability of localized spot patterns for the Brusselator RD model on the surface of the sphere. A key difficulty with a weakly nonlinear analysis of patterns on the sphere is that the Laplacian eigenfunctions exhibit a high degree of mode degeneracy. In contrast, in the context of localized pattern, spot patterns for the Brusselator on the sphere are shown to exhibit three types of instabilities: self-replication, competition, and oscillatory instabilities of the spot amplitudes. An analysis of these instabilities is given and the results are compared with full numerical computations. The study of localized patterns on the sphere has some similarity to the well-studied problem of Eulerian point vortices on the sphere.
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