Hotspot Formation and Dynamics for a Continuum Model of Urban Crime

The existence, stability, and dynamics of localized patterns of criminal activity are studied for the reaction-diffusion model of urban crime introduced by Short et. al. [Math. Models. Meth. Appl. Sci., 18, Suppl. (2008), pp. 1249–1267]. In the singularly perturbed limit of small diffusivity ratio, this model admits hotspot patterns, where criminal activity of high amplitude is localized within certain narrow spatial regions. By using a combination of asymptotic analysis and numerical path-following methods, hotspot equilibria are constructed on a finite 1-D domain and their bifurcation properties analyzed as the diffusivity of criminals is varied. It is shown, both analytically and numerically, that new hotspots of criminal activity can be nucleated in low crime regions with inconspicuous crime activity gradient when the spatial extent of these regions exceeds a critical threshold. These nucleations are referred to as “peak insertion” events, and for the steady-state problem they occur near a saddle-node bifurcation point characterizing hotspot equilibria. For the time-dependent problem, a differential algebraic (DAE) system characterizing the slow dynamics of a collection of hotspots is derived, and the results compared favorably with full numerical simulations of the PDE system. The asymptotic theory to construct hotspot equilibria, and to derive the DAE system for quasi-steady patterns, is based on the resolution of a triple-deck structure near the core of each hotspot and the identification of so-called switchback terms.