The Stability of Steady-State Hot-Spot Patterns for Reaction-Diffusion Models of Urban Crime

The existence and stability of localized patterns of criminal activity is studied for the two-component reaction-diffusion model of urban crime that was introduced by Short et.~al.~[Math. Models. Meth. Appl. Sci., 18, Suppl. (2008), pp.~1249--1267]. 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. Singular perturbation techniques are used to construct steady-state hot-spot patterns in one and two-dimensional spatial domains, and new types of nonlocal eigenvalue problems (NLEP's) are derived that determine the stability of these hot-spot patterns to O(1) time-scale instabilities. From an analysis of these NLEP's, and a further analysis of the spectrum associated with the slow translational instabilities, an explicit threshold for the minimum spacing between stable hot-spots is derived. The theory is confirmed via detailed numerical simulations of the full PDE system. Moreover, the parameter regime where localized hot-spots occur is compared with the parameter regime, studied in previous works, where Turing instabilities from a spatially uniform steady-state occur. Finally, in the 1-D context, we show how the existence and stability of hot-spot patterns is altered from the inclusion of a third component to the reaction-diffusion system that incorporates the effect of police. In the context of this extended model, the optimal strategy for the police is discussed.
Joint Work with Theodore Kolokolnikov (Dalhousie) and Juncheng Wei (Chinese U. of Hong Kong and UBC).