Asynchronous Instabilities of Crime Hotspots for a 1-D Reaction-Diffusion Model of Urban Crime with Focused Police Patrol

We analyze the existence and linear stability of steady-state localized hotspot patterns for a 1-D three-component singularly perturbed reaction-diffusion (RD) system modeling urban crime in the presence of police intervention. Our three-component RD model augments the two-component system for an attractiveness field and criminal density, as introduced by Short et al. [Math. Models Methods Appl. Sci., 18 (2008), suppl., pp. 1249-1267], by including the effect of a police deployment that exhibits a biased random walk toward maxima of the attractiveness field. In our model, the rate at which criminals are introduced is decreased by the total level of police deployment, and the strength of the bias in the police random walk toward the maxima of the attractiveness field is modeled by a patrol focus parameter, q > 0. For our three-component model, hotspot steady-state patterns are constructed asymptotically and, from a detailed derivation and analysis of certain non-local eigenvalue problems (NLEPs), phase diagrams in parameter space are obtained that characterize regions of linear stability of the steady-state pattern. In certain parameter regimes, we show that the police intervention leads to a rapid annihilation of some hotspots, whereas in other parameter regimes, notably when the police diffusivity is below a threshold value, the police intervention only displaces crime periodically to neighboring spatial regions (at least on short time-scales). Mathematically, we show that this crime displacement effect arises due to a Hopf bifurcation in the NLEP associated with certain asynchronous modes of instability of the steady-state hotspot pattern. Such robust asynchronous temporal oscillations of the hotspot amplitudes in our three-component system is a new phenomenon, which does not typically occur in two-component RD systems. The effect of a "cops-on-the-dots"" patrol strategy, corresponding to q = 2, in which the police mimic the bias of the criminals toward spatial maxima of the attractiveness, is examined through a combination of rigorous spectral results and a numerical parameterization of any Hopf bifurcation threshold. For the special choice q = 3, we show that explicit linear stability results can be readily obtained from the NLEP. Our linear stability results are validated through full numerical PDE simulations of the three-component RD system.