The Nucleation-Annihilation Behavior for Hotspot Patterns of Urban Crime with Police Deployment

Chunyi Gai

A hybrid asymptotic-numerical approach is developed to study hotspot patterns for a three-component 1-D reaction-diffusion (RD) system that models urban crime with police intervention. Our analysis is focused on a scaling regime where there are two distinct competing mechanisms for producing complex spatio-temporal dynamics of hotspot patterns; a mechanism to annihilate hotspots and a further mechanism to nucleate new hotspots from a quiescent background. The nucleation threshold for steady-state hotspot patterns arises from a saddle-node bifurcation point of hotspot equilibria. By deriving a new analytical expression for a hotspot profile, combined with a local normal form analysis, our asymptotic analysis provides a rather accurate prediction of this nucleation threshold. From a numerical computation of the spectrum of the linearization around a two-boundary hotspot pattern, we have identified instability parameter thresholds for both zero eigenvalue crossings and Hopf bifurcations. Overall, this provides a phase diagram in parameter space where distinct types of dynamical behaviors can occur from perturbations of this two-boundary hotspot steady-state solution. In one region of this phase diagram, corresponding to a small police diffusivity, a two-boundary hotspot steady-state is unstable to a subcritical asynchronous oscillatory instability in the hotspot amplitudes. This instability normally leads to the complete annihilation of one of the hotspots. However, in the regime where this instability is coincident with the non-existence of a one-hotspot steady-state, we show that hotspot patterns undergo complex nucleation-annihilation dynamics that are characterized by large-scale and persistent oscillations of the hotspot amplitudes. This case study suggests that there are parameter regimes in the three-component RD system where the problem of predicting where and when hotspots of crime will appear and then essentially disappear is largely intractable.

 Overview  Program