Chunyi Gai, Department of Mathematics, University of British Columbia

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

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.