Cops-on-the-Dots: The Linear Stability of Crime Hotspots for a 1-D Reaction-Diffusion Model of Urban Crime

Published in University of British Columbia, 2018

Abstract: In a singularly perturbed limit, we analyze the existence and linear stability of steady-state hotspot solutions foran extension of the 1-D three-component reaction-diffusion (RD) system formulated and studied numerically in Joneset. al. [Math. Models. Meth. Appl. Sci.,20, Suppl., (2010)], which models urban crime with police intervention. In ourextended RD model, the field variables are the attractivenessfield for burglary, the criminal density, and the police density,and it includes a scalar parameter that determines the strength of the police drift towards maxima of the attractivenessfield. For a special choice of this parameter, we recover the “cops-on-the-dots” policing strategy of Jones et. al., wherethepolice mimic the drift of the criminals towards maxima of the attractiveness field. For our extended model, the methodof matched asymptotic expansions is used to construct 1-D steady-state hotspot patterns as well as to derive nonlocaleigenvalue problems (NLEPs), each having three distinct nonlocal terms, that characterize the linear stability of thesehotspot steady-states toO(1) time-scale instabilities. For a cops-on-the-dots policing strategy, where some key identitiescan be used to recast these NLEPs into equivalent NLEPs with only one nonlocal term, we prove that a multi-hotspotsteady-state is linearly stable to synchronous perturbations of the hotspot amplitudes. Moreover, for asynchronousperturbations of the hotspot amplitudes, a hybrid analytical-numerical method is used to construct linear stability phasediagrams in the police versus criminal diffusivity parameterspace. In one particular region of these phase diagrams, thehotspot steady-states are shown to be unstable to asynchronous oscillatory instabilities in the hotspot amplitudes thatarise from a Hopf bifurcation. Within the context of our model,this provides a parameter range where the effect of acops-on-the-dots policing strategy is to only displace crime temporally between neighboring spatial regions. Our hybridapproach to study the equivalent NLEP combines rigorous spectral results with a numerical parameterization of anyHopf bifurcation threshold. For the cops-on-the-dots policing strategy, our linear stability predictions for steady-statehotspot patterns are confirmed from full numerical PDE simulations of the three-component RD system.

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