Abstract: In a singularly perturbed limit, we analyze the existence and linear stability of steady-state hotspot solutions for an extension of the 1-D three-component reaction-diffusion (RD) system formulated and studied numerically in Jones et. al. [Math. Models. Meth. Appl. Sci.,20, Suppl., (2010)], which models urban crime with police intervention. In our extended RD model, the field variables are the attractiveness field 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 attractiveness field. For a special choice of this parameter, we recover the “cops-on-the-dots” policing strategy of Jones et. al., where the police mimic the drift of the criminals towards maxima of the attractiveness field. For our extended model, the method of matched asymptotic expansions is used to construct 1-D steady-state hotspot patterns as well as to derive nonlocal eigenvalue problems (NLEPs), each having three distinct nonlocal terms, that characterize the linear stability of these hotspot steady-states toO(1) time-scale instabilities. For a cops-on-the-dots policing strategy, where some key identities can be used to recast these NLEPs into equivalent NLEPs with only one nonlocal term, we prove that a multi-hotspot steady-state is linearly stable to synchronous perturbations of the hotspot amplitudes. Moreover, for asynchronous perturbations of the hotspot amplitudes, a hybrid analytical-numerical method is used to construct linear stability phase diagrams in the police versus criminal diffusivity parameter space. In one particular region of these phase diagrams, the hotspot steady-states are shown to be unstable to asynchronous oscillatory instabilities in the hotspot amplitudes that arise from a Hopf bifurcation. Within the context of our model,this provides a parameter range where the effect of a cops-on-the-dots policing strategy is to only displace crime temporally between neighboring spatial regions. Our hybrid approach to study the equivalent NLEP combines rigorous spectral results with a numerical parameterization of any Hopf bifurcation threshold. For the cops-on-the-dots policing strategy, our linear stability predictions for steady-state hotspot patterns are confirmed from full numerical PDE simulations of the three-component RD system.
This publication is part of the Development of mathematical tools for (non-local) reaction-advection-diffusion equations