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Hotspot formation and dynamics for a continuum model of urban crime†

Published online by Cambridge University Press:  20 July 2015

W. H. TSE  and
M. J. WARD
W. H. TSE
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T 1Z2, Canada email: simontse@math.ubc.ca, ward@math.ubc.ca
M. J. WARD
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, B.C., V6T 1Z2, Canada email: simontse@math.ubc.ca, ward@math.ubc.ca
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Abstract

The existence, stability, and dynamics of localized patterns of criminal activity are studied for the reaction–diffusion model of urban crime introduced by Short et al. (Math. Models. Meth. Appl. Sci.18(Suppl.), (2008), 1249–1267). In the singularly perturbed limit of small diffusivity ratio, this model admits hotspot patterns, where criminal activity of high amplitude is localized within certain narrow spatial regions. By using a combination of asymptotic analysis and numerical path-following methods, hotspot equilibria are constructed on a finite 1-D domain and their bifurcation properties analysed as the diffusivity of criminals is varied. It is shown, both analytically and numerically, that new hotspots of criminal activity can be nucleated in low-crime regions with inconspicuous crime activity gradient when the spatial extent of these regions exceeds a critical threshold. These nucleations are referred to as “peak insertion” events, and for the steady-state problem, they occur near a saddle-node bifurcation point characterizing hotspot equilibria. For the time-dependent problem, a differential algebraic (DAE) system characterizing the slow dynamics of a collection of hotspots is derived, and the results compared favourably with full numerical simulations of the PDE system. The asymptotic theory to construct hotspot equilibria, and to derive the differential algebraic system for quasi-steady patterns, is based on the resolution of a triple-deck structure near the core of each hotspot and the identification of so-called switchback terms.

Keywords

fold pointpeak insertionslow hotspot dynamicsswitchback termrenormalization
Type
Papers
Information
European Journal of Applied Mathematics , Volume 27 , Issue 3: Mathematical Modelling of Crime and Security , June 2016 , pp. 583 - 624
Copyright
Copyright © Cambridge University Press 2015 

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Footnotes

†

M. J. W. gratefully acknowledges the grant support of NSERC (Canada).

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