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Cops-on-the-dots: The linear stability of crime hotspots for a 1-D reaction-diffusion model of urban crime

Published online by Cambridge University Press:  11 November 2019

Department of Mathematics, UBC, Vancouver, Canada emails:;;
Department of Mathematics and Statistics, Dalhousie, Halifax, Canada email:
Department of Mathematics, UBC, Vancouver, Canada emails:;;
Department of Mathematics, UBC, Vancouver, Canada emails:;;


In a singularly perturbed limit, we analyse 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 population density and the police population density. Our model 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) that characterise the linear stability of these hotspot steady states to ${\cal O}$(1) timescale instabilities. For a cops-on-the-dots policing strategy, we prove that a multi-hotspot steady state is linearly stable to synchronous perturbations of the hotspot amplitudes. Alternatively, for asynchronous perturbations of the hotspot amplitudes, a hybrid analytical–numerical method is used to construct linear stability phase diagrams in the police vs. 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 neighbouring spatial regions. Our hybrid approach to study the NLEPs combines rigorous spectral results with a numerical parameterisation 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.

© Cambridge University Press 2019

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Berestycki, H., Wei, J. & Winter, M. (2014) Existence of symmetric and asymmetric spikes for a crime hotspot model. SIAM J. Math. Anal., 46(1), 691719.CrossRefGoogle Scholar
Blom, J. G., Trompert, R. A. & Verwer, J. G. (1996) Algorithm 758: VLUGR 2: A vectorizable adaptive grid solver for PDEs in 2D. ACM Trans. Math. Softw., 22(3), 302328.CrossRefGoogle Scholar
Braga, A. A. (2001) The effects of hot spots policing on crime. Ann. Am. Acad. Polit. S. S., 578, 104125.CrossRefGoogle Scholar
Brantingham, P. L. & Brantingham, P. J. (1987) Crime Patterns, McMillan.Google Scholar
Camacho, A., Lee, H. R. L. & Smith, L. (2016) Modeling policing strategies for departments with limited resources. Europ. J. Appl. Math., 27(3), 479501.CrossRefGoogle Scholar
Cantrell, R., Cosner, C. & Manasevich, R. (2012) Global bifurcation of solutions for crime modeling equations. SIAM J. Math. Anal . 44(3), 13401358.CrossRefGoogle Scholar
Doelman, A., Gardner, R. A. & Kaper, T. J. (2001) Large stable pulse solutions in reaction-diffusion equations. Indiana U. Math. J., 50(1), 443507.CrossRefGoogle Scholar
Gu, Y., Wang, Q. & Yi, G. (2017) Stationary patterns and their selection mechanism of urban crime models with heterogeneous near-repeat victimization effect. Europ. J. Appl. Math., 28(1), 141178.CrossRefGoogle Scholar
Johnson, S. & Bower, K. (2005) Domestic burglary repeats and space-time clusters: The dimensions of risk. Europ. J. of Criminology, 2, 6792.Google Scholar
Jones, P. A., Brantingham, P. J. & Chayes, L. (2010) Statistical models of criminal behavior: The effect of law enforcement actions. Math. Models. Meth. Appl. Sci., 20, (Suppl.), 13971423.CrossRefGoogle Scholar
Kolokolnikov, T., Ward, M. J. & Wei, J. (2014) The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime. DCDS-B, 19(5), 13731410.CrossRefGoogle Scholar
Kolokolnikov, T., Ward, M. J. & Wei, J. (2005) The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: the low feed rate regime. Studies Appl. Math., 115(1), 2171.CrossRefGoogle Scholar
Lloyd, D. J. B. & O’Farrell, H. (2013) On localised hotspots of an urban crime model. Physica D, 253, 2339.CrossRefGoogle Scholar
van der Ploeg, H. & Doelman, A. (2005) Stability of spatially periodic pulse patterns in a class of singularly perturbed reaction-diffusion equations. Indiana Univ. Math. J., 54(5), 12191301Google Scholar
Pitcher, A. B. (2010) Adding police to a mathematical model of burglary. Europ. J. Appl. Math., 21(4–5), 401419.CrossRefGoogle Scholar
Ricketson, L. (2011) A continuum model of residential burglary incorporating law enforcement, unpublished. Retrieved from Scholar
Rodriguez, N. & Bertozzi, A. (2010) Local existence and uniqueness of solutions to a PDE model for criminal behavior, M3AS (Special Issue on Mathematics and Complexity in Human and Life Sciences), 20(1), 14251457.Google Scholar
Short, M. B., D’Orsogna, M. R., Pasour, V. B., Tita, G. E., Brantingham, P. J., Bertozzi, A. L. & Chayes, L. B. (2008) A statistical model of criminal behavior. Math. Models. Meth. Appl. Sci., 18(Suppl.), 12491267.CrossRefGoogle Scholar
Short, M. B., Bertozzi, A. L. & Brantingham, P. J. (2010) Nonlinear patterns in urban crime – hotspots, bifurcations, and suppression. SIAM J. Appl. Dyn. Sys., 9(2), 462483.CrossRefGoogle Scholar
Short, M. B., Brantingham, P. J., Bertozzi, A. L. & Tita, G. E. (2010). Dissipation and displacement of hotpsots in reaction-diffusion models of crime. Proc. Nat. Acad. Sci., 107(9), 39613965.CrossRefGoogle Scholar
Tse, W.-H. & Ward, M. J. (2016) Hotspot formation and dynamics for a continuum model of urban crime. Europ. J. Appl. Math., 27(3), 583624.CrossRefGoogle Scholar
Tse, W.-H. & Ward, M. J. (2018) Asynchronous instabilities of crime hotspots for a 1-D reaction-diffusion model of urban crime with focused police patrol. SIAM J. Appl. Dyn. Sys., 17(3), 20182075.CrossRefGoogle Scholar
Tse, W. H. (2016) Localized pattern formation in continuum models of urban crime, Ph. D thesis, Univ. of British Columbia.Google Scholar
Ward, M. J. & Wei, J. (2003) Hopf bifurcations and oscillatory instabilities of spike solutions for the one-dimensional Gierer-Meinhardt model. J. Nonlinear Sci., 13(2), 209264.CrossRefGoogle Scholar
Ward, M. J. & Wei, J. (2003) Hopf bifurcation of spike solutions for the shadow Gierer-Meinhardt model. Europ. J. Appl. Math. 14(6), 677711.CrossRefGoogle Scholar
Wei, J. & Winter, M. (2003) A nonlocal eigenvalue problem and the stability of spikes for reaction-diffusion systems with fractional reaction rates. Int. J. Bifur. Chaos 13(6), 15291543.CrossRefGoogle Scholar
Wei, J. & Winter, M. (2001) Spikes for the two-dimensional Gierer-Meinhardt system: the weak coupling case. J. Nonlinear Sci., 11(6), 415458.CrossRefGoogle Scholar
Wei, J. (1999) On single interior spike solutions for the Gierer-Meinhardt system: uniqueness and stability estimates. Europ. J. Appl. Math., 10(4), 353378.CrossRefGoogle Scholar
Wei, J. (2008) Existence and stability of spikes for the Gierer-Meinhardt system. In: Chipot, M. (editor), Handbook of Differential Equations, Stationary Partial Differential Equations, Vol. 5, Elsevier, pp. 489581.Google Scholar
Wilson, J. Q. & Kelling, G. L. (1998) Broken windows and police and neighborhood safety. Atlantic Mon., 249, 2938.Google Scholar
Zipkin, J. R., Short, M. B. & Bertozzi, A. L. (2014) Cops on the dots in a mathematical model of urban crime and police response. DCDS-B, 19(5), 14791506.CrossRefGoogle Scholar