Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Cops on the dots in a mathematical model of urban crime and police response

Pages: 1479 - 1506, Volume 19, Issue 5, July 2014      doi:10.3934/dcdsb.2014.19.1479

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Joseph R. Zipkin - Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095, United States (email)
Martin B. Short - School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, United States (email)
Andrea L. Bertozzi - Department of Mathematics, University of California Los Angeles, Los Angeles, CA, 90095, United States (email)

Abstract: Hotspots of crime localized in space and time are well documented. Previous mathematical models of urban crime have exhibited these hotspots but considered a static or otherwise suboptimal police response to them. We introduce a program of police response to hotspots of crime in which the police adapt dynamically to changing crime patterns. In particular, they choose their deployment to solve an optimal control problem at every time. This gives rise to a free boundary problem for the police deployment's spatial support. We present an efficient algorithm for solving this problem numerically and show that police presence can prompt surprising interactions among adjacent hotspots.

Keywords:  Reaction-diffusion equations, optimal control, free boundary problem, crime modeling.
Mathematics Subject Classification:  Primary: 35K57, 49M29, 65K10; Secondary: 65M06.

Received: January 2013;      Revised: January 2014;      Available Online: April 2014.