July  2014, 19(5): 1479-1506. doi: 10.3934/dcdsb.2014.19.1479

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

1. 

Department of Mathematics, University of California, Los Angeles, Los Angeles, CA 90095, United States

2. 

School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, United States

3. 

Department of Mathematics, University of California Los Angeles, Los Angeles, CA, 90095

Received  January 2013 Revised  January 2014 Published  April 2014

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.
Citation: Joseph R. Zipkin, Martin B. Short, Andrea L. Bertozzi. Cops on the dots in a mathematical model of urban crime and police response. Discrete & Continuous Dynamical Systems - B, 2014, 19 (5) : 1479-1506. doi: 10.3934/dcdsb.2014.19.1479
References:
[1]

H. Berestycki and J.-P. Nadal, Self-organised critical hot spots of criminal activity,, Eur. J. Appl. Math., 21 (2010), 371. doi: 10.1017/S0956792510000185. Google Scholar

[2]

H. Berestycki, N. Rodríguez and L. Ryzhik, Traveling wave solutions in a reaction-diffusion model for criminal activity,, Multiscale Model. Simul., 11 (2013), 1097. doi: 10.1137/12089884X. Google Scholar

[3]

D. Birks, M. Townsley and A. Stewart, Generative explanations of crime: Using simulation to test criminological theory,, Criminology, 50 (2012), 221. doi: 10.1111/j.1745-9125.2011.00258.x. Google Scholar

[4]

A. A. Braga, The effects of hot spots policing on crime,, Ann. Am. Acad. Polit. S. S., 578 (2001), 104. Google Scholar

[5]

P. J. Brantingham and P. L. Brantingham, Patterns in Crime,, Macmillan, (1984). Google Scholar

[6]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy,, J. Chem. Phys., 28 (1958), 258. doi: 10.1063/1.1744102. Google Scholar

[7]

R. S. Cantrell, C. Cosner and R. Manásevich, Global bifurcation of solutions for crime modeling equations,, SIAM J. Math. Anal., 44 (2012), 1340. doi: 10.1137/110843356. Google Scholar

[8]

S. Chaturapruek, J. Breslau, D. Yazdi, T. Kolokolnikov and S. G. McCalla., Crime modeling with Lévy flights,, SIAM J. Appl. Math., 73 (2013), 1703. doi: 10.1137/120895408. Google Scholar

[9]

J. E. Eck and D. Weisburd, Crime places in crime theory,, in Crime and Place: Crime Prevention Studies (eds. John E. Eck and David Weisburd), (1995), 1. Google Scholar

[10]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. Google Scholar

[11]

S. D. Johnson, K. Bowers and A. Hirschfield, New insights into the spatial and temporal distribution of repeat victimization,, British J. Criminol, 37 (1997), 224. doi: 10.1093/oxfordjournals.bjc.a014156. Google Scholar

[12]

Paul A. Jones, P. J. Brantingham and L. R. Chayes, Statistical models of criminal behavior: The effects of law enforcement actions,, Math. Mod. Meth. Appl. S., 20 (2010), 1397. doi: 10.1142/S0218202510004647. Google Scholar

[13]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[14]

E. F. Keller and L. A. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225. doi: 10.1016/0022-5193(71)90050-6. Google Scholar

[15]

T. Kolokolnikov, M. J. Ward and J. Wei, The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime,, to appear in Discrete Cont. Dyn.-B, (). Google Scholar

[16]

A. B. Pitcher, Adding police to a mathematical model of burglary,, Eur. J. Appl. Math., 21 (2010), 401. doi: 10.1017/S0956792510000112. Google Scholar

[17]

N. Rodríguez, On the global well-posedness theory for a class of {PDE} models for criminal activity,, Physica D, 260 (2013), 191. doi: 10.1016/j.physd.2012.08.003. Google Scholar

[18]

N. Rodríguez and A. Bertozzi, Local existence and uniqueness of solutions to a PDE model for criminal behavior,, Math. Mod. Meth. Appl. S., 20 (2010), 1425. doi: 10.1142/S0218202510004696. Google Scholar

[19]

M. B. Short, M. R. D'Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, A statistical model of criminal behavior,, Math. Mod. Meth. Appl. S., 18 (2008), 1249. doi: 10.1142/S0218202508003029. Google Scholar

[20]

M. B. Short, A. L. Bertozzi and P. J. Brantingham, Nonlinear patterns in urban crime: Hotspots, bifurcations, and suppression,, SIAM J. Appl. Dyn. Syst., 9 (2010), 462. doi: 10.1137/090759069. Google Scholar

[21]

M. B. Short, P. J. Brantingham, A. L. Bertozzi and G. E. Tita, Dissipation and displacement of hotspots in reaction-diffusion models of crime,, P. Natl. Acad. Sci. USA, 107 (2010), 3961. doi: 10.1073/pnas.0910921107. Google Scholar

[22]

H. Sun, Continuum Equations for Crime Modeling with Random Process,, unpublished report, (2010). Google Scholar

[23]

M. Townsley, S. D. Johnson and J. H. Ratcliffe, Space time dynamics of insurgent activity in Iraq,, Security J., 21 (2008), 139. doi: 10.1057/palgrave.sj.8350090. Google Scholar

[24]

Y. Yao and A. L. Bertozzi, Blow-up dynamics for the aggregation equation with degenerate diffusion,, Physica D, 260 (2013), 77. doi: 10.1016/j.physd.2013.01.009. Google Scholar

show all references

References:
[1]

H. Berestycki and J.-P. Nadal, Self-organised critical hot spots of criminal activity,, Eur. J. Appl. Math., 21 (2010), 371. doi: 10.1017/S0956792510000185. Google Scholar

[2]

H. Berestycki, N. Rodríguez and L. Ryzhik, Traveling wave solutions in a reaction-diffusion model for criminal activity,, Multiscale Model. Simul., 11 (2013), 1097. doi: 10.1137/12089884X. Google Scholar

[3]

D. Birks, M. Townsley and A. Stewart, Generative explanations of crime: Using simulation to test criminological theory,, Criminology, 50 (2012), 221. doi: 10.1111/j.1745-9125.2011.00258.x. Google Scholar

[4]

A. A. Braga, The effects of hot spots policing on crime,, Ann. Am. Acad. Polit. S. S., 578 (2001), 104. Google Scholar

[5]

P. J. Brantingham and P. L. Brantingham, Patterns in Crime,, Macmillan, (1984). Google Scholar

[6]

J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy,, J. Chem. Phys., 28 (1958), 258. doi: 10.1063/1.1744102. Google Scholar

[7]

R. S. Cantrell, C. Cosner and R. Manásevich, Global bifurcation of solutions for crime modeling equations,, SIAM J. Math. Anal., 44 (2012), 1340. doi: 10.1137/110843356. Google Scholar

[8]

S. Chaturapruek, J. Breslau, D. Yazdi, T. Kolokolnikov and S. G. McCalla., Crime modeling with Lévy flights,, SIAM J. Appl. Math., 73 (2013), 1703. doi: 10.1137/120895408. Google Scholar

[9]

J. E. Eck and D. Weisburd, Crime places in crime theory,, in Crime and Place: Crime Prevention Studies (eds. John E. Eck and David Weisburd), (1995), 1. Google Scholar

[10]

T. Hillen and K. J. Painter, A user's guide to PDE models for chemotaxis,, J. Math. Biol., 58 (2009), 183. doi: 10.1007/s00285-008-0201-3. Google Scholar

[11]

S. D. Johnson, K. Bowers and A. Hirschfield, New insights into the spatial and temporal distribution of repeat victimization,, British J. Criminol, 37 (1997), 224. doi: 10.1093/oxfordjournals.bjc.a014156. Google Scholar

[12]

Paul A. Jones, P. J. Brantingham and L. R. Chayes, Statistical models of criminal behavior: The effects of law enforcement actions,, Math. Mod. Meth. Appl. S., 20 (2010), 1397. doi: 10.1142/S0218202510004647. Google Scholar

[13]

E. F. Keller and L. A. Segel, Initiation of slime mold aggregation viewed as an instability,, J. Theor. Biol., 26 (1970), 399. doi: 10.1016/0022-5193(70)90092-5. Google Scholar

[14]

E. F. Keller and L. A. Segel, Model for chemotaxis,, J. Theor. Biol., 30 (1971), 225. doi: 10.1016/0022-5193(71)90050-6. Google Scholar

[15]

T. Kolokolnikov, M. J. Ward and J. Wei, The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime,, to appear in Discrete Cont. Dyn.-B, (). Google Scholar

[16]

A. B. Pitcher, Adding police to a mathematical model of burglary,, Eur. J. Appl. Math., 21 (2010), 401. doi: 10.1017/S0956792510000112. Google Scholar

[17]

N. Rodríguez, On the global well-posedness theory for a class of {PDE} models for criminal activity,, Physica D, 260 (2013), 191. doi: 10.1016/j.physd.2012.08.003. Google Scholar

[18]

N. Rodríguez and A. Bertozzi, Local existence and uniqueness of solutions to a PDE model for criminal behavior,, Math. Mod. Meth. Appl. S., 20 (2010), 1425. doi: 10.1142/S0218202510004696. Google Scholar

[19]

M. B. Short, M. R. D'Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, A statistical model of criminal behavior,, Math. Mod. Meth. Appl. S., 18 (2008), 1249. doi: 10.1142/S0218202508003029. Google Scholar

[20]

M. B. Short, A. L. Bertozzi and P. J. Brantingham, Nonlinear patterns in urban crime: Hotspots, bifurcations, and suppression,, SIAM J. Appl. Dyn. Syst., 9 (2010), 462. doi: 10.1137/090759069. Google Scholar

[21]

M. B. Short, P. J. Brantingham, A. L. Bertozzi and G. E. Tita, Dissipation and displacement of hotspots in reaction-diffusion models of crime,, P. Natl. Acad. Sci. USA, 107 (2010), 3961. doi: 10.1073/pnas.0910921107. Google Scholar

[22]

H. Sun, Continuum Equations for Crime Modeling with Random Process,, unpublished report, (2010). Google Scholar

[23]

M. Townsley, S. D. Johnson and J. H. Ratcliffe, Space time dynamics of insurgent activity in Iraq,, Security J., 21 (2008), 139. doi: 10.1057/palgrave.sj.8350090. Google Scholar

[24]

Y. Yao and A. L. Bertozzi, Blow-up dynamics for the aggregation equation with degenerate diffusion,, Physica D, 260 (2013), 77. doi: 10.1016/j.physd.2013.01.009. Google Scholar

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