Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-23T20:40:28.861Z Has data issue: false hasContentIssue false
  • English
  • Français

Stationary patterns and their selection mechanism of urban crime models with heterogeneous near-repeat victimization effect

Published online by Cambridge University Press:  26 May 2016

YU GU ,
QI WANG  and
GUANGZENG YI
YU GU
Affiliation:
Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130, China email: [email protected], [email protected], [email protected]
QI WANG
Affiliation:
Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130, China email: [email protected], [email protected], [email protected]
GUANGZENG YI
Affiliation:
Department of Mathematics, Southwestern University of Finance and Economics, 555 Liutai Ave, Wenjiang, Chengdu, Sichuan 611130, China email: [email protected], [email protected], [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we study two PDEs that generalize the urban crime model proposed by Short et al. (2008 Math. Models Methods Appl. Sci.18, 1249–1267). Our modifications are made under assumption of the spatial heterogeneity of both the near-repeat victimization effect and the dispersal strategy of criminal agents. We investigate pattern formations in the reaction–advection–diffusion systems with non-linear diffusion over multi-dimensional bounded domains subject to homogeneous Neumann boundary conditions. It is shown that the positive homogeneous steady state loses its stability as the intrinsic near-repeat victimization rate ε decreases and spatially inhomogeneous steady states emerge through bifurcation. Moreover, we find the wavemode selection mechanism through rigorous stability analysis of these non-trivial spatial patterns, which shows that the only stable pattern must have a wavenumber that maximizes the bifurcation value. Based on this wavemode selection mechanism, we will be able to predict the formation of stable aggregates of the house attractiveness and criminal population density, at least when the diffusion rate ε is around the principal bifurcation value. Our theoretical results also suggest that large domains support more stable aggregates than small domains. Finally, we perform extensive numerical simulations over 1D intervals and 2D squares to illustrate and verify our theoretical findings. Our numerics also demonstrate the formation of other interesting patterns in these models such as the merging of two interior spikes and the emerging of new spikes, etc. These non-trivial solutions can model the well-observed phenomenon of aggregation in urban criminal activities.

Keywords

Urban crime modelnon-linear diffusionpattern formationstability analysis
Type
Papers
Information
European Journal of Applied Mathematics , Volume 28 , Issue 1 , February 2017 , pp. 141 - 178
Copyright
Copyright © Cambridge University Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Alama, S. & Lu, Q. (2009) Compactly supported solutions to stationary degenerate diffusion equations. J. Differ. Equ. 246 (8), 3214–3240.CrossRefGoogle Scholar
[2] Anselin, L., Cohen, J., Cook, D., Gorr, W. & Tita, G. (2000) Spatial analyses of crime. Crim.Just. 4, 212–262.Google Scholar
[3] Aronson, D. G. (1980) Density-dependent interaction-diffusion systems. Dynamics and modelling of reactive systems. In: Proceedings of an Advanced Seminar Conducted by the Mathematics Research Center, the University of Wisconsin-Madison, 1979, Publ. Math. Res. Center Univ. Wisconsin, 44, Academic Press, New York, London, pp. 161–176.Google Scholar
[4] Berestycki, H. & Nadal, J.-P. (2010) Self-organised critical hot spots of criminal activity. Eur. J. Appl. Math. 21 (4–5), 371–399.CrossRefGoogle Scholar
[5] Berestycki, H., Nadal, J.-P. & Rodríguez, N. (2015) A model of riots dynamics: Shocks, diffusion and thresholds. Netw. Heterog. Media 10 (3), 443–475.Google Scholar
[6] Berestycki, H. & Rodríguez, N. (2016) Analysis of a heterogeneous model for riot dynamics: The effect of censorship of information, Eur. J. Appl. Math, 27(Special Issue 03), 554–582.CrossRefGoogle Scholar
[7] Berestycki, H., Rodríguez, N. & Ryzhik, L. (2013) Traveling wave solutions in a reaction–diffusion model for criminal activity. Multiscale Model. Simul. 11 (4), 1097–1126.Google Scholar
[8] Berthelin, F., Chiron, D. & Ribot, M. (2016) Stationary solutions with vacuum for a one-dimensional chemotaxis model with nonlinear pressure. Commun. Math. Sci. 14 (1), 147–186.CrossRefGoogle Scholar
[9] Bettencourt, L., Lobo, J., Strumsky, D. & West, G. (2010) Urban scaling and its deviations: Revealing the structure of wealth, innovation and crime across cities. PLOS One 5 (11).CrossRefGoogle ScholarPubMed
[10] Bottoms, A. & Wiles, P. (1992) Crime, policing and places: Essay in environmental criminology. Routledge 27–54.Google Scholar
[11] Brantingham, P.J. & Brantingham, P.L. (1984) Patterns in Crime. MacMillan, New York.Google Scholar
[12] Brantingham, P. (1998) Freight train graffiti: Subculture, crime, dislocation. Justice Q. 15 (4), 587–608.Google Scholar
[13] Brantingham, P. (2005) Domestic burglary repeats and space-time clusters the dimensions of risk. Eur. J. Criminol. 2 (1), 67–92.Google Scholar
[14] Cantrell, R., Cosner, C. & Manásevich, R. (2012) Global bifurcation of solutions for crime modeling equations. SIAM J. Math. Anal. 44 (3), 1340–1358.Google Scholar
[15] Carrillo, J. A., Castorina, D. & Volzone, B. (2015) Ground states for diffusion dominated free energies with logarithmic interaction. SIAM J. Math. Anal. 47 (1), 1–25.Google Scholar
[16] Chambliss, W. (1973) Functional and conflict theories of crime. MSS Modular Publications.Google Scholar
[17] Chaturapruek, S., Breslau, J., Yazdi, D., Kolokolnikov, T. & McCalla, S. (2013) Crime modeling with Lévy flights. SIAM J. Appl. Math. 73 (4), 1703–1720.Google Scholar
[18] Chayes, L., Kim, I. & Yao, Y. (2013) An aggregation equation with degenerate diffusion: Qualitative property of solutions. SIAM J. Math. Anal. 45 (5), 2995–3018.Google Scholar
[19] Cohn, S., Barkan, S. & Halteman, W. (1991) Punitive attitudes toward criminals: Racial consensus or racial conflict. Soc. Probs 38 (2), 287–296.Google Scholar
[20] Crandall, M. & Rabinowitz, P. (1971) Bifurcation from simple eigenvalues. J. Funct. Anal. 8 (2), 321–340.CrossRefGoogle Scholar
[21] Crandall, M. & Rabinowitz, P. (1973) Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch. Rational Mech. Anal. 52 (2), 161–180.CrossRefGoogle Scholar
[22] Daskalopoulos, P. & Kenig, C. Degenerate diffusions. Initial value problems and local regularity theory. EMS Tracts in Mathematics, 1. European Mathematical Society (EMS), Zrich, 2007. x+198 pp. ISBN: 978-3-03719-033-3Google Scholar
[23] Drangeid, A.-K. (1989) The principle of linearized stability for quasilinear parabolic evolution equations. Nonlinear Anal. 13 (9), 1091–1113.Google Scholar
[24] Hosono, Y. (1986) Traveling wave solutions for some density dependent diffusion equations. Japan J. Appl. Math. 3 (1), 163–196.Google Scholar
[25] Hosono, Y. & Mimura, M. (1989) Localized cluster solutions of nonlinear degenerate diffusion equations arising in population dynamics. SIAM J. Math. Anal. 20 (4), 845–869.CrossRefGoogle Scholar
[26] Horstmann, D. (2003) From 1970 until present: the Keller-Segel model in chemotaxis and its consequences I. Jahresber DMV 105 (3), 103–165.Google Scholar
[27] Lloyd, D. & O'Farrell, H. (2013) On localised hotspots of an urban crime model. Phys. D. 253, 23–39.Google Scholar
[28] Manásevich, R., Garcia–Huidobro, M. & Mawhin, J. (2013) Existence of solutions for a 1-D boundary value problem coming from a model for burglary. Nonlinear Anal. Real World Appl. 14 (5), 1939–1946.Google Scholar
[29] Hillen, T. & Painter, K. (2009) A user's guide to PDE models for chemotaxis. J. Math. Biol. 58 (1–2), 183–217.Google Scholar
[30] Garcia-Huidobro, M., Manásevich, R. & Mawhin, J. (2014) Solvability of a nonlinear Neumann problem for systems arising from a burglary model. Appl. Math. Lett. 35, 102–108.Google Scholar
[31] Johnson, S. D. (2010) A brief history of the analysis of crime concentration. Eur. J. Appl. Math. 21 (4–5), 349–370.Google Scholar
[32] Johnson, S. D., Bowers, K. & Hirschfield, A. (1997) New insights into the spatial and temporal distribution of repeat victimization. Br. J. Criminol. 37 (2), 224–241.Google Scholar
[33] Jones, P., Brantingham, P. & Chayes, L. (2010) Statistical models of criminal behavior: The effects of law enforcement actions. Math. Models Methods Appl. Sci. 20 (supple.1), 1397–1423.Google Scholar
[34] Kato, T. (1976) Perturbation theory for linear operators. Second edition. Grundlehren der Mathematischen Wissenschaften, Band 132. Springer-Verlag, Berlin-New York, xxi+619 pp. 47-XX.Google Scholar
[35] Keller, E. F. & Segel, L. A. (1970) Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26 (3), 399–415.Google Scholar
[36] Keller, E. F. & Segel, L. A. (1971) Model for chemotaxis. J. Theor. Biol. 30 (2), 225–234.Google Scholar
[37] Keller, E. F. & Segel, L. A. (1971) Traveling bands of chemotactic bacteria: A theretical analysis. J. Theor. Biol. 30 (2), 235–248.CrossRefGoogle Scholar
[38] Kolokolnikov, T., Ward, M. & Wei, J. (2014) The stability of steady–state hot-spot patterns for a reaction–diffusion model of urban crime. Discrete Contin. Dyn. Syst. Ser. B. 19 (5), 1373–1410.Google Scholar
[39] Kubrin, C. E. & Weitzer, R. (2003) New directions in social disorganization theory. J. Res. Crime Delinq. 40 (4), 374–402.Google Scholar
[40] Ma, M., Ou, C. & Wang, Z.-A. (2012) Stationary solutions of a volume-filling chemotaxis model with logistic growth and their stability. SIAM J. Appl. Math. 72 (3), 740–766.CrossRefGoogle Scholar
[41] Manásevich, R., Phan, Q. & Souplet, P. (2013) Global existence of solutions for a chemotaxis-type system arising in crime modelling. Eur. J. Appl. Math. 24 (2), 273–296.Google Scholar
[42] Nec, Y. & Ward, M. (2013) The stability and slow dynamics of two-spike patterns for a class of reaction–diffusion system. Math. Model. Nat. Phenom. 8 (5), 206–232.Google Scholar
[43] Painter, K. & Hillen, T. (2011) Spatio–temporal chaos in a chemotaxis model. Phys. D. 240 (4–5), 363–375.Google Scholar
[44] Perc, M., Donnay, K. & Helbing, D. (2013) Understanding recurrent crime as system–immanent collective behavior. PLOS One 8 (10).Google Scholar
[45] Peterson, R. D. & Krivo, L. J. (1993) Racial segregation and black urban homicide. Soc. Forces 71 (4), 1001–1026.CrossRefGoogle Scholar
[46] Pitcher, A. (2010) Adding police to a mathematical model of burglary. Eur. J. Appl. Math. 21 (4–5), 401–419.CrossRefGoogle Scholar
[47] Ricketson, L. A continuum model of residential burglary incorporating law enforcement. preprint.Google Scholar
[48] Rodríguez, N. (2013) On the global well-posedness theory for a class of PDE models for criminal activity. Phys. D. 260, 191–200.Google Scholar
[49] Rodríguez, N. & Bertozzi, A. (2010) Local existence and uniqueness of solutions to a PDE model of criminal behavior. Math. Models Methods Appl. Sci., 20 (suppl.1), 1425–1457.Google Scholar
[50] Rodríguez, N. & Ryzhik, L. (2016) Exploring the effects of social preference, economic disparity, and heterogeneous environments on segregation. Commun. Math. Sci. 14 (2), 363–387.Google Scholar
[51] Sampson, R. & Groves, W. B. (1989) Community structure and crime: Testing social–disorganization theory. Am. J. Sociol. 94 (4), 774–802.Google Scholar
[52] Sampson, R. & Raudenbush, S. (2004) Seeing disorder: Neighborhood stigma and the social construction of broken windows. Soc. Psychol. Q. 67 (4), 319–342.Google Scholar
[53] Shi, J. & Wang, X. (2009) On global bifurcation for quasilinear elliptic systems on bounded domains. J. Differ. Equ. 246 (7), 2788–2812.Google Scholar
[54] Short, M., Bertozzi, A. & Brantingham, P. (2010) Nonlinear patterns in urban crime: Hotspots, bifurcations, and suppression. SIAM, J. Appl. Dyn. Syst. 9 (2), 462–483.Google Scholar
[55] Short, M., Mohler, G., Brantingham, P. & Tita, G. (2014) Gang rivalry dynamics via coupled point process networks. Discrete Contin. Dyn. Syst. Ser. B 19 (5), 1459–1477.Google Scholar
[56] Short, M., D'Orsogna, M., Pasour, V., Tita, G., Brantingham, P., Bertozzi, A. & Chayes, L. (2008) A statistical model of criminal behavior. Math. Models Methods Appl. Sci. 18 (suppl.), 1249–1267.Google Scholar
[57] Sleeman, B., Ward, M. & Wei, J. (2005) The existence and stability of spike patterns in a chemotaxis model. SIAM J. Appl. Math. 65 (3), 790–817.Google Scholar
[58] Tse, S. & Ward, M. (2016) Hotspot formation and dynamics for a continuum model of urban crime. Eur. J. Appl. Math, 27(Special Issue 03), 583–624 Google Scholar
[59] Wilson, J. & Kelling, G. (1982) Broken windows: The police and neighborhood safety. Atlantic Mon. 249, 29–38.Google Scholar
[60] Wolfgang, M. & Ferracuti, F. The Subculture of Violence: Towards an Integrated Theory in Criminology, 406, Sage Publications, Inc.Google Scholar
[61] Woodworth, J., Mohler, G., Bertozzi, A. & Brantingham, P. (2014) Non-local crime density estimation incorporating housing information. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 372 (2028), 15.Google Scholar
[62] Zipkin, J., Short, M. & Bertozzi, A. (2014) Cops on the dots in a mathematical model of urban crime and police response. Discrete Contin. Dyn. Syst. Ser. B 19 (5), 1479–1506.Google Scholar