Research Article
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Year 2022, , 26 - 40, 31.03.2022
https://doi.org/10.53391/mmnsa.2022.01.003

Abstract

References

  • Brantingham, P. & Tita, G. Offender Mobility and crime pattern formation from first principles. (In L. Liu, & J. Eck (Ed.), Artificial Crime Analysis Systems: Using Computer Simulations,2008).
  • Short, M., D’Orsogna, M., Pasour, V., Tita, G., Brantingham, P., Bertozzi, A. & Chayes, L. A Statistical Model of Criminal Behavior. Mathematical Models And Methods In Applied Sciences, 18, 1249-1267, (2008).
  • Short, M.B., Bertozzi, A.L., & Brantingham, P.J. Nonlinear patterns in urban crime: Hotspots, bifurcations, and suppression. SIAM Journal on Applied Dynamical Systems, 9(2), 462-483, (2010).
  • Jones, P.A., Brantingham, P.J., & Chayes, L.R. Statistical models of criminal behavior: the effects of law enforcement actions. Mathematical Models and Methods in Applied Sciences, 20(supp01), 1397-1423, (2010).
  • Berestycki, H. & Nadal, J.P. Self-organised critical hot spots of criminal activity. European Journal Of Applied Mathematics, 21(4-5), 371-399, (2010).
  • Chaturapruek, S., Breslau, J., Yazdi, D., Kolokolnikov, T. & McCalla, S.G. Crime modeling with Lévy flights. SIAM Journal On Applied Mathematics, 73(4), 1703-1720, (2013).
  • Kolokolnikov, T., Ward, M. & Wei, J. The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime. Discrete And Continuous Dynamical Systems - Series B, 19, 1373, (2014).
  • Zipkin, J.R., Short, M.B. & Bertozzi, A.L. Cops on the dots in a mathematical model of urban crime and police response. Discrete And Continuous Dynamical Systems - Series B, 19(5), 1479, (2014).
  • Camacho, A., Lee, H.R.L & Smith, L.M. Modelling policing strategies for departments with limited resources. European Journal Of Applied Mathematics, 27(3), 479-501, (2016).
  • Gu, Y., Wang, Q., & Yi, G. Stationary patterns and their selection mechanism of urban crime models with heterogeneous near-repeat victimization effect. European Journal Of Applied Mathematics, 28(1), 141- 78, (2017).
  • Pan, C., Li, B., Wang, C., Zhang, Y., Geldner, N., Wang, L. & Bertozzi, A.L. Crime modeling with truncated Lévy flights for residential burglary models. Mathematical Models And Methods In Applied Sciences, 28(09), 1857-1880, (2018).
  • Wang, Q., Wang, D. & Feng, Y. Global well-posedness and uniform boundedness of urban crime models: One-dimensional case. Journal Of Differential Equations, 269(7), 6216-6235, (2020).
  • Rodríguez, N. & Winkler, M. Relaxation by nonlinear diffusion enhancement in a two-dimensional cross-diffusion model for urban crime propagation. Mathematical Models And Methods In Applied Sciences, 30(11), 2105-2137, (2020).
  • Kang, K., Kolokolnikov, T. & Ward, M. The stability and dynamics of a spike in the 1D Keller–Segel model. IMA Journal Of Applied Mathematics, 72(2), 140-162, (2007).
  • Mei, L. & Wei, J. The existence and stability of spike solutions for a chemotax is system modeling crime pattern formation. Mathematical Models And Methods In Applied Sciences, 30(09), 1727-1764, (2020).
  • Kondo, S. & Miura, T. Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation. Science, 329(5999), 1616-1620, (2010).
  • Keller, E. & Segel, L. Traveling bands of chemotactic bacteria: A theoretical analysis. Journal Of Theoretical Biology, 30(2), 235-248, (1971).
  • Biler, P. & Wu, G. Two-dimensional chemotaxis models with fractional diffusion. Mathematical Methods In The Applied Sciences, 32(1), 112-126, (2009).
  • Calvó-Armengol, A. & Zenou, Y. Social Networks and Crime Decisions: The Role of Social Structure in Facilitating Delinquent Behavior. International Economic Review, 45(3), 939-958, (2004).
  • Nec, Y. Spike-Type Solutions to One Dimensional Gierer–Meinhardt Model with Lévy Flights. Studies In Applied Mathematics, 129(3), 272-299, (2012).
  • Cruz-García, S., Martínez-Farías, F., Santillán-Hernández, A. & Rangel, E. Mathematical home burglary model with stochastic long crime trips and patrolling: Applied to Mexico City. Applied Mathematics And Computation, 396, 125865, (2021).
  • Levajkovic, T., Mena, H. & Zarfl, M. Lévy processes, subordinators and crime modeling. Novi Sad Journal Of Mathematics, 46(2), 65-86, (2016).
  • Cohen, L. & Felson, M. Social Change and Crime Rate Trends: A Routine Activity Approach. American Sociological Review, 44, 588-608, (1979).
  • Wilson, J.Q. & Kelling, G. L. Broken windows. The Atlantic, 249, 29-38, (1982).
  • Kilbas, A.A., Srivastava, H.M. & Trujillo, J. Theory and Applications of Fractional Differential Equations (Vol. 204). Elsevier, North-Holland Mathematics Studies, (2006).
  • Kopriva, D. Implementing Spectral Methods for Partial Differential Equations. Springer, (2009).
  • Osgood, B.G. Lectures on the Fourier Transform and Its Applications. SIAM, (2019).
  • Trefethen, L.N. Spectral methods in MATLAB, volume 10 of Software, Environments, and Tools. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 24, (2000).
  • Butcher, J. Numerical Methods for Ordinary Differential Equations. John Wiley, 2015.
  • Ghrist, M.L., Fornberg, B. & Reeger, J.A. Stability ordinates of Adams predictor-corrector methods. BIT Numerical Mathematics, 55(3), 733-750, (2015).
  • Lindfield, G.R. & Penny, J.E. Numerical methods: using MATLAB. Elsevier, 2012.
  • Tadjeran, C. & Meerschaert, M.M. A second-order accurate numerical method for the two-dimensional fractional diffusion equation. Journal Of Computational Physics, 220(2), 813-823, (2007).
  • Lapidus, L. & Seinfeld, J. 4 Predictor-Corrector Methods. Numerical Solution Of Ordinary Differential Equations, 74, 152-241, (1971).
  • Rodriguez, N., & Bertozzi, A. Local existence and uniqueness of solutions to a PDE model for criminal behavior. Mathematical Models and Methods in Applied Sciences, 20(supp01), 1425-1457, (2010).
  • Cantrell, R.S., Cosner, C. & Manásevich, R. Global Bifurcation of Solutions for Crime Modeling Equations. SIAM Journal On Mathematical Analysis, 44(3), 1340-1358, (2012).
  • Tse, W.H. & Ward, M.J. Hotspot formation and dynamics for a continuum model of urban crime. European Journal Of Applied Mathematics, 27(3), 583-624, (2016).
  • Rodríguez, N., Wang, Q. & Zhang, L. Understanding the Effects of On- and Off-Hotspot Policing: Evidence of Hotspot, Oscillating, and Chaotic Activities. SIAM Journal On Applied Dynamical Systems, 20(4), 1882-1916, (2021).

Bi-dimensional crime model based on anomalous diffusion with law enforcement effect

Year 2022, , 26 - 40, 31.03.2022
https://doi.org/10.53391/mmnsa.2022.01.003

Abstract

Several models based on discrete and continuous fields have been proposed to comprehend residential criminal dynamics. This study introduces a two-dimensional model to describe residential burglaries diffusion, employing Lèvy flights dynamics. A continuous model is presented, introducing bidimensional fractional operator diffusion and its differences with the 1-dimensional case. Our results show, graphically, the hotspot's existence solution in a 2-dimensional attractiveness field, even fractional derivative order is modified. We also provide qualitative evidence that steady-state approximation in one dimension by series expansion is insufficient to capture similar original system behavior. At least for the case where series coefficients have a linear relationship with derivative order. Our results show, graphically, the hotspot's existence solution in a 2-dimensional attractiveness field, even if fractional derivative order is modified. Two dynamic regimes emerge in maximum and total attractiveness magnitude as a result of fractional derivative changes, these regimes can be understood as considerations about different urban environments. Finally, we add a Law enforcement component, embodying the "Cops on dots" strategy; in the Laplacian diffusion dynamic, global attractiveness levels are significantly reduced by Cops on dots policy but lose efficacy in Lèvy flight-based diffusion regimen. The four-step Preditor-Corrector method is used for numerical integration, and the fractional operator is approximated, getting the advantage of the spectral methods to approximate spatial derivatives in two dimensions.

References

  • Brantingham, P. & Tita, G. Offender Mobility and crime pattern formation from first principles. (In L. Liu, & J. Eck (Ed.), Artificial Crime Analysis Systems: Using Computer Simulations,2008).
  • Short, M., D’Orsogna, M., Pasour, V., Tita, G., Brantingham, P., Bertozzi, A. & Chayes, L. A Statistical Model of Criminal Behavior. Mathematical Models And Methods In Applied Sciences, 18, 1249-1267, (2008).
  • Short, M.B., Bertozzi, A.L., & Brantingham, P.J. Nonlinear patterns in urban crime: Hotspots, bifurcations, and suppression. SIAM Journal on Applied Dynamical Systems, 9(2), 462-483, (2010).
  • Jones, P.A., Brantingham, P.J., & Chayes, L.R. Statistical models of criminal behavior: the effects of law enforcement actions. Mathematical Models and Methods in Applied Sciences, 20(supp01), 1397-1423, (2010).
  • Berestycki, H. & Nadal, J.P. Self-organised critical hot spots of criminal activity. European Journal Of Applied Mathematics, 21(4-5), 371-399, (2010).
  • Chaturapruek, S., Breslau, J., Yazdi, D., Kolokolnikov, T. & McCalla, S.G. Crime modeling with Lévy flights. SIAM Journal On Applied Mathematics, 73(4), 1703-1720, (2013).
  • Kolokolnikov, T., Ward, M. & Wei, J. The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime. Discrete And Continuous Dynamical Systems - Series B, 19, 1373, (2014).
  • Zipkin, J.R., Short, M.B. & Bertozzi, A.L. Cops on the dots in a mathematical model of urban crime and police response. Discrete And Continuous Dynamical Systems - Series B, 19(5), 1479, (2014).
  • Camacho, A., Lee, H.R.L & Smith, L.M. Modelling policing strategies for departments with limited resources. European Journal Of Applied Mathematics, 27(3), 479-501, (2016).
  • Gu, Y., Wang, Q., & Yi, G. Stationary patterns and their selection mechanism of urban crime models with heterogeneous near-repeat victimization effect. European Journal Of Applied Mathematics, 28(1), 141- 78, (2017).
  • Pan, C., Li, B., Wang, C., Zhang, Y., Geldner, N., Wang, L. & Bertozzi, A.L. Crime modeling with truncated Lévy flights for residential burglary models. Mathematical Models And Methods In Applied Sciences, 28(09), 1857-1880, (2018).
  • Wang, Q., Wang, D. & Feng, Y. Global well-posedness and uniform boundedness of urban crime models: One-dimensional case. Journal Of Differential Equations, 269(7), 6216-6235, (2020).
  • Rodríguez, N. & Winkler, M. Relaxation by nonlinear diffusion enhancement in a two-dimensional cross-diffusion model for urban crime propagation. Mathematical Models And Methods In Applied Sciences, 30(11), 2105-2137, (2020).
  • Kang, K., Kolokolnikov, T. & Ward, M. The stability and dynamics of a spike in the 1D Keller–Segel model. IMA Journal Of Applied Mathematics, 72(2), 140-162, (2007).
  • Mei, L. & Wei, J. The existence and stability of spike solutions for a chemotax is system modeling crime pattern formation. Mathematical Models And Methods In Applied Sciences, 30(09), 1727-1764, (2020).
  • Kondo, S. & Miura, T. Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation. Science, 329(5999), 1616-1620, (2010).
  • Keller, E. & Segel, L. Traveling bands of chemotactic bacteria: A theoretical analysis. Journal Of Theoretical Biology, 30(2), 235-248, (1971).
  • Biler, P. & Wu, G. Two-dimensional chemotaxis models with fractional diffusion. Mathematical Methods In The Applied Sciences, 32(1), 112-126, (2009).
  • Calvó-Armengol, A. & Zenou, Y. Social Networks and Crime Decisions: The Role of Social Structure in Facilitating Delinquent Behavior. International Economic Review, 45(3), 939-958, (2004).
  • Nec, Y. Spike-Type Solutions to One Dimensional Gierer–Meinhardt Model with Lévy Flights. Studies In Applied Mathematics, 129(3), 272-299, (2012).
  • Cruz-García, S., Martínez-Farías, F., Santillán-Hernández, A. & Rangel, E. Mathematical home burglary model with stochastic long crime trips and patrolling: Applied to Mexico City. Applied Mathematics And Computation, 396, 125865, (2021).
  • Levajkovic, T., Mena, H. & Zarfl, M. Lévy processes, subordinators and crime modeling. Novi Sad Journal Of Mathematics, 46(2), 65-86, (2016).
  • Cohen, L. & Felson, M. Social Change and Crime Rate Trends: A Routine Activity Approach. American Sociological Review, 44, 588-608, (1979).
  • Wilson, J.Q. & Kelling, G. L. Broken windows. The Atlantic, 249, 29-38, (1982).
  • Kilbas, A.A., Srivastava, H.M. & Trujillo, J. Theory and Applications of Fractional Differential Equations (Vol. 204). Elsevier, North-Holland Mathematics Studies, (2006).
  • Kopriva, D. Implementing Spectral Methods for Partial Differential Equations. Springer, (2009).
  • Osgood, B.G. Lectures on the Fourier Transform and Its Applications. SIAM, (2019).
  • Trefethen, L.N. Spectral methods in MATLAB, volume 10 of Software, Environments, and Tools. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 24, (2000).
  • Butcher, J. Numerical Methods for Ordinary Differential Equations. John Wiley, 2015.
  • Ghrist, M.L., Fornberg, B. & Reeger, J.A. Stability ordinates of Adams predictor-corrector methods. BIT Numerical Mathematics, 55(3), 733-750, (2015).
  • Lindfield, G.R. & Penny, J.E. Numerical methods: using MATLAB. Elsevier, 2012.
  • Tadjeran, C. & Meerschaert, M.M. A second-order accurate numerical method for the two-dimensional fractional diffusion equation. Journal Of Computational Physics, 220(2), 813-823, (2007).
  • Lapidus, L. & Seinfeld, J. 4 Predictor-Corrector Methods. Numerical Solution Of Ordinary Differential Equations, 74, 152-241, (1971).
  • Rodriguez, N., & Bertozzi, A. Local existence and uniqueness of solutions to a PDE model for criminal behavior. Mathematical Models and Methods in Applied Sciences, 20(supp01), 1425-1457, (2010).
  • Cantrell, R.S., Cosner, C. & Manásevich, R. Global Bifurcation of Solutions for Crime Modeling Equations. SIAM Journal On Mathematical Analysis, 44(3), 1340-1358, (2012).
  • Tse, W.H. & Ward, M.J. Hotspot formation and dynamics for a continuum model of urban crime. European Journal Of Applied Mathematics, 27(3), 583-624, (2016).
  • Rodríguez, N., Wang, Q. & Zhang, L. Understanding the Effects of On- and Off-Hotspot Policing: Evidence of Hotspot, Oscillating, and Chaotic Activities. SIAM Journal On Applied Dynamical Systems, 20(4), 1882-1916, (2021).
There are 37 citations in total.

Details

Primary Language English
Subjects Applied Mathematics
Journal Section Research Articles
Authors

Francisco Javier Martínez-farías This is me 0000-0002-1853-5981

Anahi Alvarado-sánchez This is me 0000-0001-7736-1056

Eduardo Rangel-cortes This is me 0000-0003-4604-6066

Arturo Hernández-hernández This is me 0000-0001-5617-0808

Publication Date March 31, 2022
Submission Date December 15, 2021
Published in Issue Year 2022

Cite

APA Martínez-farías, F. J., Alvarado-sánchez, A., Rangel-cortes, E., Hernández-hernández, A. (2022). Bi-dimensional crime model based on anomalous diffusion with law enforcement effect. Mathematical Modelling and Numerical Simulation With Applications, 2(1), 26-40. https://doi.org/10.53391/mmnsa.2022.01.003


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