具奇异敏感的趋向性犯罪模型中的警察威慑效应

数学物理学报 ›› 2025, Vol. 45 ›› Issue (2): 512-533.

具奇异敏感的趋向性犯罪模型中的警察威慑效应

李彬¹(),谢莉^2,^*()

¹宁波工程学院统计与数据科学学院浙江宁波 315211
²重庆师范大学数学科学学院重庆 401331

收稿日期:2023-11-28 修回日期:2024-10-23 出版日期:2025-04-26 发布日期:2025-04-09
通讯作者: 谢莉 E-mail:blimath@163.com;mathxieli@cqnu.edu.cn
作者简介:李彬，E-mail:blimath@163.com
基金资助:
宁波市自然基金(2022J147);重庆市科委项目(CSTB2023NSCQ-MSX0411);重庆市教委科研项目(KJZD-M202000502);重庆市教委科研项目(CXQT21014)

The Effects of Police Deployment in a Chemotaxis System with Singular Sensitivity for Criminal Activities

Bin Li¹(),Li Xie^2,^*()

¹School of Statistics and Data Science, Ningbo University of Technology, Zhejiang Ningbo 315211
²School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331

Received:2023-11-28 Revised:2024-10-23 Online:2025-04-26 Published:2025-04-09
Contact: Li Xie E-mail:blimath@163.com;mathxieli@cqnu.edu.cn
Supported by:
NSF of Ningbo Municipality(2022J147);Chongqing Science and Technology Commission Project(CSTB2023NSCQ-MSX0411);Research Project of Chongqing Education Commission(KJZD-M202000502);Research Project of Chongqing Education Commission(CXQT21014)

摘要/Abstract

摘要：

该文在有界光滑域 $\Omega\subset \mathbb{R}^n$ ( $n\geq3$ )上研究了一个带奇异灵敏度的两组分非局部模型, 该模型是三组分的 Jones-Brantingham-Chayes 趋向性模型的一个简化模型, 后者被用于模拟在警察威慑下犯罪活动的时空动态. 该文在较大趋化敏感系数范围内证明了相应初边值问题拥有全局经典解. 值得指出的是, 相较于无警察威慑效应的 Short et al 趋向性犯罪模型的相关结果, 警察威慑扩大了确保解全局存在的趋化敏感系数范围, 在某种意义下这也表明了警察威慑效应对模型解性质具有正则化效应. 注意, 先前数值结果 (Jones, Brantingham and Chayes. Math Models Methods Appl Sci, 2010) 表明警察威慑有益于镇压犯罪热点的形成, 因此该文的研究结果也是相应数值结果的一个理论支持.

关键词: 趋向性模型, 奇异敏感, 经典解, 全局存在性, 长时间行为

Abstract:

As a simplified version of a three-component chemotaxis system introduced by Jones et al to model the spatio-temporal behavior of criminal activities under the effects of police deployment, a two-component non-local system with singular sensitivity is considered over a bounded domain $\Omega\subset \mathbb{R}^n$ with $n\geq3$ . For all reasonably regular initial data, the existence of classical solution of the corresponding initial-boundary value problem is established globally in time. In particular, we enlarge the range of chemotactic sensitivity $\chi_1$ , compared to known results on the Short et al model which describes the spatio-temporal behavior of criminal activities without the effects of police deployment, which reveals that the police deployment has a regularization effect on solution. This is also a theoretical supplement to the previously numerical result unveiled by Jones, Brantingham and Chayes (Math Models MethodsAppl Sci, 2010) that the police deployment is beneficial for suppressing the formation of criminal hotspots.

Key words: chemotaxis system, singular sensitivity, classical solution, global existence, long-time behavior

中图分类号:

O175.2

李彬,谢莉. 具奇异敏感的趋向性犯罪模型中的警察威慑效应[J]. 数学物理学报, 2025, 45(2): 512-533.

Bin Li,Li Xie. The Effects of Police Deployment in a Chemotaxis System with Singular Sensitivity for Criminal Activities[J]. Acta mathematica scientia,Series A, 2025, 45(2): 512-533.

参考文献 53

[1]	Ahn J, Kang K, Lee J. Global well-posedness of logarithmic Keller-Segel type systems. J Differential Equations, 2021, 287: 185-211
[2]	Bellomo N, Colasuonno F, Knopoff D, Soler J. From a systems theory of sociology to modeling the onset and evolution of criminality. Math Models Methods Appl Sci, 2015, 10: 421-441
[3]	Bellomo N, Outada N, Soler J, et al. Chemotaxis and cross-diffusion models in complex environments: Models and analytic problems toward a multiscale vision. Math Models Methods Appl Sci, 2022, 32(4): 713-792
[4]	Berestycki H, Nadal J. Self-organised critical hot spots of criminal activity. Eur J Appl Math, 2010, 21: 371-399
[5]	Berestycki H, Wei J, Winter M. Existence of symmetric and asymmetric spikes for a crime hotspot model. SIAM J Math Anal, 2014, 46(1): 691-719
[6]	Cao X. Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces. Discrete Contin Dyn Syst, 2015, 35(5), 1891-1904
[7]	Cantrell R, Cosner C, Manásevich R. Global bifurcation of solutions for crime modeling equations. SIAM J Math Anal, 2012, 44(3): 1340-1358
[8]	D'Orsogna M, Perc M. Statistical physics of crime: A review. Phys Life Rev, 2015, 12: 1-21 doi: 10.1016/j.plrev.2014.11.001 pmid: 25468514
[9]	Espejo E, Winkler M. Global classical solvability and stabilization in a two dimensional chemotaxis-Navier-Stokes system modeling coral fertilization. Nonlinearity, 2018, 31: 1227-1259
[10]	Freitag M. Global solutions to a higher-dimensional system related to crime modeling. Math Meth Appl Sci, 2018, 41(16): 6326-6335
[11]	Fuest M, Heihoff F. Unboundedness phenomenon in a model of urban crime. Commun Contemp Math, 2021, 26(7): Article 2350032
[12]	Fujie K, Ito A, Winkler M, Yokota T. Stabilization in a chemotaxis model for tumor invasion. Discrete Contin Dyn Syst, 2016, 36: 151-169
[13]	Gu Y, Wang Q, Yi G. Stationary patterns and their selection mechanism of urban crime models with heterogeneous near-repeat victimization effect. Eur J Appl Math, 2017, 28(1): 141-178
[14]	Heihoff F. Generalized solutions for a system of partial differential equations arising from urban crime modeling with a logistic source term. Z Für Angew Math Phys, 2020, 71(3): Article 80
[15]	Hillen T, Painter K, Winkler M. Convergence of a cancer invasion model to a logistic chemotaxis model. Math Models Methods Appl Sci, 2013, 23(1): 165-198
[16]	Jiang Y, Yang L. Global solvability and stabilization in a three-dimensional cross-diffusion system modeling urban crime propagation. Acta Appl Math, 2022, 178: Article 11
[17]	Jones P, Brantingham P, Chayes L. Statistical models of criminal behavior: the effects of law enforcement actions. Math Models Methods Appl Sci, 2010, 20: 1397-1423
[18]	Kolokolnikov T, Ward M, Wei J. The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime. Discrete Contin Dyn Syst, 2014, 19(5): 1373-1410
[19]	Ladyzhenskaya O A, Ural'tseva N W. Linear and Quasilinear Elliptic Equations. New York-London: Academic Press, 1968
[20]	Ladyzhenskaya O A, Solonnikov V A, Ural'tseva N. Linear and Quasilinear Equations of Parabolic Type. Translations of Mathematical Monographs. Providence, RI: American Mathematical Society, 1968
[21]	Lankeit J, Winkler M. A generalized solution concept for the Keller-Segel system with logarithmic sensitivity: global solvability for large nonradial data. NoDEA-Nonlinear Differ Equ Appl, 2017, 24(4): Article 49
[22]	Li B, Xie L. Generalized solution to a 2D parabolic-parabolic chemotaxis system for urban crime: Global existence and large time behavior. Eur J Appl Math, 2024, 35(3): 409-429
[23]	Li B, Xie L. Generalized solution and eventual smoothness in a logarithmic Keller-Segel system for criminal activities. Math Models Methods Appl Sci, 2023, 33(6): 1281-1330
[24]	Li B, Xie L. Smoothness effects of a quadratic damping term of mixed type on a chemotaxis-type system modeling propagation of urban crime. Nonliear Anal Real World Appl, 2023, 73: Article 103912
[25]	Li B, Wang Z, Xie L. Regularization effect of the mixed-type damping in a higher-dimensional logarithmic Keller-Segel system related to crime modeling. Math Biosci Eng, 2023, 24(3): 4532-4559
[26]	Lloyd D, O'Farrell H. On localised hotspots of an urban crime model. Phys D, 2013, 253: 23-39
[27]	Lloyd D, Santitissadeekorn N, Short M. Exploring data assimilation and forecasting issues for an urban crime model. Eur J Appl Math, 2016, 27(3): 451-478
[28]	Mei L, Wei J. The existence and stability of spike solutions for a chemotax is system modeling crime pattern formation. Math Models Methods Appl Sci, 2020, 30: 1727-1764
[29]	Pitcher A. Adding police to a mathematical model of burglary. Eur J Appl Math, 2010, 21: 401-419
[30]	Qiu Z, Li B. Eventual smoothness of generalized solutions to a singular chemotaxis system for urban crime in space dimension 2. Electron Res Arch, 2023, 31(6): 3218-3244
[31]	Rodríguez N, Bertozzi A. Local existence and uniqueness of solutions to a PDE model for criminal behavior. Math Models Methods Appl Sci, 2010, 20(supp1): 1425-1457
[32]	Rodríguez N, Winkler M. On the global existence and qualitative behavior of one-dimensional solutions to a model for urban crime. Eur J Appl Math, 2022, 33(5): 919-959
[33]	Rodríguez N, Winkler W. Relaxation by nonlinear diffusion enhancement in a two-dimensional cross-diffusion model for urban crime propagation. Math Models Methods Appl Sci, 2020, 30: 2105-2137
[34]	Shen J, Li B. Mathematical analysis of a continuous version of statistical models for criminal behavior. Math Meth Appl Sci, 2020, 43(1): 409-426
[35]	Short M, D'Orsogna M, Pasour V, et al. A statistical model of criminal behavior. Math Models Methods Appl Sci, 2008, 18: 1249-1267
[36]	Short M, Brantingham P, Bertozzi A, Tita G. Dissipation and displacement of hotspots in reaction-diffusion models of crime. Proc Natl Acad Sci, 2010, 107: 3961-3965
[37]	Short M, Bertozzi A, Brantingham P. Nonlinear patterns in urban crime: hotspots, bifurcations, and suppression. SIAM J Appl Dyn Syst, 2010, 9(9): 462-483
[38]	Short M, Mohler G, Brantingham P, Tita G. Gang rivalry dynamics via coupled point process networks. Discrete Contin Dyn Syst, 2014, 19(5): 1459-1477
[39]	Tania N, Vnderleib B, Heathc J, Edelstein-Keshetc L. Role of social interactions in dynamic patterns of resource patches and forager aggregation. Proc Natl Acad Sci, 2012, 109: 1128-1133
[40]	Tao Y, Winkler M. Global smooth solutions in a two-dimensional cross-diffusion system modeling propagation of urban crime. Commun Math Sci, 2021, 19(3): 829-849
[41]	Tao Y, Winkler M. Small-signal solutions to a nonlocal cross-diffusion model for interaction of scroungers with rapidly diffusing foragers. Math Models Methods Appl Sci, 2023, 33(1): 103-138
[42]	Tse W, Ward M. Asynchronous instabilities of crime hotspots for a 1-D reaction-diffusion model of urban crime with focused police patrol. SIAM J Appl Dyn Syst, 2018, 17(3): 2018-2075
[43]	Tse W, Ward M. Hotspot formation and dynamics for a continuum model of urban crime. Eur J Appl Math, 2016, 27: 583-624
[44]	Wang D, Feng Y. Global well-posedness and uniform boundedness of a higher dimensional crime model with a logistic source term. Math Meth Appl Sci, 2022, 45(8): 4727-4740
[45]	Wang Q, Wang D, Feng Y. Global well-posedness and uniform boundedness of urban crime models: One-dimensional case. J Differential Equations, 2020, 269(7): 6216-6235
[46]	Winkler M. Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model. J Differential Equations, 2010, 248(12): 2889-2905
[47]	Winkler M. Global solutions in a fully parabolic chemotaxis system with singular sensitivity. Math Methods Appl Sci, 2010, 34(2): 176-190
[48]	Winkler M. Global solvability and stabilization in a two-dimensional cross-diffusion system modeling urban crime propagation. Ann Inst Henri Poincaré Anal Non Linéaire, 2019, 36: 1747-1790
[49]	Winkler M. The two-dimensional Keller-Segel system with singular sensitivity and signal absorption: global large-data solutions and their relaxation properties. Math Models Methods Appl Sci, 2016, 26: 987-1024
[50]	Winkler M. A quantitative strong parabolic maximum principle and application to a taxis-type migration-consumption model involving signal-dependent degenerate diffusion. Ann Inst Henri Poincaré Anal Non Linéaire, 2024, 41: 95-127
[51]	Xiang T. Finite time blow-up in the higher dimensional parabolic-elliptic-ODE minimal chemotaxis-haptotaxis system. J Differential Equations, 2022, 336: 44-72
[52]	Yang L, Yang X. Global existence in a two-dimensional nonlinear diffusion model for urban crime propagation. Nonliear Anal, 2022, 224: 113086
[53]	Zipkin J, Short M, Bertozzi A. Cops on the dots in a mathematical model of urban crime and police response. Discrete Contin Dyn Syst, 2014, 19(5): 1479-1506

Metrics

Viewed

Full text

	From	local

	Times	73
	Rate	100%

Abstract

Just accepted	Online first	Issue

0	0	17

	From	Others

	Times	17
	Rate	100%

Cited

Shared

Discussed