Hotspot formation and dynamics for a continuum model of urban crime
Hotspot formation and dynamics for a continuum model of urban crime
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DOI：
10.1017/s0956792515000376
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发表时间：
2016-06-01
影响因子：
1.9
通讯作者：
Ward, M. J.
中科院分区：
文献类型：
--
作者：
Tse, W. H.;Ward, M. J.
本文研究了由Short等人提出的城市犯罪反应扩散模型的局部化犯罪模式的存在性、稳定性和动态性。Meth.应用科学18（补编），（2008），1249-1267）。在小的扩散率比的奇摄动极限，该模型承认热点模式，高振幅的犯罪活动是局部化在某些狭窄的空间区域。通过使用渐近分析和数值路径跟踪方法相结合，热点平衡构建在一个有限的1-D域和他们的分歧性质分析罪犯的扩散率是不同的。它表明，无论是分析和数字，新的犯罪活动热点可以在低犯罪率的区域与不明显的犯罪活动梯度时，这些区域的空间范围超过一个临界阈值的核。这些成核被称为“峰值插入”事件，对于稳态问题，它们发生在表征热点平衡的鞍结分叉点附近。对于时间依赖性的问题，微分代数（DAE）系统的特点是一个收集的热点的缓慢动态推导出的，并与充分的数值模拟的PDE系统的结果相比有利。构造热点平衡点的渐近理论，并推导出准稳定模式的微分代数系统，是基于每个热点的核心附近的三层结构的分辨率和识别所谓的之字形。
The existence, stability, and dynamics of localized patterns of criminal activity are studied for the reaction-diffusion model of urban crime introduced by Short et al. (Math. Models. Meth. Appl. Sci. 18(Suppl.), (2008), 1249-1267). In the singularly perturbed limit of small diffusivity ratio, this model admits hotspot patterns, where criminal activity of high amplitude is localized within certain narrow spatial regions. By using a combination of asymptotic analysis and numerical path-following methods, hotspot equilibria are constructed on a finite 1-D domain and their bifurcation properties analysed as the diffusivity of criminals is varied. It is shown, both analytically and numerically, that new hotspots of criminal activity can be nucleated in low-crime regions with inconspicuous crime activity gradient when the spatial extent of these regions exceeds a critical threshold. These nucleations are referred to as "peak insertion" events, and for the steady-state problem, they occur near a saddle-node bifurcation point characterizing hotspot equilibria. For the time-dependent problem, a differential algebraic (DAE) system characterizing the slow dynamics of a collection of hotspots is derived, and the results compared favourably with full numerical simulations of the PDE system. The asymptotic theory to construct hotspot equilibria, and to derive the differential algebraic system for quasi-steady patterns, is based on the resolution of a triple-deck structure near the core of each hotspot and the identification of so-called switchback terms.