Statistical agent-based models for crime have shown that repeat victimization can lead to predictable crime hotspots (see e.g. 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. Models Methods Appl. Sci. 18 (2008) 1249-1267.), then a recent study in one-space dimension (S. Chaturapruek, J. Breslau, D. Yazdi, T. Kolokolnikov and S. G. McCalla, Crime modeling with Lévy flights, SIAM J. Appl. Math. 73 (2013) 1703-1720.) shows that the hotspot dynamics changes when movement patterns of the criminals involve long-tailed Lévy distributions for the jump length as opposed to classical random walks. In reality, criminals move in confined areas with a maximum jump length. In this paper, we develop a mean-field continuum model with truncated Lévy flights (TLFs) for residential burglary in one-space dimension. The continuum model yields local Laplace diffusion, rather than fractional diffusion. We present an asymptotic theory to derive the continuum equations and show excellent agreement between the continuum model and the agent-based simulations. This suggests that local diffusion models are universal for continuum limits of this problem, the important quantity being the diffusion coefficient. Law enforcement agents are also incorporated into the model, and the relative effectiveness of their deployment strategies are compared quantitatively.
|Original language||English (US)|
|Number of pages||24|
|Journal||Mathematical Models and Methods in Applied Sciences|
|State||Published - Aug 1 2018|
Bibliographical noteFunding Information:
We would like to thank Theodore Kolokolnikov, Martin Short, Scott McCalla, Sorathan Chaturapruek, and Adina Ciomaga for helpful discussions. This work is supported by NSF grant DMS-1045536, NSF grant DMS-1737770, and ARO MURI grant W911NF-11-1-0332. L. W. is also partly supported by NSF grant DMS-1620135. This work was initiated during an undergraduate research training program at UCLA in 2015.
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- Crime models
- Law enforcement agents
- Truncated Lévy flights