Crime Modeling with Lévy Flights

The UCLA burglary hotspot model, introduced in [M. B. Short, M. R. D'Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi, and L. B. Chayes, Math. Models Methods Appl. Sci., 18 (2008), pp. 1249--1267], models the formation of hotspots of criminal activity. In this paper, we extend the UCLA model to incorporate a more realistic model of human locomotion. The movement of the criminal agents follows a biased Levy flight with step sizes distributed according to a power-law distribution. The biased Brownian motion of the original model is then derived as a special case. Starting with an agent-based model, we derive its continuum limit. This consists of two equations and involves the fractional Laplacian operator. A numerical method based on the fast Fourier transform is used to simulate the continuum model; these simulations compare favorably with the direct numerical simulations of the agent-based model. A Turing-type analysis is performed to estimate how the instability of the homogeneous ste...

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