Crime Modeling with Lévy Flights

Abstract

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 Lévy 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 steady state, as well as the expected number of hotspots, depends on the system parameters and especially the exponent of the underlying power law. The assumptions of the underlying agent-based model naturally lead to a separation of scales of the diffusion coefficients in the continuum limit. Using these assumptions, we asymptotically construct the leading-order profile of the localized hotspot of criminal activity.