The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime

Theodore Kolokolnikov; Michael J. Ward; Juncheng Wei

doi:10.3934/dcdsb.2014.19.1373

Article Contents

2014, Volume 19, Issue 5: 1373-1410. Doi: 10.3934/dcdsb.2014.19.1373

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The stability of steady-state hot-spot patterns for a reaction-diffusion model of urban crime

1.
Dalhousie University, Department of Mathematics and Statistics, Halifax, Nova Scotia, B3H 3J5
2.
Mathematics Department, University of British Columbia, Vancouver, B.C. V6T 1Z2
3.
Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong

Received: January 2012

Revised: April 2012

Published: July 2014

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Abstract

The existence and stability of localized patterns of criminal activity are studied for the reaction-diffusion model of urban crime that was introduced by Short et. al. [Math. Models. Meth. Appl. Sci., 18, Suppl. (2008), pp. 1249--1267]. Such patterns, characterized by the concentration of criminal activity in localized spatial regions, are referred to as hot-spot patterns and they occur in a parameter regime far from the Turing point associated with the bifurcation of spatially uniform solutions. Singular perturbation techniques are used to construct steady-state hot-spot patterns in one and two-dimensional spatial domains, and new types of nonlocal eigenvalue problems are derived that determine the stability of these hot-spot patterns to ${\mathcal O}(1)$ time-scale instabilities. From an analysis of these nonlocal eigenvalue problems, a critical threshold $K_c$ is determined such that a pattern consisting of $K$ hot-spots is unstable to a competition instability if $K>K_c$. This instability, due to a positive real eigenvalue, triggers the collapse of some of the hot-spots in the pattern. Furthermore, in contrast to the well-known stability results for spike patterns of the Gierer-Meinhardt reaction-diffusion model, it is shown for the crime model that there is only a relatively narrow parameter range where oscillatory instabilities in the hot-spot amplitudes occur. Such an instability, due to a Hopf bifurcation, is studied explicitly for a single hot-spot in the shadow system limit, for which the diffusivity of criminals is asymptotically large. Finally, the parameter regime where localized hot-spots occur is compared with the parameter regime, studied in previous works, where Turing instabilities from a spatially uniform steady-state occur.

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Mathematics Subject Classification: Primary: 35B35, 35B20, 45M05; Secondary: 92D50, 92D25.

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