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

The existence and stability of localized patterns of criminal activityare studied for the reaction-diffusion model of urban crime that wasintroduced 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 localizedspatial regions, are referred to as hot-spot patterns and they occurin a parameter regime far from the Turing point associated with thebifurcation of spatially uniform solutions. Singular perturbationtechniques are used to construct steady-state hot-spot patterns in oneand two-dimensional spatial domains, and new types of nonlocaleigenvalue problems are derived that determine the stability of thesehot-spot patterns to O(1) time-scale instabilities. Froman analysis of these nonlocal eigenvalue problems, a criticalthreshold Kc is determined such that a pattern consisting of Khot-spots is unstable to a competition instability if K>Kc. Thisinstability, due to a positive real eigenvalue, triggers the collapseof some of the hot-spots in the pattern. Furthermore, in contrast tothe well-known stability results for spike patterns of theGierer-Meinhardt reaction-diffusion model, it is shown for the crimemodel that there is only a relatively narrow parameter range whereoscillatory instabilities in the hot-spot amplitudes occur. Such aninstability, due to a Hopf bifurcation, is studied explicitly for asingle hot-spot in the shadow system limit, for which the diffusivityof criminals is asymptotically large. Finally, the parameter regimewhere localized hot-spots occur is compared with the parameter regime,studied in previous works, where Turing instabilities from a spatiallyuniform steady-state occur.