  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.
