  In this paper, we study two PDEs that generalize the urban crime model
proposed by Short \emph{et al}. [Math. Models Methods Appl. Sci., 18 (2008),
pp. 1249-1267]. Our modifications are made under assumption of the spatial
heterogeneity of both the near-repeat victimization effect and the dispersal
strategy of criminal agents. We investigate pattern formations in the
reaction-advection-diffusion systems with nonlinear diffusion over
multi-dimensional bounded domains subject to homogeneous Neumann boundary
conditions. It is shown that the positive homogeneous steady state loses its
stability as the intrinsic near-repeat victimization rate $\epsilon$ decreases
and spatially nonconstant solutions emerge through bifurcation. Moreover, we
find the wavemode selection mechanism through rigorous stability analysis of
these nontrivial patterns, which shows that the only stable pattern must have
wavenumber that maximizes the bifurcation value. Based on this wavemode
selection mechanism, we will be able to precisely predict the formation of
stable aggregates of the house attractiveness and criminal population density,
at least when the diffusion rate $\epsilon$ is around the principal bifurcation
value. Our theoretical results also suggest that large domains support more
stable aggregates than small domains. Finally, we perform extensive numerical
simulations over 1D intervals and 2D squares to illustrate and verify our
theoretical findings. Our numerics also include some interesting phenomena such
as the merging of two interior spikes and the emerging of new spikes, etc.
These nontrivial solutions can model the well observed aggregation phenomenon
in urban criminal activities.
