Stationary patterns and their selection mechanism of urban crime models with heterogeneous near-repeat victimization effect

In this paper, we study two PDEs that generalize the urban crime model proposed by Short et al. (2008 Math. Models Methods Appl. Sci. 18, 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 non-linear 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 ε decreases and spatially inhomogeneous steady states emerge through bifurcation. Moreover, we find the wavemode selection mechanism through rigorous stability analysis of these non-trivial spatial patterns, which shows that the only stable pattern must have a wavenumber that maximizes the bifurcation value. Based on this wavemode selection mechanism, we will be able to predict the formation of stable aggregates of the house attractiveness and criminal population density, at least when the diffusion rate ε 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 demonstrate the formation of other interesting patterns in these models such as the merging of two interior spikes and the emerging of new spikes, etc. These non-trivial solutions can model the well-observed phenomenon of aggregation in urban criminal activities.