Nonlinear Patterns in Urban Crime: Hotspots, Bifurcations, and Suppression

We present a weakly nonlinear analysis of our recently developed model for the formation of crime patterns. Using a perturbative approach, we find amplitude equations that govern the development of crime "hotspot" patterns in our system in both the one-dimensional (1D) and two-dimensional (2D) cases. In addition to the supercritical spots already shown to exist in our previous work, we prove here the existence of subcritical hotspots that arise via subcritical pitchfork bifurcations or transcritical bifurcations, depending on the geometry. We present numerical results that both validate our analytical findings and confirm the existence of these subcritical hotspots as stable states. Finally, we examine the differences between these two types of hotspots with regard to attempted hotspot suppression, referencing the varying levels of success such attempts have had in real world scenarios.