On localised hotspots of an urban crime model

We investigate stationary, spatially localised crime hotspots on the real line and the plane of an urban crime model of Short et al. [M. Short, M. DÓrsogna, A statistical model of criminal behavior, Mathematical Models and Methods in Applied Sciences 18 (2008) 1249–1267]. Extending the weakly nonlinear analysis of Short et al., we show in one-dimension that localised hotspots should bifurcate off the background spatially homogeneous state at a Turing instability provided the bifurcation is subcritical. Using path-following techniques, we continue these hotspots and show that the bifurcating pulses can undergo the process of homoclinic snaking near the singular limit. We analyse the singular limit to explain the existence of spike solutions and compare the analytical results with the numerical computations. In two-dimensions, we show that localised radial spots should also bifurcate off the spatially homogeneous background state. Localised planar hexagon fronts and hexagon patches are found and depending on the proximity to the singular limit these solutions either undergo homoclinic snaking or act like “multi-spot” solutions. Finally, we discuss applications of these localised patterns in the urban crime context and the full agent-based model.