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On localised hotspots of an urban crime model. (English) Zbl 1284.91486

Summary: We investigate stationary, spatially localised crime hotspots on the real line and the plane of an urban crime model of M. B. Short et al. [Math. Models Methods Appl. Sci. 18, 1249–1267 (2008; Zbl 1180.35530)]. Extending the weakly nonlinear analysis of M. B. Short et al. [loc. cit.], 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.

MSC:

91D10 Models of societies, social and urban evolution
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
35B36 Pattern formations in context of PDEs

Citations:

Zbl 1180.35530
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References:

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