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Global Bifurcation of Solutions for Crime Modeling Equations

We study pattern formation in a quasi-linear system of two elliptic equations that was developed by Short et al. [Math. Models Methods Appl. Sci., 18 (2008), pp. 1249–1267] as a model for residential burglary. That model is based on the observation that the rate of burglaries of houses that have been burglarized recently and their close neighbors is typically higher than the average rate in the larger community, which creates hotspots for burglary. The patterns generated by the model describe the location of those hotspots. We prove that the system supports global bifurcation of spatially varying solutions from the spatially constant equilibrium, leading to the formation of spatial patterns. The analysis is based on recent results on global bifurcation in quasi-linear elliptic systems derived by Shi and Wang [J. Differential Equations, 7 (2009), pp. 2788–2812]. We show in some cases that near the bifurcation point the bifurcating spatial patterns are stable.