Understanding the effects of on- and off-hotspot policing: evidence of hotspot, oscillating, and chaotic activities. (English) Zbl 1478.35129
Summary: This paper considers a system of partial differential equations that govern the spatiotemporal dynamics of urban crime with law enforcement. The deployment of law enforcement is called on-hotspot policing if the dispatched police approach the crime hotspot gradient with the same intensity as the criminal agents, and it is called off-hotspot policing otherwise. We reveal several parameter regimes governed by the policing intensity that promote the emergence of crime hotspots. The existence and stability of these regular patterns, which can be either time-stationary or time-periodic, are proved using bifurcation theories. The results give a wavemode selection mechanism for these spatially heterogeneous solutions and suggest that hotspot policing can stabilize crime aggregates or drive them from one location to another. We also prove that this PDE system admits a unique and global-in-time solution, and the solution is uniformly bounded. However, this system can be ill-posed for both on- and off-hotspot policing as the solution dynamics do not always change continuously with respect to the initial data. Moreover, phase transitions between closed loops of solution trajectories occur for a wide range of system parameters, and this reveals another dimension of complexity of the system. Concerning the antihotspot policing strategy (i.e., law enforcement actively deployed away from hotspots in space), we show that this likely counterintuitive strategy tends to stabilize static hotspots and annihilate time-periodic aggregates. These results point to a difficulty of using this system for something like predicting crime and determining the effectiveness of hotspot policing, due to the parameter sensitivity, and they call for further advancement in the mathematical modeling and analysis of urban criminal activities with hotspot policing.
MSC:
| 35K57 | Reaction-diffusion equations |
| 35B10 | Periodic solutions to PDEs |
| 35B25 | Singular perturbations in context of PDEs |
| 35B32 | Bifurcations in context of PDEs |
| 35B36 | Pattern formations in context of PDEs |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35Q91 | PDEs in connection with game theory, economics, social and behavioral sciences |
Keywords:
urban crime model; hotspot policing; chaos; pattern formation; wavemode selection mechanismReferences:
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