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We analyze a system of partial differential equations introduced as a model for urban crime with law enforcement. This system is known to obtain spatially localized patterns representing crime hotspots. In this work, we obtain the amplitude equations that describe the hotspot pattern formation in the model using weakly nonlinear analysis techniques. In particular, we find the existence of super- and sub-critical pitchfork bifurcations. Moreover, we propose different suppression strategies and investigate numerically how the suggested approaches effectively eradicate the two types of hotspots. We highlight the challenges in weakly nonlinear analysis and the limitations of the proposed suppression mechanisms, emphasizing the need for further theoretical and practical work on this urban crime model with law enforcement.
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This content will become publicly available on March 23, 2027
Global bifurcations and pattern formation in target–offender–guardian crime models
Abstract We study a reaction–advection–diffusion model of a target–offender–guardian system designed to capture interactions between urban crime and policing. Using Crandall–Rabinowitz bifurcation theory and spectral analysis, we establish rigorous conditions for both steady-state and Hopf bifurcations. These results identify critical thresholds of policing intensity at which spatially uniform equilibria lose stability, leading either to persistent heterogeneous hotspots or oscillatory crime–policing cycles. From a criminological perspective, such thresholds represent tipping points in guardian mobility: once crossed, they can lock neighbourhoods into stable clusters of criminal activity or trigger recurrent waves of hotspot formation. Numerical simulations complement the theory, exhibiting stationary patterns, periodic oscillations and chaotic dynamics. By explicitly incorporating law enforcement as a third interacting component, our framework extends classical two-equation models. It offers new tools for analysing non-linear interactions, bifurcations and pattern formation in multi-agent social systems.
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- Award ID(s):
- 2042413
- PAR ID:
- 10684217
- Publisher / Repository:
- European Journal of Applied Mathematics
- Date Published:
- Journal Name:
- European Journal of Applied Mathematics
- ISSN:
- 0956-7925
- Page Range / eLocation ID:
- 1 to 29
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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