Crime hotspot dynamics in residential burglary models with police response

TL;DR

This paper models residential burglary dynamics incorporating police response delays, revealing how delayed feedback can cause oscillating crime hotspots and emphasizing the importance of timely crime data over police density.

Contribution

It introduces a coupled PDE-ODE model with delayed police response based on empirical data, analyzing stability and hotspot behaviors in crime modeling.

Findings

Delayed police response can destabilize stable crime equilibria.

Timely crime data access is more effective than police density in stabilizing crime.

The model predicts moving, splitting, and merging crime hotspots.

Abstract

We develop and analyze mathematical models for residential burglary that incorporates police deployment through a delayed feedback mechanism. Motivated by empirical observations from publicly available crime and policing data, we extend a well-known agent-based model by introducing a dynamic police response driven by crime information that becomes available only after a finite delay. Taking the mean-field limit, we derive a coupled continuum system consisting of three partial differential equations and one ordinary differential equation describing the interactions among criminal density, environmental attractiveness, delayed crime signal, and police deployment. Linear stability analysis of homogeneous steady states reveals that response delays can destabilize otherwise stable equilibria through Hopf bifurcations. As a result, the model predicts sustained temporal oscillations and dynamically evolving crime hotspots. Numerical simulations of both the agent-based and continuum models confirm the theoretical analysis and uncover rich spatio-temporal behaviors, including moving, splitting, and merging hotspots. Through a parametric study, we investigate the roles of police density, crime information delay, and neighborhood effects in controlling stability, hotspot size, and oscillatory behavior. Our results indicate that timely access to crime data plays a more important role than police density in stabilizing crime levels.