The mechanisms driving the nucleation, spread, and dissipation of crime hotspots are poorly understood. As a consequence, the ability of law enforcement agencies to use mapped crime patterns to design crime prevention strategies is severely hampered. We also lack robust expectations about how different policing interventions should impact crime. Here we present a mathematical framework based on reaction-diffusion partial differential equations for studying the dynamics of crime hotspots. The system of equations is based on empirical evidence for how offenders move and mix with potential victims or targets. Analysis shows that crime hotspots form when the enhanced risk of repeat crimes diffuses locally, but not so far as to bind distant crime together. Crime hotspots may form as either supercritical or subcritical bifurcations, the latter the result of large spikes in crime that override linearly stable, uniform crime distributions. Our mathematical methods show that subcritical crime hotspots may be permanently eradicated with police suppression, whereas supercritical hotspots are displaced following a characteristic spatial pattern. Our results thus provide a mechanistic explanation for recent failures to observe crime displacement in experimental field tests of hotspot policing.crime pattern formation | hotspot policing | mathematical modeling | nonlinearity | partial differential equations C rime is a ubiquitous feature of all modern cities, but not all neighborhoods are affected equally. In fact, serious crimes ranging from residential burglary to homicide are strongly patterned in time and space, forming crime "hotspots" (1-3). Studies show that policing actions directed at crime hotspots do lead to real reductions in offending and calls to the police for service (4, 5), while displacement of crime to adjacent settings may be less common than once thought (6-8). However, further gains in crime reduction are dependent upon gaining a quantitative understanding of the mechanisms that drive the emergence, spread, and dissipation of crime hotspots. Reaction-diffusion models, in which activators and inhibitors move, mix, and interact, provide a useful framework in which to investigate the formation of crime patterns and the impact of alternative policing strategies on crime hotspot stability. In this context, motivated offenders (activators) search their environment for suitable targets or victims (activators), which may also be mobile, following simple behavioral routines (9, 10). If an offender encounters a target in the absence of an effective security measure (inhibitor), then he is free to exploit that target. The immediate presence of security such as law enforcement is sufficient to deter that crime. Here we show that largescale spatial crime patterns, including the formation of stationary crime hotspots, are strongly dependent upon the local diffusion of risk, driven by offender mobility in the environment, coupled with the phenomena of repeat and near repeat victimization.