The Nucleation-Annihilation Dynamics of Hotspot Patterns for a Reaction-Diffusion System of Urban Crime with Police Deployment

A hybrid asymptotic-numerical approach is developed to study the existence and linear stability of steady-state hotspot patterns for a three-component one-dimensional reaction-diffusion (RD) system that models urban crime with police intervention. Our analysis is focused on a new scaling regime in the RD system where there are two distinct competing mechanisms of hotspot annihilation and creation that, when coincident in a parameter space, lead to complex spatio-temp oral dynamics of hotspot patterns. Hotspot annihilation events are shown numerically to be triggered by an asynchronous oscillatory instability of the hotspot amplitudes that arises from a secondary instability on the branch of periodic solutions that emerges from a Hopf bifurcation of the steady-state solution. In addition, hotspots can be nucleated from a quiescent background when the criminal diffusivity is below a saddle-node bifurcation threshold of hotspot equilibria, which we estimate from our asymptotic analysis. To investigate instabilities of hotspot steady states, the spectrum of the linearization around a two-boundary hotspot pattern is computed, and instability thresholds due to either zero-eigenvalue crossings or Hopf bifurcations are shown. The bifurcation software pde2path is used to follow the branch of periodic solutions and detect the onset of the secondary instability. Overall, these results provide a phase diagram in parameter space where distinct types of dynamical behaviors occur. In one region of this phase diagram, where the police diffusivity is small, a two-boundary hotspot steady state is unstable to an asynchronous oscillatory instability in the hotspot amplitudes. This instability typically triggers a nonlinear process leading to the annihilation of one of the hotspots. However, for parameter values where this instability is coincident with the nonexistence of a one-hotspot steady state, we show that hotspot patterns undergo complex ``nucleation-annihilation"" dynamics that are characterized by large-scale persistent oscillations of the hotspot amplitudes. In this way, our results identify parameter ranges in the three-component crime model where the effect of police intervention is to simply displace crime between adjacent hotspots and where new crime hotspots regularly emerge ``spontaneously"" from regions that were previously free of crime. More generally, it is suggested that when these annihilation and nucleation mechanisms are coincident for other multihotspot patterns, the problem of predicting the spatial-temporal distribution of crime is largely intractable.