Localized Pattern Formation in Continuum Models of Urban Crime

In this thesis, the phenomenon of localized crime hotspots in models capturing the repeat and near-victimization nature of urban crime was studied. Stability, insertion, slow movement of crime hotspots and the effect of police patrol modeled by an extra equation derived from biased random walk were studied by means of matched asymptotic expansions, nonlocal eigenvalue problem (NLEP) stability analysis, and numerical computations. In the absence of police, we confirmed the linear stability of the far-from equilibrium steady-states with crime hotspots in the original parameters regimes as observed in [47]. The results hold for both the supercritical and subcritical regimes distinguished by a Turing bifurcation (cf. [48, 49]). Moreover, the phenomenon of peak insertion was characterized by a simple nonlinear equation computable by quadratures and a normal form equation identical to that of the self-replication of Mesa patterns [28] was derived. Slow dynamics of unevenly-spaced configurations of hotspots were described by an ODE-DAE system, which was derived from resolving an intricate triple-deck structure of boundary layers formed between the hotspots and their neighbouring regions. In the presence of police, which was modeled by a simple interaction with criminals, single and multiple hotspots patterns were constructed in a near-shadow limit of criminal diffusivity. While a single hotspot was found to be unconditionally stable, the linear stability behaviour of multiple-hotspot patterns was found to depend on two thresholds, between which we also observe a novel Hopf bifurcation phenomenon leading to asynchronous oscillations. For one particular, but representative, parameter value in the model, the determination of the spectrum of the NLEP was found to reduce to the study of a quadratic equation for the eigenvalue. For more general parameter values, where this reduction does not apply, a winding number analysis on the NLEP was used to determine detailed stability properties associated with multiple steady-state hotspot solutions.