In a singularly perturbed limit, we analyse the existence and linear stability of steady-state hotspot solutions for an extension of the 1-D three-component reaction-diffusion (RD) system formulated and studied numerically in Jones et. al. [Math. Models. Meth. Appl. Sci., 20, Suppl., (2010)], which models urban crime with police intervention. In our extended RD model, the field variables are the attractiveness field for burglary, the criminal population density and the police population density. Our model includes a scalar parameter that determines the strength of the police drift towards maxima of the attractiveness field. For a special choice of this parameter, we recover the ‘cops-on-the-dots’ policing strategy of Jones et. al., where the police mimic the drift of the criminals towards maxima of the attractiveness field. For our extended model, the method of matched asymptotic expansions is used to construct 1-D steady-state hotspot patterns as well as to derive nonlocal eigenvalue problems (NLEPs) that characterise the linear stability of these hotspot steady states to ${\cal O}$(1) timescale instabilities. For a cops-on-the-dots policing strategy, we prove that a multi-hotspot steady state is linearly stable to synchronous perturbations of the hotspot amplitudes. Alternatively, for asynchronous perturbations of the hotspot amplitudes, a hybrid analytical–numerical method is used to construct linear stability phase diagrams in the police vs. criminal diffusivity parameter space. In one particular region of these phase diagrams, the hotspot steady states are shown to be unstable to asynchronous oscillatory instabilities in the hotspot amplitudes that arise from a Hopf bifurcation. Within the context of our model, this provides a parameter range where the effect of a cops-on-the-dots policing strategy is to only displace crime temporally between neighbouring spatial regions. Our hybrid approach to study the NLEPs combines rigorous spectral results with a numerical parameterisation of any Hopf bifurcation threshold. For the cops-on-the-dots policing strategy, our linear stability predictions for steady-state hotspot patterns are confirmed from full numerical PDE simulations of the three-component RD system.

中文翻译：

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Cops-on-the-dots：犯罪热点的线性稳定性，用于城市犯罪的一维反应扩散模型

在奇异扰动极限中，我们分析了稳态热点解的存在性和线性稳定性，以扩展在 Jones 等人中以数值方式制定和研究的一维三分量反应扩散 (RD) 系统。人。[数学。楷模。冰毒。应用程序。科学，20, Suppl., (2010)]，它模拟了警察干预的城市犯罪。在我们扩展的 RD 模型中，场变量是入室盗窃的吸引力场、犯罪人口密度和警察人口密度。我们的模型包括一个标量参数，它决定了警察向吸引力场最大值漂移的强度。对于此参数的特殊选择，我们恢复了 Jones 等人的“点上警察”警务策略。al.，警察模仿罪犯向吸引力场的最大值漂移。对于我们的扩展模型，匹配渐近展开的方法用于构建一维稳态热点模式，并导出表征这些热点稳态线性稳定性的非局部特征值问题 (NLEP)${\cal O}$(1) 时间尺度不稳定性。对于 cops-on-the-dots 警务策略，我们证明了多热点稳态对于热点幅度的同步扰动是线性稳定的。或者，对于热点幅度的异步扰动，使用混合分析-数值方法在警察与犯罪扩散率参数空间中构建线性稳定性相图。在这些相图的一个特定区域中，热点稳态显示出对于由 Hopf 分岔引起的热点振幅中的异步振荡不稳定性是不稳定的。在我们的模型的上下文中，这提供了一个参数范围，其中警察策略的效果是仅在相邻空间区域之间暂时取代犯罪。我们研究 NLEP 的混合方法将严格的光谱结果与任何 Hopf 分岔阈值的数值参数化相结合。对于 cops-on-the-dots 警务策略，我们对稳态热点模式的线性稳定性预测从三分量 RD 系统的全数值 PDE 模拟中得到证实。