具有异质性近似重复受害效应的城市犯罪模型的平稳模式及其选择机制
Stationary patterns and their selection mechanism of Urban crime models with heterogeneous near-repeat victimization effect
摘要：
在本文中，我们研究了两个偏微分方程，推广了城市犯罪模型的短\emph{et al}。[数学模型方法应用科学，18（2008），pp. 1249-1267]。我们的修改是在假设的空间异质性的近重复的受害效果和分散战略的犯罪代理人。本文研究了多维有界区域上具有非线性扩散的反应-对流-扩散系统在齐次Neumann边界条件下的斑图形成。结果表明，正齐次稳态失去其稳定性的内在近重复受害率$ε$减少和空间非常数的解决方案出现通过分歧。此外，我们发现波模选择机制，通过严格的稳定性分析，这些非平凡的模式，这表明，唯一的稳定模式必须有波数，最大化的分歧值。基于这种波模选择机制，我们将能够精确地预测形成稳定的聚集的房屋吸引力和犯罪人口密度，至少当扩散率$ε$是周围的主要分歧值。我们的理论结果还表明，大域支持更稳定的聚集体比小域。最后，我们进行了广泛的数值模拟在一维区间和二维正方形来说明和验证我们的理论研究结果。我们的数值还包括一些有趣的现象，如合并的两个内部尖峰和新的尖峰的出现等，这些非平凡的解决方案可以模拟城市犯罪活动中观察到的聚集现象。
英文摘要：
In this paper, we study two PDEs that generalize the urban crime model proposed by Short \emph{et al}. [Math. Models Methods Appl. Sci., 18 (2008), pp. 1249-1267]. Our modifications are made under assumption of the spatial heterogeneity of both the near-repeat victimization effect and the dispersal strategy of criminal agents. We investigate pattern formations in the reaction-advection-diffusion systems with nonlinear diffusion over multi-dimensional bounded domains subject to homogeneous Neumann boundary conditions. It is shown that the positive homogeneous steady state loses its stability as the intrinsic near-repeat victimization rate $ε$ decreases and spatially nonconstant solutions emerge through bifurcation. Moreover, we find the wavemode selection mechanism through rigorous stability analysis of these nontrivial patterns, which shows that the only stable pattern must have wavenumber that maximizes the bifurcation value. Based on this wavemode selection mechanism, we will be able to precisely predict the formation of stable aggregates of the house attractiveness and criminal population density, at least when the diffusion rate $ε$ is around the principal bifurcation value. Our theoretical results also suggest that large domains support more stable aggregates than small domains. Finally, we perform extensive numerical simulations over 1D intervals and 2D squares to illustrate and verify our theoretical findings. Our numerics also include some interesting phenomena such as the merging of two interior spikes and the emerging of new spikes, etc. These nontrivial solutions can model the well observed aggregation phenomenon in urban criminal activities.
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