THE STABILITY OF STEADY-STATE HOT-SPOT PATTERNS FOR A REACTION-DIFFUSION MODEL OF URBAN CRIME
THE STABILITY OF STEADY-STATE HOT-SPOT PATTERNS FOR A REACTION-DIFFUSION MODEL OF URBAN CRIME
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DOI：
10.3934/dcdsb.2014.19.1373
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发表时间：
2014-07-01
影响因子：
1.2
通讯作者：
Wei, Juncheng
中科院分区：
文献类型：
--
作者：
Kolokolnikov, Theodore;Ward, Michael J.;Wei, Juncheng
针对Short等人提出的城市犯罪的反应扩散模型，研究了局部化犯罪模式的存在性和稳定性。[Math. Models. Meth.应用科学，18，Suppl.（2008），pp. 1249-12671.此类模式的特征是犯罪活动集中在局部空间区域，被称为热点模式，它们发生在远离与空间均匀解的分叉相关的图灵点的参数范围内。奇异摄动技术被用来构建在一个和两个维的空间域中的稳态热点图案，和新类型的非局部本征值问题推导出确定这些热点图案的稳定性0（1）时间尺度的不稳定性。通过对这些非局部特征值问题的分析，确定了一个临界阈值Ice，使得当K > K-c时，由K个热点组成的图案不稳定为竞争不稳定。由于正的真实的特征值，这种不稳定性触发了图案中某些热点的崩溃。此外，在著名的Gierer-Meinhardt反应扩散模型的尖峰模式的稳定性结果相比，它示出的犯罪模型，只有一个相对较窄的参数范围内的热点振幅振荡不稳定性发生。这样的不稳定性，由于一个霍普夫分岔，明确研究了一个单一的热点在阴影系统的限制，罪犯的扩散率是渐近大。最后，局部化热点发生的参数制度进行比较，在以前的作品中，图灵不稳定性从一个空间均匀的稳态发生研究的参数制度。
The existence and stability of localized patterns of criminal activity are studied for the reaction-diffusion model of urban crime that was introduced by Short et. al. [Math. Models. Meth. Appl. Sci., 18, Suppl. (2008), pp. 1249-12671. Such patterns, characterized by the concentration of criminal activity in localized spatial regions, are referred to as hot-spot patterns and they occur in a parameter regime far from the Turing point associated with the bifurcation of spatially uniform solutions. Singular perturbation techniques are used to construct steady-state hot-spot patterns in one and two-dimensional spatial domains, and new types of nonlocal eigenvalue problems are derived that determine the stability of these hot-spot patterns to 0(1) time-scale instabilities. From an analysis of these nonlocal eigenvalue problems, a critical threshold Ice is determined such that a pattern consisting of K hot-spots is unstable to a competition instability if K > K-c This instability, due to a positive real eigenvalue, triggers the collapse of some of the hot-spots in the pattern. Furthermore, in contrast to the well-known stability results for spike patterns of the Gierer-Meinhardt reaction-diffusion model, it is shown for the crime model that there is only a relatively narrow parameter range where oscillatory instabilities in the hot-spot amplitudes occur. Such an instability, due to a Hopf bifurcation, is studied explicitly for a single hot-spot in the shadow system limit, for which the diffusivity of criminals is asymptotically large. Finally, the parameter regime where localized hot-spots occur is compared with the parameter regime, studied in previous works, where Turing instabilities from a spatially uniform steady-state occur.