Constable George W A, McKane Alan J
Theoretical Physics Division, School of Physics and Astronomy, The University of Manchester, Manchester M13 9PL, United Kingdom.
Phys Rev E Stat Nonlin Soft Matter Phys. 2014 Mar;89(3):032141. doi: 10.1103/PhysRevE.89.032141. Epub 2014 Mar 31.
We investigate the stochastic dynamics of entities which are confined to a set of islands, between which they migrate. They are assumed to be one of two types, and in addition to migration, they also reproduce and die. Birth and death events are later moderated by weak selection. Systems which fall into this class are common in biology and social science, occurring in ecology, population genetics, epidemiology, biochemistry, linguistics, opinion dynamics, and other areas. In all these cases the governing equations are intractable, consisting as they do of multidimensional Fokker-Planck equations or, equivalently, coupled nonlinear stochastic differential equations with multiplicative noise. We develop a methodology which exploits a separation in time scales between fast and slow variables to reduce these equations so that they resemble those for a single island, which are amenable to analysis. The technique is generally applicable, but we choose to discuss it in the context of population genetics, in part because of the extra features that appear due to selection. The idea behind the method is simple, its application is systematic, and the results are in very good agreement with simulations of the full model for a range of parameter values.
我们研究局限于一组岛屿且在岛屿间迁移的实体的随机动力学。假设它们为两种类型之一，除了迁移之外，它们还会繁殖和死亡。出生和死亡事件随后会受到弱选择的调节。属于此类的系统在生物学和社会科学中很常见，出现在生态学、群体遗传学、流行病学、生物化学、语言学、舆论动态及其他领域。在所有这些情况下，支配方程都难以处理，因为它们由多维福克 - 普朗克方程组成，或者等效地，由带有乘性噪声的耦合非线性随机微分方程组成。我们开发了一种方法，该方法利用快速变量和慢速变量之间的时间尺度分离来简化这些方程，使其类似于单个岛屿的方程，便于进行分析。该技术具有普遍适用性，但我们选择在群体遗传学的背景下进行讨论，部分原因是由于选择而出现的额外特征。该方法背后的思想很简单，其应用是系统的，并且对于一系列参数值，结果与完整模型的模拟结果非常吻合。
