论文标题
通过控制因子WKB方法来驯服扩散近似
Taming the diffusion approximation through a controlling-factor WKB method
论文作者
论文摘要
从种群遗传学到生态学和进化，扩散近似（DA）广泛用于随机种群动力学的分析。 DA是一个不受控制的近似值，它假定相关状态空间上计算出的数量的平滑度，并且在不满足此属性时会失败。在选择方向开关符号的情况下，这种故障变得严重。在这里，我们采用了WKB（大问题）方法，该方法仅需要给定数量的对数才能在其状态空间上平滑。将WKB方案与渐近匹配技术相结合，我们展示了如何以受控方式得出扩散近似以及如何产生更好的近似值，适用于更广泛的参数。我们还引入了基于WKB的数值技术的可扩展性（独立于人口大小）。该方法应用于种群遗传学和进化中的核心问题，在零和两种竞争中发现了最终固定的机会。
The diffusion approximation (DA) is widely used in the analysis of stochastic population dynamics, from population genetics to ecology and evolution. DA is an uncontrolled approximation that assumes the smoothness of the calculated quantity over the relevant state space and fails when this property is not satisfied. This failure becomes severe in situations where the direction of selection switches sign. Here we employ the WKB (large-deviations) method, which requires only the logarithm of a given quantity to be smooth over its state space. Combining the WKB scheme with asymptotic matching techniques, we show how to derive the diffusion approximation in a controlled manner and how to produce better approximations, applicable for much wider regimes of parameters. We also introduce a scalable (independent of population size) WKB-based numerical technique. The method is applied to a central problem in population genetics and evolution, finding the chance of ultimate fixation in a zero-sum, two-types competition.
