描述中性遗传多样性的随机偏微分方程
Stochastic partial differential equations describing neutral genetic diversity under short range and long range dispersal
摘要：
本文考虑了一个在包含突变、遗传漂变和短程扩散或长程扩散的空间连续体中中性遗传多样性演化的数学模型。我们考虑的模型是Barton、Etheridge和Véber提出的空间$ Λ$ -Viot过程，它通过对$ \R^d \times [0,1] $的测度来描述种群在任何时刻的状态，其中$ \R^d $是地理空间，$ [0,1] $是遗传类型空间。在这两种情况下（短程和长程扩散），我们证明了一个功能的中心极限定理的人口密度变大的过程中，在一些时空重新标度。然后，我们从这两个中心极限定理推导出一个公式的渐近概率的身份随机挑选两个人从两个给定的空间位置。在短距离扩散的情况下，我们恢复了经典的Wright-Malécot公式，该公式被广泛用于空间结构人口的人口推断。在长程扩散的情况下，我们得到了一个新的公式，它可以打开的方式，更好地评价长程扩散的推断方法。
英文摘要：
In this paper, we consider a mathematical model for the evolution of neutral genetic diversity in a spatial continuum including mutations, genetic drift and either short range or long range dispersal. The model we consider is the spatial $ Λ$-Fleming-Viot process introduced by Barton, Etheridge and Véber, which describes the state of the population at any time by a measure on $ \R^d \times [0,1] $, where $ \R^d $ is the geographical space and $ [0,1] $ is the space of genetic types. In both cases (short range and long range dispersal), we prove a functional central limit theorem for the process as the population density becomes large and under some space-time rescaling. We then deduce from these two central limit theorems a formula for the asymptotic probability of identity of two individuals picked at random from two given spatial locations. In the case of short range dispersal, we recover the classical Wright-Malécot formula, which is widely used in demographic inference for spatially structured populations. In the case of long range dispersal we obtain a new formula which could open the way for a better appraisal of long range dispersal in inference methods.
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