A Gaussian particle distribution for branching Brownian motion with an inhomogeneous branching rate

Matthew I. Roberts, Jason Schweinsberg
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引用次数: 4

Abstract

Motivated by the goal of understanding the evolution of populations undergoing selection, we consider branching Brownian motion in which particles independently move according to one-dimensional Brownian motion with drift, each particle may either split into two or die, and the difference between the birth and death rates is a linear function of the position of the particle. We show that, under certain assumptions, after a sufficiently long time, the empirical distribution of the positions of the particles is approximately Gaussian. This provides mathematically rigorous justification for results in the biology literature indicating that the distribution of the fitness levels of individuals in a population over time evolves like a Gaussian traveling wave.
具有非均匀分支速率的分支布朗运动的高斯粒子分布
为了理解种群的进化过程,我们考虑了分支布朗运动,在分支布朗运动中,粒子根据一维布朗运动漂移独立运动,每个粒子可能分裂成两个或死亡,并且出生率和死亡率之间的差异是粒子位置的线性函数。我们证明,在一定的假设下,在足够长的时间后,粒子位置的经验分布近似于高斯分布。这在数学上为生物学文献中的结果提供了严格的证明,这些结果表明,随着时间的推移,种群中个体的适应水平分布就像高斯行波一样演变。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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