Branching Brownian motion and population models

近年来，人们对使用概率论工具来解决生物学问题的兴趣越来越大。 概率论对群体遗传学领域做出了重大贡献，因为许多导致群体进化的因素最好被建模为随机事件。 本项目的目的是通过对随机演化种群的一些自然模型进行详细的数学研究，在这一领域做出进一步的贡献。 粒子根据带有漂移的布朗运动，以恒定的速率分成两部分，并在撞击原点时被吸收。 这个过程可以被看作是对经历选择的群体进行建模。 粒子位置代表个体的适应性，分支事件代表出生，原点的吸收模拟不适应个体的死亡。 我们的目标将是提供精确的估计，该过程的生存很长一段时间的概率和最右边的粒子的位置。 该项目的第二部分涉及研究一个固定规模的群体模型，其中个体获得有益的突变。 目标是证明有关进化速度、种群中个体适应度的分布以及种群谱系的结果。 该项目的第三部分涉及对β聚结的进一步研究，这一过程可用于描述大家庭人口的谱系。

In recent years, there has been growing interest in using tools from probability theory to solve biological problems. Probability theory has contributed substantially to the field of population genetics because many of the factors that are responsible for the evolution of populations are best modeled as random events. The aim of this project is to make further contributions in this area by undertaking a detailed mathematical study of some natural models of randomly evolving populations.The first part of the project focuses on branching Brownian motion with absorption. Particles move according to Brownian motion with drift, divide into two at a constant rate, and are absorbed upon hitting the origin. This process can be viewed as modeling a population undergoing selection. The particle positions represent fitnesses of individuals, the branching events represent births, and absorption at the origin models the death of unfit individuals. The goal will be to provide sharp estimates for the probability that the process survives for a long time and for the position of the right-most particle. The second part of the project involves studying a model of a population of fixed size in which individuals acquire beneficial mutations. The goal will be to prove results concerning the speed of evolution, the distribution of the fitnesses of individuals in the population, and the genealogy of the population. The third part of the project involves further study of the beta coalescent, a process which can be used to describe the genealogy of populations with large family sizes.