Evolution in spatially structured populations

对研究背景的简要描述：我们关注的是捕捉种群大小在空间和时间上波动的遗传变异的数学模型；换句话说，将生态学和遗传学结合在一起的模型。第一步是写下合理的生态模型。一类非常自然的模型概括了分枝马尔可夫过程的概念，它根据当前的种群规模来调节个体产生的后代数量。对于大的种群，人们越过了比例限制，并用带有监管的所谓的道森-渡边超过程取代了基于个体的模型。人们对这些模型知之甚少，但如果监管如此之强，以至于种群规模是恒定的，那么就可以恢复弗莱明-维奥超过程，这在分析上是容易处理的。我们的第一个目标是考虑弗莱明-维奥特极限的波动。它们将以无限维奥恩斯坦-乌伦贝克过程的形式出现。我们的中心问题将是调查它们如何影响遗传变异的模式。研究方法的创新：通常，这项技术一直是对群体遗传学中的模型进行数字验证。这里的新技术是为了通过严格的数学来确定它们的有效性。目的和目的：通过种群的总规模来调节种群并不完全是生物学上的自然现象。我们的长期目标是了解如果人口只受当地拥挤的影响会发生什么。这一规定更切合实际，因为在当地人口高度拥挤的地区，对有限资源的竞争将阻碍当地的繁殖率。第一步将是消化现有的关于测值扩散的文献；最初这将涉及研究Dawson-Watanabe和Fleming-Viot超过程。由于调节种群数量的模型知之甚少，我们将通过进行一些初步的数值实验来开始对这一领域的探索。这将指导我们对这些过程的行为的直觉。一定数量的民间传说围绕着一个关键问题，即了解活在人口中的个体的祖先谱系在太空中传播的速度有多快。由于存储祖先树的成本，这甚至不能通过幼稚的方法在数字上实现，但我们将从Hallatschek和Nelson在该模型下所称的“示踪动力学”的一些数值实验开始。这对于一篇好的论文来说已经足够了，但如果时间允许，我们将转向数学问题：限制人口模型是什么，围绕这个限制的波动是什么样子？潜在影响：这项研究的目的将是为数学种群遗传学创建模型，数学生物学家可能会根据这些模型来匹配数据。通过开发和理解种群大小受控制的模型，我们希望了解这如何影响遗传变异模式。该项目将属于EPSRC统计和应用概率研究领域。

Brief description of the context of the research:We are concerned with mathematical models that capture the genetic variation that we should see in populations whose size fluctuates in space and time; in other words models that combine ecology with genetics. The first step is to write down sensible ecological models. A very natural class of models generalises the idea of a branching Markov process by regulating the number of offspring that an individual produces according to the current population size. For large populations, one passes to a scaling limit and replaces the individual based model by a so-called Dawson-Watanabe superprocess with regulation. These models are rather poorly understood, but if the regulation is so strong that the population size is constant, then one recovers the Fleming-Viot superprocess, which is analytically tractable. Our first goal is to consider fluctuations about the Fleming-Viot limit. These will take the form of an infinite-dimensional Ornstein-Uhlenbeck process. Our central question will be to investigate how they affect patterns of genetic variation.Novelty of the research methodology:Commonly the technique has been to numerically justify models in population genetics. The novel technique here is in our aim to establish their validity through rigorous mathematics. Aims and objectives:Regulating the population by its total size is not completely biologically natural. Our longer term goal is to understand what happens if the population is regulated only by local crowding. This stipulation is more realistic since at areas of high local crowding, competition for finite resources will hinder the local reproduction rate. The first step will be to digest the existing literature on measure-valued diffusions; initially this will involve studying the Dawson-Watanabe and Fleming-Viot superprocesses. Since the models which regulate the population size are poorly understood, we will begin our exploration into this field by performing some preliminary numerical experiments. This will guide our intuition as to the behaviour of such processes. A certain amount of folklore surrounds the key question of understanding how fast ancestral lineages of individuals alive in the population spread through space. Because of the cost of storing trees of ancestors, this is not even numerically accessible through a naive approach, but we shall begin with some numerical experiments for what Hallatschek and Nelson call the `tracer dynamics' under the model. This would be enough for a good thesis, but if time permits we'll turn to the mathematical questions: what is the limiting population model and what do the fluctuations about that limit look like? Potential impact:The purpose of this research will be to create models for mathematical population genetics, which mathematical biologists may fit data to. By developing and understanding models for populations whose size is regulated, we hope to understand how this affects patterns of genetic variation. The project will fall within the EPSRC statistics and applied probability research area.