Evolution in spatially structured populations

研究背景简介：我们关注的是数学模型，这些模型捕捉了我们应该在空间和时间上波动的种群中看到的遗传变异;换句话说，模型将联合收割机生态学与遗传学结合起来。第一步是写下合理的生态模型。一类非常自然的模型通过调节个体根据当前种群规模产生的后代数量来概括分支马尔可夫过程的思想。对于大的群体，一个传递到一个标度极限，并取代了基于个人的模型，所谓的道森-渡边超过程与监管。人们对这些模型的理解相当少，但如果调节是如此之强，以至于种群规模是恒定的，那么人们就可以恢复Fleming-Viot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Fleming-Viot极限的波动。这些将采取无限维Ornstein-Uhlenbeck过程的形式。我们的中心问题将是调查它们如何影响遗传变异的模式。研究方法的新奇：通常的技术是在数量上证明群体遗传学模型的合理性。这里的新技术是我们的目标，通过严格的数学来建立它们的有效性。目的和目标：根据人口总数来调节人口并不完全符合生物学的自然规律。我们的长期目标是了解如果种群仅由局部拥挤来调节会发生什么。这一规定比较现实，因为在当地人口密集的地区，对有限资源的竞争会妨碍当地的繁殖率。第一步将是消化现有的文献测度值扩散;最初这将涉及研究道森-渡边和弗莱明-维奥特超过程。由于调节种群规模的模型知之甚少，我们将开始我们的探索到这个领域进行一些初步的数值实验。这将引导我们对这些过程的行为的直觉。一些民间传说围绕着一个关键问题，即理解种群中存活的个体的祖先谱系在空间中传播的速度有多快。由于存储祖先树的成本，这甚至不能通过简单的方法进行数值计算，但是我们将开始对Hallatschek和纳尔逊称之为模型下的“示踪动力学”进行一些数值实验。这对于一篇好论文来说已经足够了，但是如果时间允许的话，我们将转向数学问题：什么是极限种群模型，关于这个极限的波动是什么样的？潜在影响：这项研究的目的将是创建数学群体遗传学模型，数学生物学家可以将数据拟合到其中。通过开发和理解规模受到调控的种群模型，我们希望了解这如何影响遗传变异的模式。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.