Mathematical Sciences: Fleming - Viot Processes with Selection

数学科学：弗莱明 - Viot 过程与选择

• 批准号: 9311984
• 负责人: Stewart Ethier
• 金额: $ 9万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1993
• 资助国家: 美国
• 起止时间: 1993-07-01 至 1996-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9311984&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Fleming; Viot; Processes

Fleming--Viot (FV) processes are probability-measure-valued diffusion processes that model the evolution of the distribution of types in a population. This proposal concerns three topics related to the infinitely-many-alleles model with selection. The first topic involves the simplest FV process with selection, and the aim is to obtain an explicit formula for the transition function (albeit less explicit than in the neutral case). The second topic concerns a model introduced by Tachida (referred to by him as a nearly neutral mutation model), which can be formulated as an FV process with an unbounded selection intensity function. The aim is establish uniqueness and study the ergodic properties of the model, with the hope of explaining some of Tachida's simulation findings. The third topic involves the concept of allelic genealogy introduced by Takahata and Nei. One can formulate an FV process that keeps track of such genealogies in the infinitely-many-alleles setting. There are several open problems in the neutral case, but the challenge is to analyze the selective models in a more rigorous way. This proposal is concerned with certain stochastic (i.e., random) processes in population genetics. Each individual in a population is assumed to be of some genetic "type." The processes model the evolution of the frequencies of the various types in the population, taking into account such genetic factors as random mating, mutation, and natural selection. Letting the set of possible types be arbitrary (in particular, infinite) allows one to model the ``infinitely many alleles'' assumption. It also allows one to incorporate historical or genealogical information into the description of an individual's type. Three problems, concerning three specific models (each from the biological literature), are proposed. The problems are concerned with providing a better mathematical understanding of the models and possibly explaining certain phenomena discovered empirically by biologists. Each model involves selection, which complicates it considerably, but makes it of greater interest than simply the neutral case would be. The difficulty with selection is that one cannot obtain closed systems of linear equations for the quantities of interest. Nevertheless, several ideas are proposed that should help to overcome this.

Fleming-Viot（FV）过程是一种概率测度值扩散过程，它模拟了种群中类型分布的演化。 这个建议涉及三个主题的无限多等位基因模型与选择。 第一个主题涉及最简单的具有选择的FV过程，目的是获得过渡函数的显式公式（尽管不如中性情况下显式）。 第二个主题是关于由Tachida引入的模型（他称之为近中性突变模型），该模型可以被表述为具有无界选择强度函数的FV过程。 目的是建立唯一性和研究遍历性的模型，希望解释一些立田的模拟结果。 第三个主题涉及到由Takahata和Nei引入的等位基因谱系的概念。 人们可以制定一个FV过程，在无限多等位基因的情况下跟踪这样的系谱。 在中立的情况下有几个开放的问题，但挑战是以更严格的方式分析选择性模型。 该建议涉及某些随机（即，群体遗传学中的随机过程。 群体中的每个个体都被假定为某种遗传"类型"。“这些过程模拟了种群中各种类型频率的演变，同时考虑了随机交配、突变和自然选择等遗传因素。 让可能类型的集合是任意的（特别是，无限的）允许人们模拟“无限多个等位基因”的假设。 它还允许人们将历史或家谱信息纳入对个人类型的描述中。 提出了三个问题，涉及三个具体的模型（每个从生物学文献）。 这些问题涉及提供一个更好的数学理解的模型，并可能解释某些现象发现经验 生物学家。 每个模型都涉及选择，这使它变得相当复杂，但比简单的中性情况更令人感兴趣。 选择的困难在于，人们不能得到关于感兴趣的量的封闭的线性方程组。 然而，提出了一些想法，应有助于克服这一点。