Problems in Probability Arising from Ecology and Genetics

Durrett Durrett建议研究概率论中的各种问题，这些问题源于生态学和遗传学中的问题。在随机空间模型领域，他将研究多类型选民模型的渐近行为，在这些模型中，选民有几种观点，更有可能模仿有相似观点的邻居。他将与泰德·考克斯合作，将他们关于垫脚石模型的结论推广到大小为k 2的样本，并将研究当两个等位基因中的每一个在一半空间中受到青睐时，杂交区的平衡性质。他将与埃德·珀金斯一起研究粒子系统的测量价值极限，以产生不同类型粒子之间相互作用的极限过程。然而，杜雷特研究的并不是所有的系统都是空间的：他将与Jason Schweinsberg合作开发涉及多碰撞结合的遗传搭便车的近似，将研究具有高突变率的病毒属性的准物种模型中的相变，并将继续与Vlada Limic就考夫曼的NK模型进行研究，这是具有大量局部极大值的适应度图景的典型例子。杜雷特的研究受到各种应用的推动。杜雷特将与泰德·考克斯合作，扩展他们在空间遗传学模型上的工作，以调查人类基因组中令人惊讶的大量连锁不平衡(遗传相关性)的解释。了解DNA序列中模式的可能来源对于使用关联图谱来定位基因和识别正选择的足迹是重要的。关于后一个问题，Durrett将与Jason Schweinsberg合作，开发一种与固定有利突变发生的遗传搭便车的近似值，这将允许推导出目前只能通过模拟才能理解的量的分析结果。杜雷特研究的一个主要主题是竞争者的空间分布对系统均衡性质的影响。他将与西蒙·莱文一起研究一组社会规范演变的随机空间模型。他将与泰德·考克斯一起研究杂交区，在那里，密切相关但基因分化的种群共存于近距离。他将与埃德·珀金斯一起研究随机空间模型的比例极限，以构建相互作用的度量值扩散的新的有趣例子，并获得大范围粒子系统的近似。

0202935Durrett Durrett proposes to study a variety of problems in probability theory that arise from questions in ecology and genetics. In the area of stochastic spatial models, he will study the asymptotic behavior of multitype voter models where voters have several opinions and are more likely to imitate neighbors with similar opinions. With Ted Cox he will work to extend their conclusions for the stepping stone model to samples of size k 2, and will study equilibrium properties of a hybrid zone which results when each of two alleles is favored in one half of the space. With Ed Perkins he will study measure valued limits of particle systems to produce limit processes with interactions between different types of particles. Not all of the systems Durrett studies will be spatial however: he will work with Jason Schweinsberg to develop approximations of genetic hitchhiking that involve the coalescent with multiple collisions, will study the phase transition in a quasi-species model which models properties of viruses with high mutation rates, and will continue his work with Vlada Limic on Kauffmann's NK model, which is a prototypical example of a fitness landscape with a large number of local maxima. Durrett's research is motivated by a variety of applications. Durrett will work with Ted Cox to extend their work on spatial genetics models to investigate explanations for the surprisingly large amount of linkage disequilibrium (genetic correlation) in the human genome. Understanding the possible sources of patterns in DNA sequences is important for using association mapping to locate genes and to identify footprints of positive selection. In relation to the latter question, Durrett will work with Jason Schweinsberg to develop an approximation to the genetic hitchhiking that occurs with the fixation of favorable mutation, which will allow for the derivation of analytical results for quantities that can currently be understood only by simulation. A major theme of Durrett's research is the influence of a spatial distribution of competitors on equilibrium properties of a system. With Simon Levin, he will study a collection of stochastic spatial models for the evolution of social norms. With Ted Cox, he will investigate hybrid zones where closely related but genetically differentiated populations coexist in close proximity. With Ed Perkins he will investigate scaling limits of stochastic spatial models, in order to construct new interesting examples of interacting measure-valued diffusions and to obtain approximations for particle systems with large range.