Stochastic Models in Population Genetics and Ecology

这些研究项目属于数学种群遗传学和相互作用粒子系统的领域。统一的推力是在种群水平上研究生物系统的数学模型。各种遗传影响之间的相互作用，如突变、选择和种群结构，有时会扰乱一个人得出有效推理的能力。第一个目标是开发从各种结构的种群中产生的复杂分子遗传数据的估计和推断方法。研究人员探索了存在这些混杂效应时的抽样分布和似然方法。第二个目标涉及研究受各种选择力量、地理隔离或其他等级结构影响的种群的谱系过程(即联合理论)。抽样理论和合并方法都为马尔可夫链、蒙特卡罗和重要性抽样等计算方法提供了一个框架。第三个目标是使用相互作用的粒子系统研究生态学和流行病学中出现的空间相互作用。所解决的问题包括几个新的具有簇破裂特征的非线性选民模型，以及关于两阶段接触过程的新工作，以及在集合种群模型中的应用。以前的自然种群生态学和遗传学模型都是建立在许多简化假设的基础上的。现在，重要的是提出更现实的模式，以解决影响社会的生态和进化问题。除了真实生物系统固有的随机性外，空间结构往往是至关重要的。这些问题的例子包括传染病的进化和传播、外来物种的入侵以及物种在支离破碎的栖息地中的生存。此外，基因测序的技术革命正在产生巨大的数据集，这些数据集只能在复杂的数学框架中破译。生物信息学的新领域在很大程度上依赖于进化论的观点。这项跨学科工作是开发和采用新的数学方法的持续努力的一部分，目的是在人口一级解决这些复杂的生物学问题。

These research projects fall within the areas of mathematical population genetics and interacting particle systems. The unifying thrust is the study of mathematical models of biological systems at the population level. The interplay between various genetic influences such as mutation, selection, and population structure can sometimes confound one's ability to draw valid inference. The first goal is to develop methods of estimation and inference for complex molecular genetic data arising from populations with various kinds of structure. The investigators explore sampling distributions and likelihood methods in the presence of these confounding effects. The second goal involves the study of genealogical processes (i.e., coalescent theory) for populations which are subject to various selective forces, geographical segregation, or other hierarchical structuring. Both sampling theory and coalescent methods provide a framework for computational methods such as Markov Chain Monte Carlo and Importance Sampling. The third goal is to study spatial interactions which arise in ecology and epidemiology using interacting particle systems. The problems addressed include several new nonlinear voter models with a cluster-breaking feature and new work on the two-stage contact process, with applications to metapopulation models. Previous models of the ecology and genetics of natural populations have been based on many simplifying assumptions. It is now important to produce more realistic models to address ecological and evolutionary problems that impact society. In addition to the inherent randomness in real biological systems, spatial structure is often of critical importance. Examples of these problems include the evolution and spread of infectious diseases, invasions of exotic species, and survival of species in fragmented habitats. Furthermore, the technical revolution in gene sequencing is producing huge data sets that can only be deciphered in a sophisticated mathematical framework. The new field of bioinformatics relies heavily on an evolutionary perspective. This interdisciplinary work is part of an ongoing effort to develop and employ new mathematical methods for the purpose of addressing these complex biological problems at the population level.