AMC-SS Spatial models for populations with variable offspring laws

这个项目涉及空间随机粒子模型的分析，对于其子代分布是状态依赖的或相对奇异的。对于前者，研究的重点是使(特定类型的)小种群具有优势的后代分布。将研究各种(非线性)自我调节机制以及粒子种群的空间分布，包括具有选择的投票者模型的版本、具有固定局部种群大小的多类型模型以及分支粒子模型及其大的种群扩散极限。核心问题之一将是确定长期生存的参数制度，而不会出现人口规模的爆炸性增长，以及--如果最初存在多种类型的粒子--各种或所有类型的长期共存。该项目的另一部分涉及描述具有单一子代分布的空间分布的粒子种群，该模型模拟具有潜在大家庭的种群。最近，随着不同类别的随机过程之间的一些深层联系被发现和利用--尽管是在非空间环境中--这些问题引起了人们的极大关注。这项工作的重点将集中在粒子质量的正向演化以及由测值过程、随机微分方程和空间合并过程描述的谱系及其扩散极限的向后演化。还将考虑与这两个研究方向有关的基因突变、选择和重组的多类型模型。了解大粒子系统的行为对于从物理、化学、生物和计算机科学等领域的微观规律推断宏观性质具有重要意义。在本研究中，我们构建并分析了模拟自我调节现象的空间随机过程。这种现象在粒子群的许多物理系统中都能观察到。其目的是确定不同类型的生存和共存等各种宏观行为发生的潜在机制和条件。这些问题是流行病学、生态学和遗传学中生物种群特别感兴趣的问题。空间随机模型的繁殖种群，其中单个个体的后代有时可能取代一个相当大的比例或种群也被考虑。例如，这种行为被认为发生在某些基因具有选择性优势的种群中。因此，这些模型在遗传学上具有特别重要的意义。对基因进化的现实模型的分析和系谱的描述对于正确地定量分析和解释现在无处不在的基因序列数据是重要的。应用范围从重建人类的起源和历史，到定位和识别基因组上导致疾病的基因。

This project is concerned with the analysis of spatial stochastic particle models for which the offspring distribution is state dependent or relatively singular. For the former, the study is focused on offspring distributions that give small populations (of a particular type) an advantage. Various (non-linear) self-regulation mechanisms as well as spatial distributions of the particle population will be investigated, including versions of voter models with selection, multi-type models with fixed local population size and branching particle models alongside their large population diffusion limits. One of the central questions will be to determine parameter regimes for long-term survival without explosion of the population size and -if multiple types of particles are present initially- of long-term coexistence of various or all types. Another part of this project is concerned with the description of spatially distributed particle populations with singular offspring distributions that model populations with potentially large individual families. These have attracted much recent attention as several deep connections between different classes of stochastic processes have been discovered and exploited - albeit in a non-spatial setting. The focus of this work will be on the forward evolution of particle mass as well as the backward evolution of genealogies and their diffusion limits as described by measure-valued processes, stochastic differential equations and spatial coalescent processes. Multi-type models for genes with mutation, selection and recombination that have ties to both lines of research will also be considered. Understanding the behavior of large particle systems is of importance for deducing macroscopic properties from microscopic rules in a wide variety of fields such as Physics, Chemistry, Biology and Computer Science. In this research, spatial stochastic processes that model self-regulation phenomena are constructed and analyzed. Such phenomena are observed in many physical systems of particle populations. The aim is to identify the underlying mechanisms and conditions for various macroscopic behaviors such as survival and coexistence of different types to occur. These questions are in particular of interest for biological populations in epidemiology, ecology and genetics. Spatial stochastic models for reproducing populations in which the offspring of single individuals may on occasion replace a significant proportion or the population are also considered. This kind of behavior is thought to occur, for example, in populations in which certain genes carry a selective advantage. These models are therefore of particular relevance in genetics. The analysis of realistic models for the evolution of genes and the description of genealogies is of importance for the correct quantitative analysis and interpretation of the now ubiquitous gene sequence data. Applications range from reconstructing the origin and history of humans to locating and identifying genes on the genome that are causative factors for diseases.