AMC-SS Spatial models for populations with variable offspring laws

本计画主要研究子体分布为状态相依或相对奇异之空间随机粒子模型之分析。对于前者，该研究的重点是后代分布，使小种群（特定类型）的优势。各种（非线性）的自我调节机制以及空间分布的粒子人口将进行调查，包括版本的选民模型的选择，多类型的模型与固定的本地人口规模和分支粒子模型旁边的大人口扩散限制。其中一个核心问题将是确定参数制度的长期生存没有爆炸的人口规模和-如果多种类型的粒子是目前最初-长期共存的各种或所有类型。该项目的另一部分是关于描述空间分布的粒子种群与奇异的后代分布模型的人口与潜在的大家庭。这些都吸引了最近的关注，因为不同类别的随机过程之间的几个深连接已被发现和利用-虽然在非空间设置。这项工作的重点将是粒子质量的前向演化，以及由测度值过程，随机微分方程和空间聚结过程描述的谱系及其扩散极限的后向演化。也将考虑与这两条研究线有联系的突变、选择和重组基因的多类型模型。了解大粒子系统的行为对于从物理、化学、生物学和计算机科学等广泛领域的微观规则推导宏观性质具有重要意义。 在这项研究中，空间随机过程模型的自我调节现象的构建和分析。这种现象在许多粒子群的物理系统中都可以观察到。其目的是确定各种宏观行为的潜在机制和条件，如不同类型的生存和共存。这些问题是特别感兴趣的流行病学，生态学和遗传学的生物种群。空间随机模型的繁殖种群中，单个个体的后代可能会取代一个显着的比例或人口也被认为是。例如，这种行为被认为发生在某些基因具有选择优势的群体中。因此，这些模型在遗传学中特别相关。 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普遍存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的致病基因。

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.