Stochastic Spatial Processes

本提案将解决相互作用粒子系统和随机空间过程理论中的问题。这些随机过程是具有许多相互作用的“组件”（细胞、个体、粒子、植物等）的大系统的模型。这些系统所模拟的现象的一些例子有：物种竞争、流行病、人口增长、遗传性状的进化。该领域研究的一个主要目标是了解大型系统的宏观行为如何依赖于组件之间的个体相互作用。本文将研究两种竞争模型，即空间随机Lotka-Volterra模型和多类型接触过程。对于Lotka-Volterra模型，目标是确定对应于一个物种生存和两个物种共存的参数区域。该项目是研究者正在努力理解和利用相互作用的粒子系统和测量值扩散之间的缩放关系的一部分。对于多类型接触过程，关注的问题是普通树木的生存与灭绝。研究者期望找到新的相依赖的生存和消失现象，不存在于晶格情况下。该建议的第二个一般领域解决了由群体遗传学引起的问题。在地理结构的种群中，遗传特征的进化通常是用木村的垫脚石模型来建模的，在木村的垫脚石模型中，关于亲属关系度量的问题被重新表述为关于合并随机漫步系统行为的问题。研究者之前的研究工作包括在具有环面或环绕边界条件的大型二维晶格集上这些系统的极限定理。这里的目标是展示这些定理在广泛的边界条件下的鲁棒性，证明它们在群体遗传学中的应用。这个建议涉及到相互作用粒子系统和随机空间过程的理论研究。这些随机过程是具有许多相互作用成分（细胞、个体、粒子、植物等）的大系统的模型。本研究的目的是对相互作用粒子系统能很好地模拟的各种复杂现象，如物种竞争、流行病、种群增长、遗传性状进化的模型，有更好的定性理解。该领域研究的一个主要目标是了解大型系统的宏观行为如何依赖于组件之间的个体相互作用。几个特定的模型将被研究，包括一个众所周知的物种间竞争模型的随机空间版本。除了研究特定模型外，研究者还将尝试扩展为某些特定模型建立的一些近似定理的有效性，以处理更一般的模型，从而证明它们在应用中的使用。

This proposal will address questions in the theory of interacting particle systems and stochastic spatial processes. These stochastic processes are models for large systems with many interacting ``components'' (cells, individuals, particles, plants, etc.). Some examples of the phenomena these systems model are: competition of species, epidemics, population growth, evolution of genetic traits. A principal goal of research in this area is to understand how the macroscopic behavior of large systems depends on the individual interactions between components. Two competition models will be studied, a spatial stochastic Lotka-Volterra model and a multitype contact process. For the Lotka-Volterra model, the objectives are to determine the parameter regions which correspond to survival of one species and coexistence of both species. The project is part of the investigator's ongoing efforts to understand and exploit the scaling relationship between interacting particle systems and measure-valued diffusions. For the multitype contact process, the questions of interest concern survival versus extinction on regular trees. The investigator expects to find new phase dependent survival and extinction phenomena that do not exist in the lattice case. A second general area of this proposal addresses problems motivated by population genetics. The evolution of genetic traits in geographically structured populations is often modeled with Kimura's stepping stone model, where questions about measures of kinship are reformulated in terms of questions about the behavior of coalescing random walk systems. The investigator's previous research includes work on limit theorems for these systems on large, two-dimensional lattice sets with torus or wrap-around boundary conditions. The goal here is to show robustness of these theorems over a wide range of boundary conditions, justifying their use in population genetics.This proposal involves research in the theory of interacting particle systems and stochastic spatial processes. These stochastic processes are models for large systems with many interacting components (cells, individuals, particles, plants, etc.). The goal of this research is to obtain a better qualitative understanding of various complex phenomena that interacting particles systems model well, such as models of: competition of species, epidemics, population growth, evolution of genetic traits. A principal goal of research in this area is to understand how the macroscopic behavior of large systems depends on the individual interactions between components. Several specific models will be studied, including a stochastic spatial version of a well known model for competition between species. In addition to work on specific models, the investigator will try to extend the validity of some approximation theorems established for some specific models to handle more general ones, thus justifying their use in applications.