Stochastic Spatial Processes

这项研究涉及随机空间过程的理论和应用中的几个主题。这类过程是具有许多相互作用的部件的大系统的随机模型。这些系统模型中的现象包括物种竞争、流行病、遗传特征的传播和催化反应。将考虑对几个具体问题的研究。第一个项目涉及一个相互催化分支的模型。其目标是了解两个空间分布的种群经历随机运动和繁殖的同时进化，其中一个种群的个体只有在另一个种群的个体存在的情况下才能繁殖和死亡。第二个项目涉及相互作用粒子系统的测量值比例限制。我们的目标是进一步研究最近才发现的现象，即某些具有强局部相互作用的晶格系统在适当缩放时收敛到超布朗运动。特别是，从两个或多个维度详细描述选民模型的这一现象是一个主要目标；将研究空间结构和谱系结构。第三个项目涉及混合区数学模型研究中出现的问题。这里的目标是描述具有空间相关选择机制的模型的大尺度行为，特别是当选择优势参数较小时。本研究涉及随机空间过程的理论和应用的几个课题。这项研究的目的是更好地定性地了解由大量相互作用的组件组成的系统所表现出的一些复杂现象，这些相互作用具有随机或随机元素。这样的系统出现在许多应用中，其中一个值得注意的领域是理论生物学和生态学，人们对改进不允许明确的空间相关性的旧的数学模型非常感兴趣(就像这些模型一样)。随机空间模型被用来研究涉及捕食者-猎物相互作用、物种共存和遗传多样性的问题。这些模型通常有几十个参数和相互作用规则；因此，它们通常不适用于严格的数学分析。研究人员计划对几个特定的模型进行研究，这些模型保留了更复杂模型的有趣的定性行为，但足够“简单”，可以进行数学分析。通过研究这些模型，研究者试图给出严格的数学论证来解释观察到的模型行为。研究人员还计划研究两种明显不同类型的随机空间模型(强相互作用的离散模型和弱相互作用的连续模型)之间似乎存在的密切联系。

9971868Cox This research deals with several topics in the theory and application of stochastic spatial processes. Such processes are stochastic models for large systems with many interacting components. Among the phenomena these systems model are competition of species, epidemics, spread of genetic traits, and catalytic reactions. Research on several specific problems will be considered. The first project involves a model for mutually catalytic branching. The goal is to understand the simultaneous evolution of two spatially distributed populations undergoing random motion and reproduction, where individuals from one population reproduce and die only in the presence of individuals from the other population. The second project involves measure-valued scaling limits of interacting particle systems. The goal is to study further the phenomenon, only recently discovered, that certain lattice systems with strong local interactions converge, when scaled appropriately, to super-Brownian motion. In particular, a detailed description of this phenomenon for the voter model in two or more dimensions is a primary goal; both spatial and genealogical structure will be studied. The third project involves questions which arise in the study of mathematical models of hybrid zones. The goal here is to describe the large scale behavior of models with a spatially dependent selection mechanism, especially when the selective advantage parameter is small. This research involves several topics in the theory and application of stochastic spatial processes. The goal of this research is to obtain a better qualitative understanding of some of the complex phenomena exhibited by systems made up of a large number of interacting components, with the interaction having a stochastic or random element. Such systems occur in many applications, one notable area being theoretical biology and ecology, where there is great interest in improving older mathematical models which do not allow for explicit spatial dependence (as do these models). Stochastic spatial models are used to study questions involving predator-prey interactions, species coexistence, and genetic diversity. These models often have dozens of parameters and interaction rules; consequently, they are generally not amenable to rigorous mathematical analysis. The investigator plans research on several specific models which retain the interesting qualitative behavior of more complicated models, but are "simple" enough to allow for the possibility of mathematical analysis. By studying these models, the investigator seeks to give rigorous mathematical arguments which explain the observed behavior of the models. The investigator also plans to study what appears to be an intimate connection between two apparently different types of stochastic spatial models (discrete models with strong interactions and continuous models with weak interactions).