Stochastic Spatial Processes

随机空间过程

• 批准号: 9971868
• 负责人: J. Theodore Cox
• 金额: $ 7.5万
• 依托单位: Syracuse University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-06-15 至 2002-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971868&HistoricalAwards=false
• 关键词: Stochastic; Spatial; Processes

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).

9971868 Cox本研究涉及随机空间过程理论和应用中的几个主题。这些过程是具有许多相互作用的组件的大系统的随机模型。这些系统模型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进行研究。第一个项目涉及相互催化分支的模型。我们的目标是了解两个空间分布的人口进行随机运动和繁殖，其中一个人口的个体繁殖和死亡，只有在存在的个人从其他人口的同时进化。第二个项目涉及相互作用粒子系统的测量值标度极限。我们的目标是进一步研究的现象，最近才发现，某些晶格系统与强局部相互作用收敛，当适当缩放，超布朗运动。特别是，在两个或两个以上的维度的选民模型的这种现象的详细描述是一个主要目标;空间和谱系结构将进行研究。第三个项目涉及在混合区的数学模型研究中出现的问题。这里的目标是描述具有空间依赖选择机制的模型的大尺度行为，特别是当选择优势参数很小时。本研究涉及随机空间过程理论与应用的若干课题。本研究的目标是获得一个更好的定性理解的一些复杂的现象所表现出的系统由大量的相互作用的组件，具有随机或随机元素的相互作用。这样的系统出现在许多应用中，一个值得注意的领域是理论生物学和生态学，其中有很大的兴趣，在改进旧的数学模型，不允许明确的空间依赖性（如这些模型）。随机空间模型用于研究捕食者-猎物相互作用、物种共存和遗传多样性等问题。这些模型通常有几十个参数和相互作用规则;因此，它们通常不适合严格的数学分析。研究者计划研究几个特定的模型，这些模型保留了更复杂模型的有趣的定性行为，但又足够“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模型的观察行为。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）之间的密切联系。