Stochastic Spatial Models

9628290考克斯摘要本课题涉及相互作用粒子系统理论与应用中的若干课题的研究。 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、催化化学反应以及许多其他现象。 调查员将对几个具体问题进行研究。 第一个项目涉及抛物安德森模型作为一个相互作用的扩散，和相关的临界现象。 第二个项目关注一维界面区域的行为。 这将继续研究不同物种混合区或“狭窄”共存区的模型。 第三个项目试图用相互作用粒子系统模型模拟物种丰度分布。 这是建立在相关工作的物种数量和面积之间的关系建模。 第四个项目涉及最近发现的相互作用系统的比较技术的可能扩展。 由于精确的计算通常是困难的或不可能的，比较方法是分析新系统的重要工具。 该计划涉及相互作用粒子系统理论和应用中的几个主题的研究。. 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、催化化学反应以及许多其他现象。 本研究的目标是获得一个更好的定性理解的各种复杂的现象，相互作用的粒子系统模型很好。例如，一个简单的物种进化和灭绝模型，研究物种丰度问题，适合这个框架。 它基于个体的出生、死亡、运动和繁殖，突变率很小。 研究人员希望表明，这种类型的模型可以解释在该领域观察到的数据模式。 部分拟议的研究涉及非常具体的模型和问题，如这一个，部分拟议的研究将旨在开发通用的数学技术来处理这种类型的模型。

9628290 Cox ABSTRACT This proposal involves research on several topics in the theory and application of interacting particle systems. Among the phenomena these systems model are: competition of species, epidemics, spread of genetic traits, catalytic chemical reactions, as well as many others. The investigator will pursue research on several specific problems. The first project involves the parabolic Anderson model as an interacting diffusion, and associated critical phenomena. The second project concerns the behavior of one-dimensional interfaces regions. This continues work on a model for hybrid zones or "narrow" coexistence zones of different species. The third project attempts to model species abundance distributions with an interacting particle system model. This builds on related work modeling the relationship between species number and area. The fourth project involves possible extensions of a recently discovered comparison technique for interacting systems. Since exact calculations are usually difficult or impossible to make, comparison methods are important tools for analyzing new systems. This proposal involves research on several topics in the theory and application of interacting particle systems. . Among the phenomena these systems model are: competition of species, epidemics, spread of genetic traits, catalytic chemical reactions, as well as many others. The goal of this research is to obtain a better qualitative understanding of various complex phenomena that interacting particle systems model well. For example, a simple model for the evolution and extinction of species, to study species abundance questions, fits into this framework. It is based on the birth, death, movement, and reproduction, with a small mutation rate, of individuals. The investigator hopes to show that this type of model can explain the data patterns observed in the field. Part of the proposed research concerns very specific models and questions such as this one, and part of the proposed researc h will aim at developing general mathematical techniques for handling models of this type.