Stochastic Spatial Models

摘要本课题涉及相互作用粒子系统的理论与应用研究。这些系统模拟的现象包括：物种竞争、流行病、遗传性状的传播、催化化学反应以及许多其他现象。研究者将对几个具体问题进行研究。第一个项目涉及作为相互作用扩散的抛物型安德森模型，以及相关的临界现象。第二个项目关注一维界面区域的行为。这将继续研究杂交区或不同物种“狭窄”共存区模型。第三个项目试图用相互作用粒子系统模型来模拟物种丰度分布。这建立在对物种数量和面积之间的关系进行建模的相关工作的基础上。第四个项目涉及最近发现的用于交互系统的比较技术的可能扩展。由于精确的计算通常很难或不可能进行，比较方法是分析新系统的重要工具。本提案涉及相互作用粒子系统的理论和应用方面的几个主题的研究。这些系统模拟的现象包括：物种竞争、流行病、遗传性状的传播、催化化学反应以及许多其他现象。本研究的目的是为了更好地定性地理解相互作用粒子系统所能很好地模拟的各种复杂现象。例如，研究物种丰度问题的物种进化和灭绝的一个简单模型就适用于这个框架。它以个体的出生、死亡、运动和繁殖为基础，具有很小的突变率。研究者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。拟议研究的一部分涉及非常具体的模型和诸如此类的问题，而拟议研究的一部分将致力于开发处理这类模型的通用数学技术。

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.