Stochastic Spatial Processes

(1)这个建议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"组件“（细胞，个体，粒子，植物等）的大型系统的模型。 从理论生态学和生物学的现象，这些系统模型的一些例子是：物种的竞争，流行病，人口增长，遗传性状的进化。这一领域研究的一个主要目标是了解大系统的宏观行为如何依赖于组件之间的个体相互作用。 感兴趣的主要模型是竞争模型，随机空间Lotka-Volterra模型。 目标是确定对应于一个物种的生存和/或两个物种共存的参数区域。该项目是研究人员正在进行的努力的一部分，以了解和利用相互作用的粒子系统和测量值扩散之间的标度关系。研究人员将使用流体力学的技术来补充这种方法，以试图证明以前获得的生存和共存条件是尖锐的。第二个要研究的竞争模型是多类型接触过程。这里的主要问题也是竞争类型的生存和共存问题。另一个有待研究的模型是基因从改良作物流入自然或野生种群的模型。这里的目标是了解修改类型入侵野生种群所需的时间长度。从数学上讲，这就变成了二维随机游动系统的行为问题。(2)这项建议涉及相互作用粒子系统和随机空间过程理论的研究。 这些过程是具有许多相互作用的组件（细胞，个体，粒子，植物等）的大型系统的模型。 本研究的目标是更好地定性了解相互作用粒子系统模型的各种复杂现象，例如：物种竞争，流行病，人口增长，遗传性状进化。 这一领域研究的主要目标是了解这些系统的大尺度行为如何依赖于小尺度的个体相互作用规则。几个具体的模型将进行研究，包括一个众所周知的物种之间的竞争模型的随机空间版本。除了在特定模型上的工作外，研究人员还将尝试扩展为一些特定模型建立的一些近似技术的有效性，以处理更一般的模型，从而证明它们在应用中的使用是合理的。

(1) 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 from theoretical ecology and biology 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. The primary model of interest is a model for competition, a stochastic spatial Lotka-Volterra model. The objectives are to determine the parameter regions which correspond to survival of one species and/or 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. The investigator will supplement this approach using techniques from hydrodynamics to try to show that the previously obtained survival and coexistence conditions are sharp. A second competition model to be studied is the multitype contact process. The main questions here are also the issues of survival and coexistence of competing types. Another model to be studied is one of gene flow from modified crops into a natural or wild population. The goal here is to understand the length of the time it takes for the modified type to invade the wild population. Mathematically, this becomes a question of the behavior of coalescing random walks systems in two dimensions.(2) This proposal involves research in the theory of interacting particle systems and stochastic spatial processes. These 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 large scale behavior of these systems depends on the small scale, individual interaction rules. 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 techniques established for some specific models to handle more general ones, thus justifying their use in applications.