Stochastic Spatial Models

随机空间模型

• 批准号: RGPIN-2014-04081
• 负责人: Perkins, Edwin
• 金额: $ 2.77万
• 依托单位: University of British Columbia
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=661953
• 关键词: Stochastic; Spatial; Models

We study random processes distributed over space which evolve in time. Examples include populations undergoing random reproduction and migration, diseases spreading through a population, species competing for resources, or oil flowing through a porous medium (percolation). Although the local rules of evolution can appear complicated, often it is easier to analyze the process at large space and time scales where these processes can exhibit universal behaviour. Much of my research is to find, justify and analyze a mathematically precise picture of this large scale evolution. **Certain epidemic models turn out to be mathematically equivalent to certain kinds of percolation. Both models have a critical parameter (infection probability or percolation probability) above which the epidemic can survive in the long term, or oil will percolate. One objective is to find how this critical parameter can behave in a certain regime, corresponding say to a disease which can perform long range infection.**For competing species our results can sometimes tell us when can they co-exist in equilibrium or when one type can drive the other out. The general mathematical methods designed to handle this question also tell us how cooperative behaviour can evolve out of simple Darwinian selection--a central problem in modern evolutionary theory. From a mathematical perspective it turns out that these questions can be harder in a 2-dimensional environment (which is most relevant) than in higher dimensions and we will try to address this question in this challenging and crucial setting. For the evolution of cooperation we will consider multi-type models where the coexistence of many types (clearly present in nature) should be possible.**We also study the limiting models that arise from viewing these stochastic spatial models in the large scale. A central question is whether or not these processes capture all the underlying information or if there are hidden variables which require a richer structure. For the continuum epidemic models, we believe the disease will survive in stochastic waves and we will study the qualitative and quantitative nature of these waves. **A final objective will be to tighten the connection between many of the original spatial models and their scaling continuum limits. For some models we know that if we take snapshots at fixed times then the probabilities of the rescaled spatial models will converge to those of a certain well-known universal limit called super-Brownian motion. We want to show that the entire time evolution of the rescaled models converge to that of this ubiquitous limit.

我们研究分布在空间中的随机过程，这些过程会随着时间的推移而发展。 例子包括接受随机繁殖和迁移的种群，通过人群传播的疾病，争夺资源的物种或流经多孔培养基的石油（渗滤）。 尽管局部进化规则看起来很复杂，但通常在大时空和时间尺度上分析这些过程可以表现出普遍行为的过程更容易。 我的大部分研究是找到，证明和分析这种大规模演变的数学精确图片。 **某些流行病模型在数学上等同于某些类型的渗透。两种模型都有一个临界参数（感染概率或渗透率），从而长期可以生存，否则油将渗透。 一个目的是找到该关键参数如何在某个制度中行为，对应于可以执行远距离感染的疾病的说法。旨在处理这个问题的一般数学方法还告诉我们，合作行为如何从简单的达尔文主义选择中进化，这是现代进化论中的核心问题。 从数学角度来看，事实证明，在二维环境（最相关）中，这些问题可能比在更高的维度上更难，我们将尝试在这个挑战和关键的环境中解决这个问题。 对于合作的演变，我们将考虑多类模型，在这些模型中，许多类型的共存（在自然界中显然存在）是可能的。 一个核心问题是这些过程是否捕获了所有基础信息，或者是否存在需要更丰富结构的隐藏变量。 对于连续流行模型，我们认为该疾病将在随机波中生存，我们将研究这些波的定性和定量性质。 **最终目标是收紧许多原始空间模型与它们的缩放连续限制之间的连接。对于某些模型，我们知道，如果我们在固定的时间拍摄快照，那么重新验证的空间模型的概率将融合到某些众所周知的称为“超棕色运动”的通用限制的概率。 我们想证明，重新制定模型的整个时间演变都融合到了无处不在的极限。