Interacting Particle Systems on Lattices and on Graphs

格子和图上相互作用的粒子系统

• 批准号: 1505215
• 负责人: Richard Durrett
• 金额: $ 30万
• 依托单位: Duke University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-08-01 至 2018-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1505215&HistoricalAwards=false
• 关键词: Interacting; Particle; Systems; Lattices; Graphs

This project concerns spatial models for ecological and social interactions motivated by various applications. The theme of this research is to see how predictions change when systems previously studied under the assumption that each individual interacts with all the others are made more realistic by incorporating space. The four main examples are the following: (i) the Staver-Levin forest model, which predicts that forest and savannah (grassland with isolated trees) are alternative stable states; (ii) evolutionary games, which have long been used in ecology to explain phenomena such as the persistence of altruistic behavior; (iii) Axelrod's model, which studies the spread of opinions when individuals interact with a probability based on the number of the number of opinions they share; (iv) the latent voter model, which studies the spread of technology in a social network when consumers who have just acquired a new product will wait some time before they are willing to purchase a new one. The general goal of studying these idealized models is to understand how properties of the equilibrium of the system depend on the details of the interactions. When each individual interacts with all the others, the system is an ordinary differential equation and is easily studied. However, when space is explicitly taken into account the problems become very difficult. This project has the following specific goals: (i) show that in the Staver-Levin model, the direction of movement of a boundary between forest and savannah indicates the one state that is the true equilibrium in the spatial model; (ii) show that evolutionary games have three separate weak selection regimes that can lead to a PDE, ODE, or a regime in which Tarnita's formulas are valid; (ii) complete Junchi Li's thesis work studying Axelrod's model in the situation in which there are a large number of issues about which there are a large number of opinions (this would provide the first rigorous result for that model in two dimensions); (iv) show that even if latent period is brief, it changes the dynamics so that there is only one nontrivial stationary distribution, in contrast to the one parameter family in the voter model without latency.

该项目涉及由各种应用激发的生态和社会互动的空间模型。这项研究的主题是查看预测在先前研究的系统与所有其他人相互作用的假设中如何通过合并空间而变得更现实。四个主要示例是：（i）Staver-Levin森林模型，该模型预测森林和萨凡纳（带有孤立的树木的草原）是替代的稳定状态； （ii）长期以来在生态学中使用的进化游戏来解释现象，例如利他行为的持久性； （iii）Axelrod的模型，该模型根据他们共享的观点数量与概率互动时研究意见的传播； （iv）潜在选民模型，该模型在社交网络中研究了技术的传播，刚刚获得新产品的消费者会等待一段时间才能购买新产品。研究这些理想化模型的一般目标是了解系统平衡的特性如何取决于相互作用的细节。当每个人与其他所有人互动时，系统是一个普通的微分方程，并且很容易研究。但是，当明确考虑空间时，问题变得非常困难。该项目具有以下具体目标：（i）在Staver-Levin模型中表明，森林和萨凡纳之间边界运动的方向表明了一种是空间模型中真正平衡的状态； （ii）表明进化游戏具有三个独立的弱选择制度，可以导致PDE，ODE或TARNITA公式有效的制度； （ii）完成研究Axelrod的模型的完整论文工作，在存在大量问题的情况下，有很多意见（这将为该模型在二维中提供第一个严格的结果）； （iv）表明，即使潜在时期简短，它也会改变动力学，因此与没有延迟的选民模型中的一个参数家族相比，只有一个非平凡的固定分布。