Ecology, Evolution, and Random Graphs

生态学、进化和随机图

• 批准号: 1005470
• 负责人: Richard Durrett
• 金额: $ 34.97万
• 依托单位: Cornell University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-08-01 至 2010-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1005470&HistoricalAwards=false
• 关键词: Ecology; Evolution; Random; Graphs

Research will be carried out on three topics: stochastic spatial models, processes taking place on random graphs, and questions related to the evolution of biological systems. Three of the proposed questions in the first topic concern "When can species coexist", while a fourth concerns the possibility that the quadratic contact process in two dimensions can have two phase transitions one for the existence of stationary distributions and a larger one for survival from a finite set. In the second topic one interesting mathematical problem concerns "explosive percolation" conjectured to have a discontinuous transition, while the more biologically important question concerns how the outcomes of epidemics and ecological competitions change when they take place on random graphs, which arguably provide better models of the real social networks. The third topic concern situations in which the characteristics of individuals in ecological competitions are also not static but evolve in response to their environment. My first steps in the area "adaptive dynamics" were taken in a study of predator-prey systems with John Mayberry. Here, we propose to study more complex examples that lead to evolutionary cycling and a second problem on the evolution of virulence, which leads to consideration of the role of spatial structure in increasing the virulence of diseases.Many interesting mathematical questions arise from biology. Here we address some questions that arise from ecology and evolution. Three examples should illustrate the nature of our work. (1) At the turn of the century, observations of social networks revealed that we live in a small world in which everyone on the planet is separated by six degrees of separation. Now we need to understand how this geometry of social networks effects the spread of epidemics and the other biological and social processes. (2) The world shows much more biodiversity than mathematical models predict, so it is important to understand the mechanisms which allow for species coexistence. More generally, we will also be interested in how spatial structure changes the outcome of ecological competition. (3) In most situations the characteristics of individuals involved in competition with other species or with infectious agents are not static but evolve in time. For example, in most cases diseases evolve to be less virulent, but in a spatially structured population the opposite may occur. Co-evolution of hosts and parasites can lead to interesting evolutionary cycling, sometimes called ?Red Queen Dynamics after the character in Alice in Wonderland who has to keep running to stay in the same place.

研究将针对三个主题进行：随机空间模型，随机图上发生的过程以及与生物系统进化有关的问题。第一个主题中提出的三个问题涉及“何时可以共存”，而第四个问题则涉及二维二次接触过程的可能性可以具有两种相变，一个用于存在固定分布的存在，而从有限组中生存的可能性更大。在第二个主题中，一个有趣的数学问题涉及“爆炸性的渗透”，该问题猜想是不连续的过渡，而更重要的重要问题涉及流行病和生态竞争的结果在随机图上发生时如何改变，这可以说是更好地提供真实社交网络的更好模型。第三个主题涉及的情况是，生态竞争中个人的特征也不是静态的，而是响应其环境而发展的情况。我在“自适应动力学”领域的第一步是在与约翰·梅伯里（John Mayberry）对捕食者捕食系统的研究中采取的。在这里，我们建议研究导致进化循环的更复杂的例子，以及关于毒力进化的第二个问题，这导致考虑了空间结构在增加疾病毒力中的作用。许多有趣的数学问题来自生物学。在这里，我们解决了生态和进化引起的一些问题。三个例子应该说明我们作品的性质。 （1）在世纪之交，对社交网络的观察表明，我们生活在一个小世界中，在这个世界中，地球上的每个人都以六个分离程度分开。现在，我们需要了解社交网络的几何形状如何影响流行病的传播以及其他生物学和社会过程。 （2）世界比数学模型预测的生物多样性要多得多，因此了解允许物种共存的机制很重要。更普遍地，我们还将对空间结构如何改变生态竞争的结果感兴趣。 （3）在大多数情况下，与其他物种竞争或传染剂竞争的人的特征不是静态的，而是及时发展的。例如，在大多数情况下，疾病演变成较少的毒力，但在空间结构的种群中可能发生了相反的情况。主机和寄生虫的共同发展可能会导致有趣的进化骑行，有时被称为“红皇后动态”，在爱丽丝梦游仙境中的角色之后，他必须继续奔跑才能留在同一地点。