Ecology, Evolution, and Random Graphs

研究将进行三个主题：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。在第一个主题中提出的三个问题涉及“物种何时可以共存”，而第四个问题则涉及二维二次接触过程可能有两个相变的可能性，一个是存在固定分布的相变，另一个是从有限集合中生存的相变。在第二个主题中，一个有趣的数学问题涉及“爆炸性渗流”，它被证明具有不连续的过渡，而生物学上更重要的问题则涉及流行病和生态竞争的结果在随机图上发生时如何变化，这可以说是真实的社交网络的更好模型。第三个主题涉及的情况下，在生态竞争中的个人的特点也不是静态的，但在响应他们的环境演变。我在“适应性动力学”领域的第一步是在与约翰·梅伯里（John Mayberry）一起研究捕食者-猎物系统时迈出的。在这里，我们建议研究更复杂的例子，导致进化循环和第二个问题的毒力的演变，这导致考虑的作用，空间结构在增加疾病的毒力。在这里，我们解决一些问题，从生态学和进化。三个例子可以说明我们工作的性质。(1)在世纪之交，对社交网络的观察显示，我们生活在一个小世界里，地球上的每个人都被六度分隔。现在我们需要了解这种社交网络的几何结构如何影响流行病的传播以及其他生物和社会过程。(2)世界上的生物多样性比数学模型预测的要多得多，因此了解物种共存的机制非常重要。更一般地说，我们也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，参与与其他物种或感染因子竞争的个体的特征不是静态的，而是随着时间的推移而演变的。例如，在大多数情况下，疾病的毒性会降低，但在空间结构化的人群中，情况可能正好相反。宿主和寄生虫的共同进化可以导致有趣的进化循环，有时称为？红皇后动力学在爱丽丝梦游仙境中的角色之后，他必须保持奔跑才能留在同一个地方。

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.