Ecology, Evolution, and Random Graphs

将在三个主题上进行研究：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“物种何时可以共存”，而第四个问题涉及二维二次接触过程可能存在两个相变，一个是关于平稳分布的存在，另一个是关于有限集的生存。在第二个主题中，一个有趣的数学问题与“爆炸性渗流”有关，被推测为具有不连续的过渡，而更具有生物学意义的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变化，这可以说提供了真实社会网络的更好模型。第三个专题涉及生态竞争中个体的特征也不是一成不变的，而是随着环境的变化而演变的情况。我在“适应动力学”领域的第一步是在与约翰·梅伯里一起研究捕食者-猎物系统时迈出的。在这里，我们建议研究导致进化循环的更复杂的例子和关于毒力进化的第二个问题，这导致考虑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，我们将讨论一些来自生态学和进化论的问题。三个例子应该说明我们工作的性质。(1)在世纪之交，对社交网络的观察表明，我们生活在一个小小的世界里，地球上的每个人都被六度分隔。现在，我们需要了解这种社会网络的几何形状如何影响流行病的传播以及其他生物和社会过程。(2)世界显示的生物多样性比数学模型预测的要多得多，因此理解允许物种共存的机制是很重要的。更广泛地说，我们也会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，与其他物种或与病原体竞争的个体的特征不是静止的，而是随着时间的推移而演变的。例如，在大多数情况下，疾病会进化成毒性较低的疾病，但在空间结构的人群中可能会发生相反的情况。宿主和寄生虫的共同进化可以导致有趣的进化循环，有时被称为红皇后动力，这是以《爱丽丝梦游仙境》中的角色命名的，他必须不停地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.