Stochastic population processes: metastability, asymptotics and phase transitions

我的研究计划的主要目标是开发和分析相互作用的种群的随机模型。一个相互作用的种群模型包括一组个体（人、动物、植物等），个体之间交互频率的规范（例如，社交网络）和一组用于更新个体交互时状态的规则。一个例子是流行病模型：有一组个体，每个人要么健康，要么感染流感。生活或工作在一起的人比不这样做的人更经常互动。当一个健康的人遇到流感患者时，有一定的概率，健康的人会感染流感。请注意，也可以包括非交互式事件：例如，流感患者可能在一段时间后自行恢复。“随机”元素意味着模型中存在一些随机性。因此，在上面的例子中，为了在计算机上进行模拟，你可以在交互的瞬间进行随机选择（就像抛硬币一样），以确定健康的人是否感染了流感。研究这些模型的一种方法是观察他们的平均行为。即使每个单独的事件是随机的，只要我们知道所涉及的概率，只要个体以相当均匀的方式相互作用（即，对于所有成对的个体来说，相互作用的频率接近相同），当模型包括大量个体时，可以相当简单地估计（无需实际运行模拟）每个时刻处于每个状态的个体比例。通过诉诸这些平均值，我们得到一个更简单的，所谓的确定性系统。然而，这种简化的代价是丢失了关于随机波动的重要信息。幸运的是，有数学工具可以用来研究波动;主要思想是描述这些波动的“速率”，作为系统在每个时刻的平均状态的函数。我打算使用这些工具来直接研究这些模型的波动，而不需要求助于计算机模拟。一个特别感兴趣的现象是相变，这是一种从一种类型的全局行为到另一种类型的变化，因为模型的参数是变化的。例如，在流行病模型中，随着感染传播率的增加，感染可以从迅速消失到引起大规模爆发。简单的例子表明，在相变点附近，波动往往主导动力学;换句话说，它们在相变附近最明显。如果我们能够获得这些波动的详细描述，我们就可以根据对一个物种种群随时间变化的观察，预测该物种何时面临灭绝的危险。这只是我们可以使用相互作用种群的随机模型研究的许多场景之一。

The main goal of my research program is to develop and analyze stochastic models of interacting populations. A model of an interacting population comprises a set of individuals (people, animals, plants, etc.), a specification of the frequency of interaction between individuals (for example, a social network) and a set of rules for updating the state of individuals when they interact. An example is an epidemic model: there are a set of individuals, each either healthy or infected with the flu. Individuals that live or work close to one another interact more often than individuals that do not. When a healthy person encounters someone with the flu, with some probability, the healthy person contracts the flu. Note that non-interactive events can also be included: for example, a person with the flu may recover on their own after some time. The “stochastic” element means there is some randomness in the model. Thus in the above example, to simulate on a computer you would make a random choice (like flipping a coin) at the moment of interaction to determine whether the healthy person contracts the flu.One way to study these models is by looking at their average behaviour. Even if each separate event is random, as long as we know the probabilities involved, and as long as individuals interact with each other in a fairly uniform way (i.e., the frequencies of interactions are close to the same for all pairs of individuals), when the model includes a large number of individuals it is reasonably straightforward to estimate (without actually running a simulation) what proportion of individuals are in each state at each moment in time. By appealing to these averages we obtain a simpler, so-called deterministic, system. However, the cost of this simplification is that important information concerning random fluctuations is lost. Fortunately, there are mathematical tools that can be used to study the fluctuations; the main idea is to describe the “rate” of these fluctuations, as a function of the system's average state at each moment in time. I intend to use these tools to study the fluctuations of these models directly, without needing recourse to computer simulation.A phenomenon of particular interest is a phase transition, which is a change from one type of global behaviour to another, as parameters of the model are varied. For example, in the epidemic model, the infection can go from dying out quickly to causing a large outbreak, as the transmission rate of the infection is increased. Simple examples have shown that near the transition point, fluctuations tend to dominate the dynamics; in other words, they are most noticeable near a phase transition. If we can obtain a detailed description of these fluctuations, we may be able predict when a species is in danger of extinction, based on observations of its population over time. This is just one of many scenarios that we can study using stochastic models of interacting populations.