Stochastic population processes: metastability, asymptotics and phase transitions

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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.