Evolution in spatially structured populations

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Brief description of the context of the research:We are concerned with mathematical models that capture the genetic variation that we should see in populations whose size fluctuates in space and time; in other words models that combine ecology with genetics. The first step is to write down sensible ecological models. A very natural class of models generalises the idea of a branching Markov process by regulating the number of offspring that an individual produces according to the current population size. For large populations, one passes to a scaling limit and replaces the individual based model by a so-called Dawson-Watanabe superprocess with regulation. These models are rather poorly understood, but if the regulation is so strong that the population size is constant, then one recovers the Fleming-Viot superprocess, which is analytically tractable. Our first goal is to consider fluctuations about the Fleming-Viot limit. These will take the form of an infinite-dimensional Ornstein-Uhlenbeck process. Our central question will be to investigate how they affect patterns of genetic variation.Novelty of the research methodology:Commonly the technique has been to numerically justify models in population genetics. The novel technique here is in our aim to establish their validity through rigorous mathematics. Aims and objectives:Regulating the population by its total size is not completely biologically natural. Our longer term goal is to understand what happens if the population is regulated only by local crowding. This stipulation is more realistic since at areas of high local crowding, competition for finite resources will hinder the local reproduction rate. The first step will be to digest the existing literature on measure-valued diffusions; initially this will involve studying the Dawson-Watanabe and Fleming-Viot superprocesses. Since the models which regulate the population size are poorly understood, we will begin our exploration into this field by performing some preliminary numerical experiments. This will guide our intuition as to the behaviour of such processes. A certain amount of folklore surrounds the key question of understanding how fast ancestral lineages of individuals alive in the population spread through space. Because of the cost of storing trees of ancestors, this is not even numerically accessible through a naive approach, but we shall begin with some numerical experiments for what Hallatschek and Nelson call the `tracer dynamics' under the model. This would be enough for a good thesis, but if time permits we'll turn to the mathematical questions: what is the limiting population model and what do the fluctuations about that limit look like? Potential impact:The purpose of this research will be to create models for mathematical population genetics, which mathematical biologists may fit data to. By developing and understanding models for populations whose size is regulated, we hope to understand how this affects patterns of genetic variation. The project will fall within the EPSRC statistics and applied probability research area.