Alison Etheridge: Some mathematical models of evolving populations (3rd lecture)

How can we explain the patterns of genetic variation in the world around us? The genetic composition of a population can be changed by natural selection, mutation, mating, and other genetic, ecological and evolutionary mechanisms. How do they interact with one another, and what was their relative importance in shaping the patterns that we see today? This question lies at the heart of theoretical population genetics. Whereas the pioneers of the field could only observe genetic variation indirectly, by looking at traits of individuals in a population, researchers today have direct access to DNA sequences, but making sense of this wealth of data presents a major scientific challenge and mathematical models play a decisive role.

In these lectures we'll discuss some of the ways in which we can distill our understanding into workable mathematical models. Although our main focus is on spatially distributed populations, we'll begin with perhaps the most classical model, the Wright–Fisher model, and use it to introduce some of the underlying principles of modelling in this area. We then illustrate Felsenstein's `pain in the torus', which captures some of the challenges of incorporating spatial structure into our models and introduce the spatial Lambda–Fleming–Viot model, which goes some way to overcoming the pain in the torus. We shall see that even obviously unrealistic models, when viewed over the long timescales of evolution, capture some of the key features of naturally evolving populations.

In the second lecture, we shall use a mixture of differential equations and the spatial Lambda–Fleming–Viot model to investigate the interactions between spatial structure, natural selection, and the randomness inherent in reproduction in a finite population.

In the final lecture, we describe a new, and rather general, class of population models that aims to capture something of the heterogeneity of real populations. Population densities fluctuate in space and time as individuals compete for space and resources. A particular novelty is that we subdivide reproduction of offspring into a juvenile phase and a maturation phase. This leads to novel limits when we scale the population model, for example replacing the diffusion term in classical reaction diffusion equations by an operator of porous medium type. We make some very preliminary remarks on the effect that this will have on patterns of genetic variation.

Each lecture will be self contained.