Spatial population models
Abstract: Mathematical models provide an invaluable tool for understanding how different forces of evolution interact for different parameter regimes and timescales. In these lectures, we introduce a range of classical, and not so classical, models that can be used to capture the interplay between the randomness due to reproduction in a finite population (known as genetic drift) and the spatial structure of that population. We shall then introduce a very simple form of selection into our models, in which an individual's relative fitness is determined by which of two possible genetic types it carries at a particular locus, and investigate the way in which spatial structure influences the effectiveness of natural selection.
Modelling hybrid zones: travelling waves in the Allen-Cahn equation (and how to stop them)
Abstract: Natural selection can take many forms. In this lecture we continue to restrict our attention to a single genetic locus, but now each individual carries two copies of a gene that has two possible forms. Individuals that carry different genetic types are disadvantaged relative to those that carry two copied of the same type. This leads us to investigate so-called `hybrid zones' using a special form of the (stochastic) Allen-Cahn equation. In earlier work we modelled populations that are distributed across the whole of Euclidean space; here we investigate the interplay between the motion of the hybrid zone and the shape of the habitat, in particular asking what happens if the population passes through an isthmus.
Meeting ID: 872 3198 7829