Fitness in Expanding Populations

Population genetics can be thought of as the field of biology which studies the interplay between genetics and evolutionary biology. Questions frequently asked in this field are how natural selection impacts on genetic diversity and vice-versa, as well as what are the mechanisms that promote genetic diversity in a population. One of the crucial methods often applied to investigate the questions above is the development of theoretical mathematical models which simulate the evolution of a population over time. Importantly, the rigorous analysis of these theoretical models often requires many tools from probability theory and stochastic analysis. From a mathematical perspective, the theoretical models may focus either on discrete interacting particle systems or on continuous processes, such as (stochastic) differential equations. Note that often the differential equations, either stochastic or deterministic, may arise as a scaling limit of discrete processes. This relationship is interesting as it can be used to prove properties of both the discrete process and its continuous counterpart.

In this probability theory PhD project, we are going to prove rigorous mathematical results about different models that were introduced to understand the interplay between the spatial structure of a population and the evolution of its genetic diversity. In order to understand why this interaction is non-trivial, we note that during evolution, a population can acquire and accumulate mutations, i.e., changes in the DNA sequence which in transmitted across generations. Importantly, mutations can affect the ability of a living organism to survive and reproduce. This ability is called fitness, and it is expected that deleterious mutations (i.e., mutations that decrease fitness) have lower probability of being transmitted through generations. In spatially structured populations, however, the probability an individual survives depends not only on the fitness of its genotype, but also on the local population density in the neighbourhood of this individual. Roughly speaking, a high local population density may either increase the survival probability due to cooperation or decrease it due to competition. This interaction may increase the probability of mutations arising in the front of a spatial population expansion spreading over a large unoccupied habitat, a phenomenon called gene surfing.

We will be interested in rigorously analysing different models of gene surfing that were introduced in the biology literature in recent years. We are particularly interested in proving a conjecture regarding a model introduced by Foutel-Rodier and Etheridge in 2020 to study the gene surfing of deleterious mutations; we aim to show that the scaling limit of this discrete model is given by the solution of a system of partial differential equations.