Repulsion in Spatial Population Models

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Mathematical population genetics is the development and subsequent study of models that describe the evolution of a population's genetic composition due to effcts such as natural selection, mutation, and random genetic drift, with the goal of explaining and analysing phenomena like adaption and speciation. Finding and understanding such models gives fundamental insights in the driving forces of evolution, and can be used to answer questions about past and future of humans and other species. For example, Speidel et al. (2019) have recently used data from the 1000 Genomes Project to make statistical inferences about thegenealogical history of human populations on dfferent continents, including, amongst other things, predictions about population sizes up to several million years in the past.The earliest models-developed by Wright (1931) and Fisher (1930, 1958)-regard a population as an abstract collection of genes and impose rules on how they are passed on between generations. Although successful at describing small, spatially contained populations, these models fail to predict evolutionary phenomena that cannot be explained without taking the spatial distribution of a population into account: Already the simple fact that offspring are born at the location of their parent gives the spread of mutations and genetic adaption a spatial inertness. This becomes especially impactful when a population is spread over a large area, where this effect may cause speciation-the formation of different subspecies.One of the key contributions leading to the modern study of spatial population models was made by Watanabe (1968) and Dawson (1977, 1979), who introduced what we now call superprocesses, a model prototype that describes the evolution of a spatially evolving population in the limit of large sizes. One of the assumptions underlying the original model is that individuals move and reproduce independently of each other, which is convenient mathematically, but unrealistic. One effect is that there is no mechanism in place that stops large populations from growing further, leading to indefinitely growing population sizes andthe frequent formation of so-called clumps of huge population density. In real populations, individuals depend on common, finite resources such as food, water, and living space. Hence, in areas of high population density, individuals will have shorter living spans, reproduce less, and tend to migrate to areas of lower population density. A lot of recent effort has been on incorporating the former two effects into the superprocess model by making the expected number of offspring of an individual decrease with the local population density. The latter effect, however, has received little attention. The goal of my research is to augment the superprocess model with a drift that pushes individuals away from areas of high population density, by introducing a repelling force between individuals. I seek to understand under which assumptions this addition leads to stable population dynamics, and how its effects differ from and interplay with the aforementioned models that control reproduction depending on local population density. Furthermore, this approach forms a natural first step towards extending the superprocess model to account for different sexes, which could be achieved by having individuals of opposite and same sex respectively attract and repel each other. Studying such a model and comparing it with the original one may lead to insights into diffrences between the evolution of sexually and asexually reproducing populations, and shed light on the puzzling observed success of the former.This project falls within the EPSRC Mathematical sciences research area.