Repulsion in Spatial Population Models

数学群体遗传学是对模型的发展和后续研究，这些模型描述了由于自然选择、突变和随机遗传漂变等影响而导致的群体遗传组成的进化，目的是解释和分析适应和物种形成等现象。发现和理解这样的模型可以为进化的驱动力提供基本的见解，并且可以用来回答关于人类和其他物种的过去和未来的问题。例如，Speidel等人（2019）最近使用1000基因组计划的数据对不同大陆上人类种群的宗谱历史进行了统计推断，其中包括对过去数百万年人口规模的预测。最早的模型——由Wright（1931）和Fisher（1930, 1958）提出——将种群视为基因的抽象集合，并对它们如何在代之间传递施加规则。尽管这些模型成功地描述了小的、空间上受限制的种群，但却无法预测那些不考虑种群的空间分布就无法解释的进化现象：后代出生在他们父母的地方这个简单的事实已经使突变的传播和遗传适应具有空间惰性。当一个种群分布在一个很大的区域时，这种影响会变得特别大，在那里这种影响可能会导致不同亚种的形成。Watanabe（1968）和Dawson（1977、1979）对现代空间种群模型的研究做出了重要贡献，他们引入了我们现在所说的超过程（superprocesses），这是一种模型原型，描述了在大尺度限制下空间种群进化的过程。原始模型的基础假设之一是个体相互独立地移动和繁殖，这在数学上很方便，但不现实。其中一个影响是，没有适当的机制来阻止大量人口的进一步增长，导致人口规模无限增长，并频繁形成所谓的人口密度巨大的团块。在真实的人口中，个体依赖于共同的、有限的资源，如食物、水和生存空间。因此，在人口密度高的地区，个体的寿命较短，繁殖较少，并倾向于迁移到人口密度低的地区。最近的许多研究都是通过使个体的预期后代数量随着当地种群密度的减少而减少，将前两种效应结合到超过程模型中。然而，后一种影响却很少受到关注。我的研究目标是通过引入个体之间的排斥力，将个体从高人口密度区域推开，从而增强超过程模型。我试图理解在哪些假设下，这种增加会导致稳定的种群动态，以及它的影响与上述根据当地人口密度控制繁殖的模型有何不同和相互作用。此外，这种方法形成了将超过程模型扩展到考虑不同性别的自然的第一步，这可以通过异性和同性个体相互吸引和排斥来实现。研究这样一个模型，并将其与原始模型进行比较，可能会使我们深入了解有性繁殖和无性繁殖种群的进化差异，并阐明前者观察到的令人困惑的成功。该项目属于EPSRC数学科学研究领域。

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.