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演变，其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提供进化驱动力的基本见解，并可用于回答有关人类和其他物种的过去和未来的问题。例如，Speidel等人（2019）最近使用来自1000个基因组计划的数据对不同大陆人类种群的系谱历史进行统计推断，其中包括对过去数百万年的人口规模的预测。（1930年，1958年）-把一个人口作为一个抽象的基因集合，并强加规则，他们是如何在世代之间传递。尽管这些模型成功地描述了空间上封闭的小种群，但它们无法预测不考虑种群空间分布就无法解释的进化现象：后代出生在父母所在地这一简单事实已经使突变和遗传适应的传播具有空间惰性。当一个种群分布在一个大的区域时，这种影响尤其重要，在那里这种影响可能导致物种形成-不同亚种的形成。Watanabe（1968）和Dawson对现代空间种群模型的研究做出了关键贡献（1977，1979），他介绍了我们现在所说的超过程，一个模型原型，描述了在大尺寸的限制下空间演化种群的演化。原始模型的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。一个影响是，没有任何机制可以阻止大量人口的进一步增长，导致人口规模无限增长，并经常形成所谓的巨大人口密度集群。在真实的种群中，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较低的地区。近年来，人们通过使个体的期望后代数随着局部种群密度的增加而减少，将前两种效应引入到超过程模型中。然而，后一种影响很少受到关注。我的研究目标是通过在个体之间引入排斥力，使个体远离人口密度高的区域，从而增强超过程模型。我试图了解在哪些假设下，这种增加会导致稳定的人口动态，以及它的影响如何不同于上述模型，并与这些模型相互作用，这些模型根据当地人口密度控制繁殖。此外，这种方法形成了扩展超过程模型以解释不同性别的自然第一步，这可以通过让异性和同性的个体分别相互吸引和排斥来实现。研究这一模型并将其与原模型进行比较，将有助于了解有性生殖种群与无性生殖种群的进化差异，并有助于解释有性生殖种群的成功性，本项目属于EPSRC数学科学研究领域的福尔斯.

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.