Genealogies in spatially structured populations --- Universality and extensions of the Brownian net

• Applicant: Dr. Frederic Alberti
• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 519713930

The aim of mathematical population genetics is to understand how evolutionary forces such as selection, mutation and recombination shape the genetic composition of populations. An important aspect is the investigation of genealogical structures via stochastic processes. These "ancestral processes" are also of independent mathematical interest. They exhibit an important phenomenon called "universality" which means that, after appropriate rescaling of time and space, the multitude of possible models reduces to a few typical representatives, or "universal scaling limits". One universal scaling limit for the collection of all ancestral lineages in a spatially structured, neutral population is the "Brownian web". Simply put, it is an uncountable collection of Brownian motions, started from every point in time and space, which coalesce upon collision. In addition to being a universal scaling limit in many contexts, the Brownian web is of great theoretical significance and an important object of investigation in modern probability theory. An important extension of the Brownian web is the "Brownian net", in which the colliding Brownian motions also branch. In contrast to the Brownian web, however, convergence to the Brownian has, to date, only been proved in a few very simple cases. The first goal of this project therefore is to provide additional examples for the universality of the Brownian net. To this end, we will consider scaling limits of the genealogies in one-dimensional contact processes evolving under selection or under recombination. It is well known that both these forces lead to branching of ancestral lineages. Under recombination, the set of ancestral lineages exhibits a natural hierarchical structure, due to the consistency of the model with respect to restriction to different parts of the genome (marginalisation consistency). We will investigate how this structure carries over to the limit, expecting that this will lead to new insights into the structure and representation of the Brownian net. Moreover, tracking additional information, e.g. by distinguishing ancestral lines of different genetic sequence positions, might lead to an interesting and natural extension. We expect that we can also relate our results on the genealogies back to the evolution forward in time.

• DFG Programme: WBP Position