### Anna Kraut, U Michigan, Ann Arbor: Multi-scale limits in a population model with moderately rare mutations (Oberseminar Mathematische Stochastik)

##### Wednesday, 24.11.2021 17:00 per ZOOM: 61828242813

To study the evolution of heterogeneous populations, we can consider individual-based Markov processes that describe the microscopic interactions between single individuals. Driven by birth, death, competition and mutation rates, the state of the system on a macroscopic population level evolves over time towards traits of higher fitness. The typical behaviour of such processes can be studied by looking at limits of large populations and rare mutations. Depending on the interplay of these two quantities, as well as the structure of the space of possible traits, different phenomena and evolutionary pathways can be observed. In this talk, I will give an introduction to the multiple scaling parameters and time scales that are involved in the above model. We will then focus on the parameter regime of moderately rare mutations, where multiple new mutant traits are present at the same time and compete to invade the population. On the population level, we can prove convergence to a deterministic jump process that transitions between different equilibrium states of coexisting subpopulations. This process reaches a final state when no fit trait is accessible within a certain distance. However, when considering an even more accelerated time scale, metastable - now again random - transitions across fitness valleys can be observed. I will outline the main ideas of constructing the limiting multi-scale processes and then demonstrate some interesting phenomena in the case of easy examples. The talk is based on a collaboration with Loren Coquille and Charline Smadi, as well as recent work with Manuel Esser.

Angelegt am 29.10.2021 von Anita Kollwitz

Geändert am 17.11.2021 von Anita Kollwitz

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