## Graduate Seminar SS 2023:

**Stochastic models in evolution**

**Prof.Dr. Anton Bovier, Manuel Essser**

## Time and place: Tuesdays, 14h ct, **New** ROOM 2.040

**Registration is closed.**

The seminar will deal with various aspects of so-called stochastic individual based models of adaptive dynamics, These models are used to describe and understand features of populations of divers biological individuals (cells, bacteria, plants, animals etc) that evolve under the key evolutionary mechanisms: birth, death, mutation/migration, and competition (cooperation). The idea is to show that under reasonable assumptions, one should be able to see the emergence of key features of Darwinian evolution as they are described in the biological theory of adaptive dynamics. Mathematically, these models are measure valued Markov processes in continuous time. The overall goal is to derive scaling limits under different assumptions on the rescaling of parameters of the model and of time. The seminar will deal cover both more theoretical and more applied, model specific aspects, depending also on the interests of the participants.

Some specific subtopics could be:

- Construction ot the Markov processes, martingale problems, criteria for existence
- Large propulation limit, law of large numbers
- Diffusion limits
- Numerical simulation algorithms: Gillespie, tau-leap, hybrid algorithms
- Champagnat scaling, trait-substitution sequence, canonical equation
- Galton-Watson processes
- Higher mutation rates, crossing of fitness valleys, metastability
- Phenotypic plasticity, dormancy, multi-type branching processes
- ....................

**Literature: A basic source are the lecture notes by me and Anna Kraut (partly under construction), and references therein.**