## Stochastics models in evolution

**Advanced topics in applied probability**

**Time and place: Tuesdays and Thursdays, 10-12, Room 1.008**

In this lecture we will introduce and discuss what has become known as "stochastic individual based models of adaptive dynamics". Mathematically, these are measure valued continuous time Markov processes that describe the dynamics of populations of individuals that are subject to the evolutionary mechanisms of birth, death, mutation and competition/interaction. They have been introduced to furnish simple mathematical models that allow to derive basic ferature of the biological theory of adaptive dynamics. This will mainly involve the derivation of scaling limits with respect to population size, mutation rates, and mutations step size.

**Literature:**

**I will write lecture notes as we go.**

**Further literature:**

**1) **Ethier, Stewart N.; Kurtz, Thomas G. Markov processes. Characterization and convergence. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. *John Wiley & Sons, Inc., New York,*

**2) **Bansaye, Vincent; Méléard, Sylvie Stochastic models for structured populations. Scaling limits and long time behavior. Mathematical Biosciences Institute Lecture Series. Stochastics in Biological Systems, 1.4. *Springer, Cham; *

**3)**Dawson, Donald.** **Introductory Lectures on Stochastic Population Systems. 2017.

For a popularised historical survey, the following is great reading:

**4)** Mukherjee, Siddartha. The Gene. Bodley, 2016.

**Requirements: Good knowledge of probability theory on the level of Stochastic Processes or better Markov processes**

**Complementary lecture series "What medicine wants from mathematics"**

A series of talks from members of the faculty of medecine will take place on **Thursdays, 9h s.t**. before the main lecture. The idee is to get an impression of what questions people form the life sciences have where they may expect help from mathematicians, The series will begin on **April 18 ** in Room 1.008 with a talk by Prof. Dr. Michael Hölzel. More details will be posted as soon as possible.

**The lecture on Tueday, April 23, has to be cancelled!**