Probabilistic approaches to the evolution of some biological systems

07.02.2018, 14:15  –  Haus 9, Raum 2.22
Institutskolloquium

Jean Bertoin (Universität Zürich), Anton Wakolbinger (Goethe-Universtität Frankfurt)

The colloquium begins at 14:15  with a talk

Large populations over many generations:

from Wright and Fisher to experimental evolution

by Anton Wakolbinger (Goethe University, Frankfurt)

Abstract:
The mathematical modelling of biological evolution has become a challenge as well as a source of inspiration for probability theory ever since the pioneering work of Wright, Fisher and Haldane in the first half of the past century. For instance, branching processes and dynamical laws of large numbers allow to model and analyze the effects and the interplay of evolutionary forces like  mutation, selection and random reproduction over various time scales. We illustrate this along a picture-book version of biological evolution provided by the famous Long Term Evolution Experiment of Lenski. The talk is based on joint work with A. Gonzáles Casanova, N. Kurt, L. Yuan, E. Baake and S. Probst.


At 15:15 there will be a coffee break followed by the second talk

A probabilistic approach to growth and fragmentation of particles

  by Jean Bertoin (University of Zürich)

Abstract:
The growth-fragmentation equation describes a system of growing and  dividing particles, and arises in models of cell division, protein polymerisation and even telecommunications protocols. Several important questions about the equation concern the asymptotic
 behaviour of solutions at large times: at what rate do they converge to zero or infinity, and what does the asymptotic profile of the solutions look like? Does the rescaled solution converge to its  asymptotic profile at an exponential speed? These questions have
traditionally been studied using analytic techniques such as entropy methods or splitting of operators. In this talk, we present a probabilistic approach to the study of this asymptotic behaviour. We   characterise the rate of decay or growth in terms of a Markov process. We then identify the Malthus exponent and the asymptotic profile in terms of a related Markov process, and give a spectral interpretation in terms of the growth-fragmentation operator and its dual.

This talk is based on a joint work with Alex Watson, Manchester University.