MARK KAC SEMINAR

This seminar will present results obtained in collaboration with Pierre-Emmanuel Jabin (Univ. Maryland), Sylvie Méléard (Ecole Polytechnique) and Gaël Raoul (CEFE, Univ. Montpellier 2).

Our goal is to describe a mathematical approach to study the long time evolution of biological populations. The principle of the method can be applied to various structured birth-death-mutation stochastic models, but we will focus in this talk on an evolutionary model of population with competition for resources through a chemostat-type model, where individuals consume several common resources which are constantly supplied. This model describes for example the adaptation of bacteria interacting with their environment composed of resources. Bacteria are characterized by continuous traits describing their intake efficiency for each resource.

The population of bacteria is assumed to follow a discrete (multitype birth and death) Markovian stochastic dynamics, and the dynamics of resource concentrations is governed by deterministic ordinary differential equations (ODEs). Three parameter scalings are involved in the analysis of the model, dealing with large population, rare mutations and small effect of mutations. We first consider a combination of rare mutations and large population, with appropriate time scaling (i.e. fast resource and birth and death dynamics and a slow mutation rate). We prove that the population behaves on the mutation time scale as a jump process describing successive fast mutant invasions between evolutionary equilibria. Next, in the small mutation steps limit, we prove that this process converges to an ODE known as the canonical equation of adaptive dynamics, and we are able to characterize the trait values where the population diversifies through a process known as evolutionary branching. A key role is played in the analysis by general results on the long-time behavior of deterministic chemostat systems of ODEs with arbitrary number of species.

We consider a growth model on a graph G where clusters stop growing (freeze) as soon as their diameter reaches N, which is a parameter of the model. We investigate the large N behaviour of the process for the cases where G is a binary tree and the triangular lattice.

We find two quite different behaviours: on the binary tree the N-parameter models converge to the model of Aldous, where clusters freeze as soon as they become infinite. However, Benjamini and Schramm showed that this infinite parameter process does not exist on the triangular lattice. Hence the analogue of the convergence result above cannot hold for the triangular lattice. It turns out that, as N goes to infinity, the density of the frozen clusters goes to 0. Moreover we show, roughly speaking, that the frozen clusters give a tiling of the lattice where the typical non-frozen parts have diameter less than N, but of order N.

In the talk we discuss the results above, motivate their proofs, and in the case of the triangular lattice we show connections to critical and near-critical percolation and to scaling limits.