N. Berestycki (Cambridge), N. Champagnat (Sophia-Antipolis), P. Collet (Palaiseau), V. Limic (Marseille), H. Metz (Leiden), A. Wakolbinger (Frankfurt)

We try to describe the effect of migratory fluxes and spatial structure on the genealogy of a population. This question leads to the study of systems of particles performing simple random walk on a given graph, and where particles coalesce according to a certain mechanism (typically, Kingman's coalescent) when they are on the same site. We obtain various asymptotic results for this process, at both small and large time scales, which are of interest in the context of population genetics. We will also discuss some conjectures related to the hydrodynamic limiting behavior of these systems. Joint work with Omer Angel, Alan Hammond and Vlada Limic.

Nicolas Champagnat (Sophia-Antipolis): A study of evolutionary branching in a logistically regulated population.

We are trying to give a mathematical basis to the notion of evolutionary branching introduced by Metz et al. in 1996 in the framework of adaptive dynamics, which studies the long-term evolution of ecologically explicit populations. It is said that evolutionary branching occurs in a population when the evolutionary dynamics drives the population from an (essentially) monotype population to an (essentially) 2-types population. We consider a finite stochastic population with birth, mutation, death and selection due to a logistic-type competition (nonlinearity in the death rates), where individuals are characterized by a finite dimensional phenotypic trait (such as body size, rate of food intake,...). We first consider the combination of the limits of large population and rare mutation, which allows one to simplify the dynamics as a jump process over the trait space describing the successive invasions of mutants, and which is based on a time-scale separation. This process allows transitions from monomorphic populations (with only one trait) to dimorphic populations. This process is therefore a natural one to study evolutionary branching. In particular, when the size of mutations in the trait space converges to 0, we are able to justify the branching criterion proposed by biologists, to precisely describe the transition from a monomorphic population to a dimorphic one, and to explicitely compute the time-scale of mutation.
This is joint work partly with Michel Bénaïaut;m and Sylvie Méléard, and partly with Anton Bovier.

Pierre Collet (Palaiseau): Quasi stationary distributions and applications to diffusions in population dynamics.

I will briefly review the notions of quasi stationary distributions and Yaglom limits from the point of view of stochastic processes and dynamical systems. I will then describe some results on the existence of these measures with a particular application to a diffusion dynamics with absorption coming from population dynamics.

Vlada Limic (Marseille): Coalescent with rebirth.

The goal of the talk is to motivate the construction of the process called the {\em coalescent with rebirth} that arises naturally in the asymptotic study of spatial Moran models and similar particle systems evolving on two-dimensional lattice.

The duality of the spatial coalescent to the spatial Moran models turns out to be a very powerful tool for studying their asymptotic behavior, due to the fact that the partitions formed from coalescing random walks started at ``sparse'' configurations converge to the Kingman coalescent,on appropriate scale. The critical recurrent dimension $d=2$, which is also interesting from the perspective of applications in biology, seems to be the most difficult one to handle. Here the monochrome cluster containing the origin has size on the order of $n^\alpha$ at time $n$, where $\alpha$ is a random quantity.

The coalescent with rebirth emerges in the study of the asymptotic behavior of the spatial coalescent on varying spatial scales in two dimensions. It evolves formally according to the following dynamics: starting with infinitely many particles/blocks, each pair of blocks coalesces at rate 1, and immediately upon each coalescing event (in which two blocks merge to one) a new particle is reborn/created. Its predecessor is the ``frozen coalescent'' of Cox and Griffeath (1986).

Based on a joint work with Andreas Greven and Anita Winter.

Hans Metz (Leiden): Relating the effective population sizes of adaptive dynamics and random genetic drift.

Adaptive dynamics is a practically very effective mathematical framework for dealing with long term biological evolution in "realistic" ecological settings. It differs from more classical approaches to modelling evolutionary change, which assume constant fitnesses, by its focus on the population dynamical basis for those fitnesses, and hence on their inevitable change over evolutionary time.

The greater realism at the ecological end is bought by making a different set of (this time genetically unrealistic) simplifying assumptions, the two main ones being: (i) a separation of the population dynamical and mutational time scales, and (ii) clonal inheritance. One of the more powerful tools of AD is its so-called Canonical Equation, a differential equation describing how the trait (vector) dominating the population changes over evolutionary time. The derivation of the CE is based on still one more simplifying assumption: (iii) small mutational step size. (Luckily, when these step sizes are small the time scale separation assumption, which then is easily flauted, becomes less of importance.) Its main components are the so-called selection gradient, the mutational covariance matrix, and an effective population size that is the product of the ordinary population size and a scalar depending on the life history of the species.

The CE has been derived by Dieckmann & Law and rigorously underpinned by Champagnat within the context of simple ODE population models. However, with little effort the result can be extended to physiologically and spatially structured as well as Mendelian populations. In all cases the equation can be written in the same form by the ploy of introducing that effective population size.

Intriguingly, when individuals are born equal (i.e. there is no need to distinguish different individual states at birth such as spatial location or phase of an environmental cycle), for various simple life histories the effective population size appearing in the canonical equation turns out to be equal to the effective population size introduced in population genetics to describe neutral genetic drift. This equality can also be deduced using a standard population genetical formula obtained from a diffusion approximation (but through a limit that still appears to lack a rigorous basis).

As a bonus the CE view on effective population sizes also gives simple unambiguous results for overlapping generation models, for which the population genetics literature so far only contains heuristic proposals based on far from clear biological assumptions. A next question is whether a similar relation holds when there is more than one birth state. I hope to have at least some results on this issue by the time of the meeting.

Anton Wakolbinger (Frankfurt): How often does the ratchet click?

In an asexually reproducing population where (slightly) deleterious mutations accumulate along the individual lineages and the individual selection disadvantage is assumed to be proportional to the number of accumulated mutations, the current bes class will eventually disappear from the population, a phenomenon known as Muller's ratchet. A question which is simple to ask but hard to answer is: 'How fast is the best type lost'? (or 'How many times does the ratchet klick?') We highlight the underlying mathematical problem from different points of view, review various diffusion approximations and discuss rigorous results in the case of a simplified model. This is joint work with Alison Etheridge and Peter Pfaffelhuber.