Bernard Derrida (ENS, Paris): Statistical properties of genealogical trees

One can associate to the genealogical tree of each individuals the distribution of repetitions of his ancestors, at a number g of generations in the past. By solving a simple model of a population of constant size N with random mating, one can show that this distribution of repetitions reaches a stationnary shape, with a non trivial power law which can be calculated analytically.
By comparing the genealogical trees of two distinct individuals, one can also show that the two trees become identical within a number log N of generations.

Paulien Hogeweg (Bioinformatics, Utrecht): Evolution of Morphogenesis

Biotic systems are preeminently multilevel systems in which processes ad different space and time scales interact, Moreover this property seems in many cases to be essential for the behavior.
In this talk I will introduce a modeling approach aimed at exploring and exploiting the multilevel nature of biotic systems, in the context of studying morphogenesis. We use the 2 scale CA-like CPM model to represent cells and their adhesion, and extend this representation on the one hand with differential gene regulation, and on the other hand with evolution. This way we can identify 'generic' morphogenetic mechanisms and their coupling to cell differentiation, cell growth and division and cell death. Moreover we study the evolutionary dynamics which occurs under this highly non-linear genotype-phenotype mapping.

Antal Jarai (CWI, Amsterdam): Thermodynamic limit of the Abelian sandpile model in Z^d

The Abelian sandpile model (ASM) was introduced by Bak, Tang and Wiesenfeld as a model exhibiting so called self-organised critical behaviour. Roughly speaking, self-organised criticality arises when a stochastic dynamics drives a system towards a state that is characterised by power laws. The mathematical study of the ASM, initiated by Dhar, has revealed a rich structure, making the analysis of the ASM more tractable than other models, and mathematically interesting in itself. By a deep observation of Dhar and Majumdar, the ASM can be mapped onto the uniform spanning tree, that, at least heuristically, explains the appearance of power laws.
In this talk I will describe the correspondence between the ASM and spanning trees, and discuss the existence of the infinite volume limit for the stationary measure of the model. This is joint work with S. R. Athreya. I will also mention ongoing research joint with F. Redig, which concerns the construction of dynamics in the infinite volume.

Vadim Malyshev (INRIA, Rocquencourt): Stochastic chemical kinetics: history and new problems

There are N_j molecules of types j=1,...,J and a list of possible reactions. Each molecule has extra degree of freedom: the energy. Collisions are assumed to have some rates (thus defining continuous time Markov chain) and the energies of the products are defined by some probability kernel. In some cases invariant measures for this process can be found for finite N_j and for the limiting Boltzman equation. Relations with nonequilibrium chemical thermodynamics are given as well.

Ronald Meester (Vrije Universiteit, Amsterdam): A mathematical analysis of the Bak-Sneppen model for evolution

The Bak-Sneppen model has been proposed as a simple model for evolution and is supposed to exhibit self-organised critical behaviour. I will introduce the model, explain the terms and discuss some rigorous mathematical results, which come close to proving some conjectures by phycisists. In particular, I will discuss the limit of the stationary distribution as the system size tends to infinity.

Jacek Miekisz (Wydzial Matematyki, Warszawa): Long-run behavior in stochastic evolutionary games

Socio-economic systems can be viewed as systems of many interacting agents or players. We will explore similarities and differences between systems of many interacting players maximizing their individual payoffs and particles minimizing their interaction energy. In particular, we will compare the notion of a Nash equilibrium in game theory with that of a ground state in classical lattice-gas models.i
We will study the long-time behavior of a discrete-time dynamics in spatial games in which players adapt to their neighbors by choosing with a high probability the strategy which maximizes the sum of the payoffs obtained from individual games and with a small probability, representing the noise of the system, they make mistakes. We will investigate such stochastic dynamics in the limits of an infinite number of players and zero noise and present examples where the order of taking these limits is essential.

Livio Triolo (Dipartimento di Matematica, Roma): Space structures and different scales for many-component biosystems

The right scale of description, and more generally, the matching between different scales, in modelling physical or biological systems, presents a substantial interest and related difficulties.
This applies for instance to competition/diffusion models of different species, or to the invasion process by an aggressive population against a native one, like tumor cells in a normal tissue.
The main point in these investigations is to understand the fate of the system in the long run.
It may be possible that under some circumstances the evolution strongly depends on the space structures: a space homogeneous situation behaves in a very different way than a space-patterned one. Moreover if singularities, like cusps or filaments, are macroscopically present, a local matching asymptotics may be needed to reveal their structure.
At more fundamental level, in the evolution of a many-component system like the ones introduced above, clusters may form: this can be considered as a strategy of survival as the evolution runs at very different pace, providing a bridge between different regimes. Some concrete examples from biomathematical modelling will be given.