# Théophile DE TRUCHIS DE LAYS

**Phd student**

**Research group : **Mathematical physics

**Thesis**

"Stochastic metastability of high-dimensionnal ecosystems"**Advisor: **Jean-René Chazottes

**Research interests**

*game theory, rare transitions, Boltzmann weights, non-markovian process, macrostate clustering, virtual energy*

Existing metastability models for evolution usually describe individual species or traits (or a few of them) ; rarely are they interested in the dynamics of entire ecosystems. In particular, there is currently a lack of diverse well-defined theoretical models to describe rare and fast transitions between periods of relatively stable behaviour (for long times). The purpose here is then to use a family of models that come from game theory and especially developped by W. Sandholm, that is population games, where a finite number of agents following different strategies gain some payoffs given their encounters with other agents. Such a model enables us to relax some global hypothesis (agents' rationnality, equilibrium knowledge, process reversibility...). Given a thermal parameter, agents can opt at specific times to change strategies toward ones that are better suited, with rates proportionnal to Boltzmann weights of the payoffs of accessible strategies. This thus defines for the vector of the popupation frequencies a Markovian process in the state space of population vectors. This state space having a an exponential size in the number of different strategies, there is a need to study a lower-dimensionnal problem to caracterise metastability, that is, defining macrostates. Thus, a method is used that come from Molecular Dynamics (where molecules stay in or near a given conformation for macroscopic times before switching conformation rapidly and rarely) : Markov State Models. Here, the state space is partitioned in a limited number of macrostates, thus defining an inherited stochastic process between these macrostates, coming directly from the underlying Markov process. This inherited process is, in general, not markovian. However, under certain conditions (on these macrostates and the lag time of the inherited process) it is possible to approximate this inherited process by a markovian one - it is then primordial to define these macrostates well. To that purpose, it is possible to define on the finite state space a virtual energy, by analogy with a Gibbs distribution. This virtual energy is defined on each state as the low-temperature limit of the ratio of the log of the underlying process' invariant law, and the inverse temperature. Finally, defining macrostates as neighbourhoods of minima of this virtual energy enables, a priori, to validate the appropriate conditions and then give a good markovian approximation of the inherited (and thus the underlying) process.

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