Population dynamics and genealogies

By Amaury Lambert

We start with stochastic models of genealogies in discrete time, distinguishing models where the population size is fixed (models of Cannings, Wright-Fisher and Moran; compositions of bridges) with models where the population size fluctuates randomly (processes of Bienaymé-Galton-Watson, Jirina; birth-and-death processes, logistic branching process).

Scaling limits of these models can be seen as genealogies of continuous populations. The continuous analogue of models with fixed size is the stochastic flow of bridges of Bertoin and Le Gall. That of branching models is the continuous-state branching process. Both processes have diffusion limits, called respectively the Fisher-Wright diffusion and the Feller diffusion. Connections between the two kinds of models are also studied, and special attention is given to extinction (probability, expected time, conditioning).

When (sub)populations are bound to extinction, the distribution of their size conditional on extinction, either not to have occurred yet or to occur much later, converges in many situations. The limits are called quasi-stationary distributions and provide a rigorous notion for the dynamics of populations that seem stable, at least at the human timescale. We show quasi-stationarity for all models introduced previously, tackling the problem of uniqueness of quasi-stationary distributions, and comparing different conditionings (Q-process).

In a slightly different setting, we prove that the contour process of splitting trees with general lifetime is a killed Lévy process, and use this observation to derive a certain number of properties of these trees, including connections between Lévy processes and branching processes (one by Le Gall and Le Jan and another one by Bertoin and Le Gall), and a generalization of the coalescent point process of Aldous and Popovic. These two kinds of results open up to more general problems such as Ray-Knight type theorems on the one hand, and coalescent processes (Kingman, Möhle and Sagitov, Pitman, Schweinsberg…) on the other hand.

If time allows, we would also like to give a flavour of the vast literature on structured populations, where individuals are given a type which is inherited, faithfully or not (mutations). This would include the Chinese restaurant process, Ewens’ sampling formula, and the GEM distribution.