When a probabilist embarks in the quest of exploring the natural selection
notably through the conditioning of probability distributions
keywords : quasi-stationary distribution (QSD), survival capacity, Q-process, Large deviation, adaptation, selection, variation
Unique Quasi-Stationary Distribution, with a possibly stabilizing extinction
We establish sufficient conditions for exponential convergence to a unique quasi-stationary distribution in the total variation norm. These conditions also ensure the existence and exponential ergodicity of the Q-process, the process conditioned to never be absorbed. The technique relies on a coupling procedure that is related to Doeblin's type conditions. The main novelty is that we modulate each coupling step depending both on a final horizon of time --for survival-- and on the initial distribution. By this way, we could notably include in the convergence a dependency on the initial condition. As an illustration, we consider a continuous-time birth-death process with catastrophes and a diffusion process describing a (localized) population adapting to its environment.
for processes with discontinuous trajectories
This paper establishes new results which allow to deduce the exponential convergence to a unique quasi-stationary distribution in the total variation norm for quite general strong Markov processes. Specifically, we can treat non-reversible processes with discontinuous trajectories, which seems to be a substantial breakthrough. Considering jumps driven by Poisson Point Processes in four different applications, we intend to illustrate the potential of these results and motivate a quite technical criterion. Such criterion is expected to be much easier to verify than an implied property which is crucial in our proof, namely a comparison of the asymptotic extinction rate between different initial conditions.
Adaptation of a population to a changing environment
under the light of quasi-stationarity
We consider a model of diffusion with jumps intended to illustrate the adaptation of a population to the variation of its environment. Assuming that our deterministic environment is changing regularly towards a constant direction, we obtain the existence and uniqueness of the quasi-stationary distribution, the associated survival capacity and the Q-process. Our approach provides moreover several results of exponential convergence (in total variation for the measures). From these summary information, we can characterize the efficiency at which adaptation occurs, and see if this adaptation is rather internal (renewal of the population from the invasions of mutants) or external (survival would be too low otherwise).
Two level natural selection with a quasi-stationarity approach
In a view for a simple model where natural selection at the individual level is confronted to selection effects at the group level, we consider some individual-based models of some large population subdivided in a large number of groups. We then obtain the convergence to the law of a stochastic process with some Feynman-Kac penalization. To analyze the limiting behavior of this law, we use a recent approach, designed for the convergence to quasistationary distributions, that generalizes the principles of Harris recurrence. We are able to deal with the fixation of the stochastic process and relate the convergence to equilibrium to the one where fixation implies extinction. We notably establish different regimes of convergence. Besides the case of an exponential rate (the rate being uniform over the initial condition), critical regimes with convergence in 1/t are also to notice. We finally address the relevance of such limiting behaviors to predict the long-time behavior of the individual-based model.
joint work in progress with M. Mariani and E. Pardoux
Quasi-stationarity in the Müller's ratchet model
Müller's ratchet model enables to analyse the adaptation stability of a population when deleterious mutations regularly affects new individuals. Here, we condition on the fact that the fittest individuals are still present in the population. Our purpose is to ensure the exponential quasi-ergodicity of the processes (with or without an upper-bound on the number of accumulating deleterious mutations). We also expect to ensure that individuals with too many deleterious mutations play a negligible role both for the survival of the fittest subpopulation and for the quasi-ergodicity of the population on a whole.
Quasi-stationarity and descent from infinity for a population regulated through a local competition
We consider a model of population where the evolution of all individuals is described specifically on an unbounded domain that can represent space and/or their phenotype. Assuming a bounded birth rate, we show that a local competition between too many close individuals and a death rate with polynomial power on the limits of the domain are sufficient to ensure the descent from infinity of the number of individuals. With the further assumption that the individuals move in a diffusive pattern (possibly with interactions), we can then ensure the exponential quasi-ergodicity of the model.
project with the support of D. Kim
Large deviation in the history of suviving processes
I wish to justify that Large Deviation results can be established as an extension of quasi-ergodicity for a process conditioned upon the fact that it is not extinct. A specific motivation is to precise the stability of the profile of mutations for a model of a population facing successfully a progressive environmental change.
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