Metastability between the clicks of the Muller ratchet

Mauro Mariani, Etienne Pardoux, Aurélien Velleret

Published 2020-07-29Version 1

We prove the existence and uniqueness of a quasi-stationary distribution for three stochastic processes derived from the model of the Muller ratchet. This model has been originally introduced to quantify the limitations of a purely asexual mode of reproduction in preventing, only through natural selection, the fixation and accumulation of deleterious mutations. As we can see by comparing the proofs, not relying on the discreteness of the system or an imposed upper-bound on the number of mutations makes it much more necessary to specify the behavior of the process with more realistic features. The third process under consideration is clearly non-classical, as it is a stochastic diffusion evolving on an irregular set of infinite dimension with hard killing at an hyperplane. We are nonetheless able to prove results of exponential convergence in total variation to the quasi-stationary distribution even in this case. The parameters in this last result of convergence are directly related to the core parameters of the Muller ratchet effect (although the relation is very intricate). The speed of convergence to the quasi-stationary distribution deduced from the infinite dimensional model extends to the approximations with a large yet finite number of potential mutations. Likewise, we have uniform upper-bounds of the empirical distribution of mutations in the population under quasi-stationarity.