The tournament ratchet's clicktime process, and metastability in a Moran model

Muller's ratchet, in its prototype version, models a haploid, asexual population whose size~$N$ is constant over the generations. Slightly deleterious mutations are acquired along the lineages at a constant rate, and individuals carrying less mutations have a selective advantage. In the classical variant, an individual's selective advantage is proportional to the difference between the population average and the individual's mutation load, whereas in the ratchet with {\em tournament selection} only the signs of the differences of the individual mutation loads matter. In a parameter regime which leads to slow clicking (i.e. to a loss of the currently fittest class at a rate $\ll 1/N$) we prove that the rescaled process of click times of the tournament ratchet converges as $N\to \infty$ to a Poisson process. Central ingredients in the proof are a thorough analysis of the metastable behaviour of a two-type Moran model with selection and deleterious mutation (which describes the size of the fittest class up to its extinction time) and a lower estimate on the size of the new fittest class at a clicktime.