Muller’s ratchet clicks in finite time

We consider the accumulation of deleterious mutations in an asexual population, a phenomenon known as Muller’s ratchet, using the continuous time model proposed by Alison et al. (2009) [4]. We show that for any parameter λ>0λ>0 (the rate at which mutations occur), for any α>0α>0 (the toxicity of the mutations) and for any size N>0N>0 of the population, the ratchet clicks a.s. in finite time. That is to say the minimum number of deleterious mutations in the population goes to infinity a.s.