Survival under high mutation rates

We consider a stochastic model for an evolving population. We show that in the presence of genotype extinctions the population dies out for a low mutation probability but may survive for a high mutation probability. This turns upside down the widely held belief that above a certain mutation threshold a population cannot survive. 1 A model with genotype extinctions There seems to be a consensus in theoretical biology that mutations are helpful for the survival of a population but that too many mutations are not, see Nowak and May (2000), Eigen (2002) and Manrubia et al. (2010). We propose to use stochastic models to challenge this belief. In fact, we will show that there are situations where the population survives for large mutation probability but dies out for small mutation probability! That is, we propose to turn upside down the idea that too many mutations are necessarily bad. We now describe our model. Let μ be a continuous probability distribution with support contained in [0,∞) and let r ∈ [0, 1]. Start with one individual at time 0, and sample a birth rate λ from the distribution μ. Individuals give birth at rate λ and die at rate 1. Every time there is a birth the new individual: (i) with probability 1 − r keeps the same birth rate λ as its parent, and (ii) with probability r is given a new birth rate λ, sampled independently of everything else from the distribution μ. Furthermore, every time a new genotype appears (i.e. a new λ) we associate the genotype to a time T (independently of everything else) sampled from a fixed distribution ν. At time T all the individuals of this genotype are killed and the genotype disappears from the population. We think of r as the mutation probability and the birth rate of an individual as representing the genotype of the individual. Since μ is assumed to be continuous, a genotype cannot appear more than once in the evolution of the population. Genotype extinctions happen when all the individuals with the same genotype die. These so called background extinctions have been going on since the beginning of life, see for instance Mayr (2001). We say that the population survives if there is a strictly positive probability of having at all times at least one individual alive. Note that no genotype can survive forever so the population may only survive if it generates infinitely many genotypes. Our first result is a necessary and sufficient condition for survival of this population. We start the population with a single individual. D R A FT Theorem 1. The population survives if and only if m(r) = rE[Λ ∫ T