Fixation probability of beneficial mutations in a fluctuating population

We compute an accurate approximation to the probability of fixation for a beneficial mutation in a population fluctuating with a stationary distribution of population size. The population dynamics are described by the theta-logistic model with environmental variance, assuming that the population size is large enough to ignore demographic variance. We show that stochastic fluctuations of population size reduce the probability of fixation. However, it is not the magnitude of the population fluctuations per se that creates this reduction. Only the environmental variance has a substantial effect on the probability of fixation. The strength of density dependence (or expected return time to equilibrium) and the functional form of density-regulation, given by the parameter theta in the theta-logistic model, have little effect on the fixation probability. Effective population size based on harmonic mean population size will therefore underestimate the expected fixation rate of beneficial mutations in fluctuating populations.