Limits to selection on standing variation in an asexual population

We consider how a population responds to directional selection on standing variation, with no new variation from recombination or mutation. Initially, there are N individuals with trait values z ₁ , …, z _N ; the fitness of individual i is proportional to . The initial values are drawn from a distribution ψ with variance V ₀ ; we give examples of the Laplace and Gaussian distributions. When selection is weak relative to drift , variance decreases exponentially at rate 1 /N ; since the increase in mean in any generation equals the variance, the expected net change is just NV ₀ , which is the same as Robertson’s (1960) prediction for a sexual population. In contrast, when selection is strong relative to drift , the net change can be found by approximating the establishment of alleles by a branching process in which each allele competes independently with the population mean and the fittest allele to establish is certain to fix. Then, if the probability of survival to time of an allele with value z is P ( z ), with mean , the winning allele is the fittest of survivors drawn from a distribution . When N is large, there is a scaling limit which depends on a single parameter ; the expecte d ultimate change is for a Gaussian distribution, and for a Laplace distribution (where 𝒲is the product log function). This approach also reveals the variability of the process, and its dynamics; we show that in the strong selection regime, the expected genetic variance decreases as ∼ t ⁻³ at large times. We discuss how these results may be related to selection on standing variation that is spread along a linear chromosome.