“Limits to Selection on Standing Variation in an Asexual Population”, Nick Barton, Himani Sachdeva2023-05-14 (cloning, natural selection)⁠:

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 z₁, …, z_N; the fitness of individual i is proportional to e^z_i. The initial values are drawn from a distribution ψ with variance V₀; we give examples of the Laplace and Gaussian distributions.

When selection is weak relative to drift (N√V₀≪1), variance decreases exponentially at rate 1/N; since the increase in mean in any generation equals the variance, the expected net change is just NV₀, which is the same as Robertson1960’s prediction for a sexual population. In contrast, when selection is strong relative to drift (N√V₀≫1), 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 t~1∕√V₀ of an allele with value z is P(z), with mean P̄, the winning allele is the fittest of NP̄. survivors drawn from a distribution ψP∕P̄. When N is large, there is a scaling limit which depends on a single parameter N√V₀; the expected ultimate change is ~√2 log(1.15N√V₀) for a Gaussian distribution, and ~(W [0.36/ (N√V₀)]-1)/√2 for a Laplace distribution (where W is 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.