No abstract

For a panmictic population of constant size evolving under neutrality, Kingman's coalescent describes the genealogy of a population sample in equilibrium. However, for genealogical trees under selection, not even expectations for most basic quantities like height and length of the resulting random tree are known. Here, we give an analytic expression for the distribution of the total tree length of a sample of size n under low levels of selection in a two-alleles model. We can prove that trees are shorter than under neutrality under genic selection and if the beneficial mutant has dominance h < 1/2, but longer for h > 1/2. The difference to neutrality is O(α 2 ) for genic selection with selection intensity α and O(α) for other modes of dominance. * Genic selectionConsider a Moran model of size N, where every individual has type either • or •, selection is genic, type • is advantageous with selection coefficient α, and mutation is bi-directional. In other words, consider a population of N (haploid) individuals with the following transitions:1. Every pair of individuals resamples at rate 1; upon such a resampling event, one of the two involved individuals dies, the other one reproduces.

We study a mutation-selection model with a fluctuating environment. More precisely, individuals in a large population are assumed to have a modifier locus determining the mutation rate u ∈ [0, ϑ] at a second locus with types v ∈ [0, 1]. In addition, the environment fluctuates, meaning that individual types change their fitness at some high rate. Fitness only depends on the type of the second locus. We obtain general limit results for the evolution of the allele frequency distribution for rapidly fluctuating environments. As an application, we make use of the resulting Fleming-Viot process and compute the fixation probabilities for higher mutation rates in the special case of two bi-allelic loci in the limit of small fitness differences at the second locus.