PUB - Publikationen an der Universität Bielefeld

In a Moran model with population size N, two types, mutation and selection, let h(k)(N) be the probability that the common ancestor is fit, given that the current number of fit individuals is k. First, we express h(k)(N) in terms of the tail probabilities of an appropriate random variable L-N. Next, we show that, when N tends to infinity (without any rescaling of parameters or time), L-N converges to a geometric random variable. We also obtain a formula for h(x), the limit of h(k)(N) when k/N tends to x is an element of (0, 1). In a second step, we describe two ways of pruning the ancestral selection graph (ASG) leading to the notions of relevant ASG and of pruned lookdown ASG (pruned LD-ASG). We use these objects to provide graphical derivations of the aforementioned results. In particular, we show that L-N is distributed as the asymptotic number of lines in the relevant ASG and as the stationary number of lines in the pruned LD-ASG. Finally, we construct an asymptotic version of the pruned LD-ASG providing a graphical interpretation of the function h. (C) 2016 Elsevier B.V. All rights reserved.