Common ancestor type distribution: A Moran model and its deterministic limit
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In a Moran model with population size NN, two types, mutation and selection, let hkN be the probability that the common ancestor is fit, given that the current number of fit individuals is kk. First, we express hkN in terms of the tail probabilities of an appropriate random variable LNLN. Next, we show that, when NN tends to infinity (without any rescaling of parameters or time), LNLN converges to a geometric random variable. We also obtain a formula for h(x)h(x), the limit of hkN when k/Nk/N tends to x∈(0,1)x∈(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 LNLN 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 hh.