Ancestral processes with selection: Branching and Moran models

We consider two versions of stochastic population models with mutation andselection. The first approach relies on a multitype branching process; here,individuals reproduce and change type (i.e., mutate) independently of eachother, without restriction on population size. We analyze the equilibriumbehaviour of this model, both in the forward and in the backward direction oftime; the backward point of view emerges if the ancestry of individuals chosenrandomly from the present population is traced back into the past. The second approach is the Moran model with selection. Here, the populationhas constant size N. Individuals reproduce (at rates depending on their types),the offspring inherits the parent's type, and replaces a randomly chosenindividual (to keep population size constant). Independently of thereproduction process, individuals can change type. As in the branching model,we consider the ancestral lines of single individuals chosen from theequilibrium population. We use analytical results of Fearnhead (2002) todetermine the explicit properties, and parameter dependence, of the ancestraldistribution of types, and its relationship with the stationary distribution inforward time.

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