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