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 mutate independently of each other, without restriction on population size. 
We analyze the equilibrium behaviour of such models, 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 stationary
state of the reversed process, the so-called  ancestral distribution, turns out as a key 
for the study of  mutation-selection balance. In particular, a general variational principle 
is derived that relates the present population, the ancestral population, and the mean fitness; 
it relies on the theory of large deviations as applied to the type process on individual lines.

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, i.e., mutate. 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.

This is joint work with Robert Bialowons