  We consider the mutation--selection differential equation with pairwise
interaction (or, equivalently, the diploid mutation--selection equation) and
establish the corresponding ancestral process, which is a random tree and a
variant of the ancestral selection graph. The formal relation to the forward
model is given via duality. To make the tree tractable, we prune branches upon
mutations, thus reducing it to its informative parts. The hierarchies inherent
in the tree are encoded systematically via tripod trees with weighted leaves;
this leads to the stratified ancestral selection graph. The latter also
satisfies a duality relation with the mutation--selection equation. Each of the
dualities provides a stochastic representation of the solution of the
differential equation. This allows us to connect the equilibria and their
bifurcations to the long-term behaviour of the ancestral process. Furthermore,
with the help of the stratified ancestral selection graph, we obtain explicit
results about the ancestral type distribution in the case of unidirectional
mutation.
