  We reconsider the deterministic haploid mutation-selection equation with two
types. This is an ordinary differential equation that describes the type
distribution (forward in time) in a population of infinite size. This paper
establishes ancestral (random) structures inherent in this deterministic model.
In a first step, we obtain a representation of the deterministic equation's
solution (and, in particular, of its equilibrium) in terms of an ancestral
process called the killed ancestral selection graph. This representation allows
one to understand the bifurcations related to the error threshold phenomenon
from a genealogical point of view. Next, we characterise the ancestral type
distribution by means of the pruned lookdown ancestral selection graph and
study its properties at equilibrium. We also provide an alternative
characterisation in terms of a piecewise-deterministic Markov process.
Throughout, emphasis is on the underlying dualities as well as on explicit
results.
