Mutation-selection models: Branching, ancestry and maximum principle

We consider the evolution of populations under the joint action of mutation and selection. To this end, mutation and reproduction are modelled as a multi-type branching process, of which we consider both the forward and the backward direction of time. The stationary state of the reversed process is the ancestral distribution, which turns out as a key for the study of mutation-selection balance. For the single-step mutation model in the continuous limit, the equilibrium properties of both the present and the ancestral population are obtained from a maximum principle. In particular, exhaustive criteria are available for the presence and absence of the error-threshold phenomenon.