A mutation-selection model for general genotypes with recombination

We investigate a continuous time, probability measure-valued dynamical system
that describes the process of mutation-selection balance in a context where the
population is infinite, there may be infinitely many loci, and there are weak
assumptions on selective costs. Our model arises when we incorporate very
general recombination mechanisms into a previous model of mutation and
selection from Steinsaltz, Evans and Wachter (2005) and take the relative
strength of mutation and selection to be sufficiently small. The resulting
dynamical system is a flow of measures on the space of loci. Each such measure
is the intensity measure of a Poisson random measure on the space of loci: the
points of a realization of the random measure record the set of loci at which
the genotype of a uniformly chosen individual differs from a reference wild
type due to an accumulation of ancestral mutations. Our motivation for working
in such a general setting is to provide a basis for understanding
mutation-driven changes in age-specific demographic schedules that arise from
the complex interaction of many genes, and hence to develop a framework for
understanding the evolution of aging.
We establish the existence and uniqueness of the dynamical system, provide
conditions for the existence and stability of equilibrium states, and prove
that our continuous-time dynamical system is the limit of a sequence of
discrete-time infinite population mutation-selection-recombination models in
the standard asymptotic regime where selection and mutation are weak relative
to recombination and both scale at the same infinitesimal rate in the limit.