  We study a continuous-time dynamical system that models the evolving
distribution of genotypes in an infinite population where genomes may have
infinitely many or even a continuum of loci, mutations accumulate along
lineages without back-mutation, added mutations reduce fitness, and
recombination occurs on a faster time scale than mutation and selection. Some
features of the model, such as existence and uniqueness of solutions and
convergence to the dynamical system of an approximating sequence of discrete
time models, were presented in earlier work by Evans, Steinsaltz, and Wachter
for quite general selective costs. Here we study a special case where the
selective cost of a genotype with a given accumulation of ancestral mutations
from a wild type ancestor is a sum of costs attributable to each individual
mutation plus successive interaction contributions from each $k$-tuple of
mutations for $k$ up to some finite ``degree''. Using ideas from complex
chemical reaction networks and a novel Lyapunov function, we establish that the
phenomenon of mutation-selection balance occurs for such selection costs under
mild conditions. That is, we show that the dynamical system has a unique
equilibrium and that it converges to this equilibrium from all initial
conditions.
