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We study a continuous-time dynamical system that models the evolvingdistribution of genotypes in an infinite population where genomes may haveinfinitely many or even a continuum of loci, mutations accumulate alonglineages without back-mutation, added mutations reduce fitness, andrecombination occurs on a faster time scale than mutation and selection. Somefeatures of the model, such as existence and uniqueness of solutions andconvergence to the dynamical system of an approximating sequence of discretetime models, were presented in earlier work by Evans, Steinsaltz, and Wachterfor quite general selective costs. Here we study a special case where theselective cost of a genotype with a given accumulation of ancestral mutationsfrom a wild type ancestor is a sum of costs attributable to each individualmutation plus successive interaction contributions from each $k$-tuple ofmutations for $k$ up to some finite ``degree''. Using ideas from complexchemical reaction networks and a novel Lyapunov function, we establish that thephenomenon of mutation-selection balance occurs for such selection costs undermild conditions. That is, we show that the dynamical system has a uniqueequilibrium and that it converges to this equilibrium from all initialconditions.

Mutation-selection balance with recombination: convergence to equilibrium for polynomial selection costs