Mathematical properties of mutation-selection models

This article provides an overview of the mathematical properties of various deterministic mutation-selection models. Many, even basic, results about models with more than two alleles per locus have only been obtained in the past two decades and are scattered throughout the literature. I review the fundamental results about existence and stability of equilibria for classical mutation-selection models with a finite number of alleles, for models like the stepwise-mutation model, and for the continuum-of-alleles model. For all these models, first- and second-order approximations for the equilibrium mean fitness and the mutation load are presented. Then I examine mutation-stabilizing selection balance. As an application of the results on the mutation load, second-order approximations for the equilibrium genetic variance can be derived. Various approximations for this equilibrium variance, which have been obtained in the literature, are concisely reviewed and their ranges of validity are compared. In particular, the dependence of the equilibrium variance on the parameters of the underlying genetic system, such as the form of the mutation distribution or the linkage map, is investigated.