Mutation-Selection Models in Population Genetics and Evolutionary Game Theory

The investigation of models including the effects of selection and mutation has a long history in population genetics, which can be traced back to Fisher, Haldane and Wright. The reason is that selection and mutation are important forces in evolution. Selection favours an optimal type and tends to eliminate genetic variation, in particular in quantitative traits. Mutation, on the other hand, is the ultimate source of genetic variability. Traditionally, models with only two alleles per locus have been treated. At the end of the fifties the first general results for multi-allele models with selection but without mutation were proved. In particular, conditions for the existence of a unique and stable interior equilibrium were derived and Fisher’s Fundamental Theorem of Natural Selection was proved to be valid (e.g. Mulholland and Smith, 1959; Scheuer and Mandel, 1959; Kingman, 1961a). It teils that in a one-locus multi-aliele diploid model mean fitness always increases. It is well known now that in models with two loci or more this is wrong in general.