Some Mathematical Models in Evolutionary Genetics

Since most traits of evolutionary or economic importance are determined by several or many genes, a proper understanding of the evolution of such traits requires the study of multilocus models. Accordingly, we present the basic models and the most fundamental results about the evolutionary dynamics of a population in which selection acts on many gene loci. First, important aspects of the classical case, when selection acts on a single diploid locus with multiple alleles, are highlighted. Then the general model with selection on a finite number of recombining multiallelic loci is treated, with the focus on asymptotic and convergence results, including generalizations of Fisher's Fundamental Theorem. Extensions dealing with migration or frequency-dependent selection are briefly outlined. Finally, the selection response and the evolution of (multivariate) quantitative traits is investigated.