The stability of symmetric solutions to polygenic models

Analysis of multilocus evolution is usually intractable for more than n approximately 10 genes, because the frequencies of very large numbers of genotypes must be followed. An exact analysis of up to n approximately 100 loci is feasible for a symmetrical model, in which a set of unlinked loci segregate for two alleles (labeled "0" and "1") with interchangeable effects on fitness. All haploid genotypes with the same number of 1 alleles can then remain equally frequent. However, such a symmetrical solution may be unstable: for example, under stabilizing selection, populations tend to fix any one genotype which approaches the optimum. Here, we show how the 2(n)x2(n) stability matrix can be decomposed into a set of matrices, each no larger than nxn. This allows the stability of symmetrical solutions to be determined. We apply the method to stabilizing and disruptive selection in a single deme and to selection against heterozygotes in a linear cline.