Inference from clines stabilized by frequency-dependent selection

Frequency-dependent selection against rare forms can maintain clines. For weak selection, s, in simple linear models of frequency-dependence, single locus clines are stabilized with a maximum slope of between square root of s/square root of 8 sigma and square root of s/square root of 12 delta, where sigma is the dispersal distance. These clines are similar to those maintained by heterozygote disadvantage. Using computer simulations, the weak-selection analytical results are extended to higher selection pressures with up to three unlinked genes. Graphs are used to display the effect of selection, migration, dominance, and number of loci on cline widths, speeds of cline movements, two-way gametic correlations ("linkage disequilibria"), and heterozygote deficits. The effects of changing the order of reproduction, migration, and selection, are also briefly explored. Epistasis can also maintain tension zones. We show that epistatic selection is similar in its effects to frequency-dependent selection, except that the disequilibria produced in the zone will be higher for a given level of selection. If selection consists of a mixture of frequency-dependence and epistasis, as is likely in nature, the error made in estimating selection is usually less than twofold. From the graphs, selection and migration can be estimated using knowledge of the dominance and number of genes, of gene frequencies and of gametic correlations from a hybrid zone.