This article outlines theoretical models of clines in additive polygenic traits, which are maintained by stabilizing selection towards a spatially varying optimum. Clines in the trait mean can be accurately predicted, given knowledge of the genetic variance. However, predicting the variance is difficult, because it depends on genetic details. Changes in genetic variance arise from changes in allele frequency, and in linkage disequilibria. Allele frequency changes dominate when selection is weak relative to recombination, and when there are a moderate number of loci. With a continuum of alleles, gene flow inflates the genetic variance in the same way as a source of mutations of small effect. The variance can be approximated by assuming a Gaussian distribution of allelic effects; with a sufficiently steep cline, this is accurate even when mutation and selection alone are better described by the ‘House of Cards’ approximation. With just two alleles at each locus, the phenotype changes in a similar way: the mean remains close to the optimum, while the variance changes more slowly, and over a wider region. However, there may be substantial cryptic divergence at the underlying loci. With strong selection and many loci, linkage disequilibria are the main cause of changes in genetic variance. Even for strong selection, the infinitesimal model can be closely approximated by assuming a Gaussian distribution of breeding values. Linkage disequilibria can generate a substantial increase in genetic variance, which is concentrated at sharp gradients in trait means.
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