Spatial patterns in the distributions of polygenic characters

The spatial patterns in the mean and variance of a quantitative character that result from the interaction of spatially varying, optimizing selection and gene flow are considered. The model analyzed is an extension of those of Kimura (1965) and Lande (1976) for the distribution of a quantitative character maintained in a population by independent mutations. For weak selection, it is shown that there is only a small effect of gene flow on the variance of the character and that the mean value changes on a length scale that is large compared to the average dispersal distance. As in models of clines in allele frequencies, it is possible to define a “characteristic length” in terms of the average dispersal distance and strength of selection. The characteristic length is the smallest length scale environmental change to which the mean value of the character can significantly respond. It is also shown that, for weak selection, an asymmetry in dispersal can result in a significant shift in location of a cline. By considering an infinite linear cline in optimal values, it is shown that gene flow can increase the variance only when there is sufficient mixing in each generation of individuals from locations with different means. A model of selection in different niches is also considered. There is an increase in variance due to the effective weakening of the intensity of selection because of the differences in optimal values in different niches.The implications of the different models for maintenance of genetic polymorphism are discussed. Under some conditions gene flow can produce a significant increase in heterozygosity. It is also argued that spatial variation in selection on a polygenic character can be much more effective in increasing heterozygosity than temporal variation because of the potentially greater increase in phenotypic variance. The difference between some of the results for polygenic characters from those of similar models of one and two locus systems is accounted for by the fact that for normally distributed polygenic characters, changes in the variance are effectively decoupled from changes in the mean.