Selection and Polygenic Characters

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Abstract
A mathematical model is developed which describes the effect of selection on polygenic or continuously varying phenotypic characters. For the simplest case the standard deviation of a phenotypic character in a population cannot be greater than √2β where β is the expected deviation of the offspring from the average of the parents' types. This is found to carry over to density-dependent selection, and it indicates that there are limits to the amount of phenotypic diversity, and therefore niche breadth, which a population can maintain. These limits, however, are also subject to selection.

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