On the opportunity for sexual selection, the Bateman gradient and the maximum intensity of sexual selection.

Bateman's classic paper on fly mating systems inspired quantitative study of sexual selection but also resulted in much debate and confusion. Here, I consider the meaning of Bateman's principles in the context of selection theory. Success in precopulatory sexual selection can be quantified as a "mating differential," which is the covariance between trait values and relative mating success. The mating differential is converted into a selection differential by the Bateman gradient, which is the least squares regression of relative reproductive success on relative mating success. Hence, a complete understanding of precopulatory sexual selection requires knowledge of two equally important aspects of mating patterns: the mating differential, which requires a focus on mechanisms generating covariance between trait values and mating success, and the Bateman gradient, which requires knowledge of the genetic mating system. An upper limit on the magnitude of the selection differential on any sexually selected trait is given by the product of the standard deviation in relative mating success and the Bateman gradient. This latter view of the maximum selection differential provides a clearer focus on the important aspects of precopulatory sexual selection than other methods and therefore should be an important part of future studies of sexual selection.