  An elementary biostatistical theory based on a selectivity-variability
principle is proposed to address a question raised by Charles Darwin, namely,
how one sex of a sexually dimorphic species might tend to evolve with greater
variability than the other sex. Briefly, the theory says that if one sex is
relatively selective then from one generation to the next, more variable
subpopulations of the opposite sex will generally tend to prevail over those
with lesser variability. Moreover, the perhaps less intuitive converse also
holds: if a sex is relatively non-selective, then less variable subpopulations
of the opposite sex will prevail over those with greater variability. This
theory requires certain regularity conditions on the distributions, but makes
no assumptions about differences in means between the sexes, nor does it
presume that one sex is selective and the other non-selective. Two mathematical
models of the selectivity-variability principle are presented: a discrete-time
one-step probabilistic model of short-term behavior with an example using
normally distributed perceived fitness values; and a continuous-time
deterministic model for the long-term asymptotic behavior of the expected sizes
of the subpopulations with an example using exponentially distributed fitness
levels.
