An Evolutionary Theory for the Variability Hypothesis

Abstract: 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.

• The paper presents a framework demonstrating how mating selectivity influences variability in sexually dimorphic species.
• It employs both discrete-time and continuous-time models to analyze short-term and long-term evolutionary dynamics.
• The study offers robust mathematical and biological insights, highlighting the heritability of phenotypic variability.

An Evolutionary Theory for the Variability Hypothesis: A Professional Overview

Theodore P. Hill's academic paper presents a nuanced biostatistical framework centered around the "selectivity-variability" principle, addressing a question first raised by Charles Darwin regarding the evolutionary development of variability differences between sexes in sexually dimorphic species. This principle proposes that the prevalence of variability in one sex relative to the selectivity of the opposite sex influences evolutionary outcomes, interpreting variability without assuming inherent mean differences or selectivity.

Key Theoretical Contributions

Underpinning the theory is a dual-fold principle: if sex A is selective in mating, subpopulations of sex B with greater variability will prevail across generations; conversely, a non-selective sex A supports the ascendance of less variable subpopulations within sex B. This theoretical proposition is mathematically framed through:

• A Discrete-Time Model: Utilizing normally distributed fitness values, this model evaluates short-term evolutionary dynamics, determining overrepresentation in successive generations based on selectivity thresholds.
• A Continuous-Time Model: This deterministic approach explores asymptotic behaviors in population sizes over longer timelines, leveraging ODEs reminiscent of those used in evolutionary game theory.

Analytical Implications

These formulations imply that variability can be mediated through selective pressures, presenting a plausible evolutionary explanation for observable sex differences in variability. The models hinge on the implicit assumption that variability is heritable, supported by evidence from genetic and animal husbandry domains, suggesting a heritable component to phenotypic variability.

Practical and Theoretical Implications

The research provides a mathematical grounding to potential mechanisms explaining Darwin's observation of greater male variability noted across empirical studies in numerous species. This insight proposes that evolutionary pressures, mediated by selectivity differences tied to parental investment and reproductive rates, drive variability differences observed between sexes.

Future Directions and Challenges

The theoretical model introduced by Hill invites numerous extensions, such as incorporating temporal variability shifts in desirability, modeling multivariate desirability factors, or exploring non-biological analogs of the selectivity-variability principle in other scientific domains. These extensions could provide a more comprehensive understanding of the interplay between evolutionary pressures and variability.

Further empirical validation and refinement of the selectivity-variability principle are essential, particularly in clarifying the precise mechanisms by which variability is inherited and exploring its implications within contemporary evolutionary contexts, including human cultural influences.

Concluding Thoughts

Hill's mathematical exposition on evolutionary variability offers significant contributions to the understanding of sex differences, invoking fundamental evolutionary principles with robust mathematical approach. While the paper refrains from claiming universality, it strategically opens academic pathways for deepening the exploration of variability from both evolutionary biology and mathematical modeling perspectives. These insights invite continued inquiry and empirical testing, with substantial potential to inform and refine evolutionary theories concerning sex-based variability.