Knowledge-independent Invariance Rules for Copula Duration in Dungflies

Abstract

In dung flies, copula duration \((\bar t)\) decreases, and the proportional rate of sperm transfer \((\bar c)\) increases, as male body size becomes larger. From a marginal value approach to optimal copula duration, we show that these relationships result in the product, \(\bar c \cdot \bar t\), remaining approximately constant across the range of male body size. The expected proportion of a female's eggs fertilized by a copulating male, equivalent to 〈1 − e−c·t〉, is likewise invariant with male body size. (Overbars and 〈 〉 refer to the averages over female sizes). We assume that the information or cues a male can perceive about females forms a set of discrete “recognition categories”, each of which is uniquely recognizable by a male, but within which he cannot discriminate. There are then likely to be different male optima for the product \(\bar c \cdot \bar t\) between categories. But the invariance rules still hold within categories, independently of exactly what the recognition categories are, provided that all males perceive the same categories. For example, suppose that males of all sizes categorise females as either 'large', 'medium', or 'small'. Then though the optimal male strategy (product \(\bar c \cdot \bar t\)) for (say) 'large' females may differ from the corresponding optima for the other two categories, it remains constant with male size across all the 'large' females. Further, the product \(\bar c \cdot \bar t\) should remain constant for all male sizes if we take the average across all females, or across any subset of recognition categories. We believe that these conclusions have general applicability and implications for optimal foraging under the marginal value theorem, and demonstrates how we can sometimes make predictions (e.g. the relation between copula duration and male body size in dungflies) without determining exactly what a forager 'knows'.