Dimensionless invariants from foraging theory's marginal value theorem.

Copula duration (t) decreases, and proportional rate of sperm transfer (c) increases, with larger male body size in dung flies, so their dimensionless product (c. t) is approximately constant (approximately 2.2). The most recent copulating male fertilizes about 89% of the eggs laid (= 1 - e(-c.t) = 1 - e(-2.2)), independent of his body size. The conditions under which natural selection favors this phenotypic invariance are studied with fitness optimization models. The dimensionless rules for optimal patch residence times are then generalized to cover phenotypic variation in other foraging cases.