Sperm competition games: a comparison of loaded raffle models and their biological implications

Our main aim is to compare the additive model, due to Mesterton-Gibbons, and the multiplicative model, due to Parker, of sperm allocation under sperm competition, when other influences are treated in the same way. We first review these (and other) models and their foundations, leading to a generalization of the multiplicative model. Sperm is assumed to cost energy, and this constraint is incorporated differently in the two models. These give the same results in the random-roles situation when the males occupy roles (of first and second to mate) randomly: the number of sperm ejaculated in the favoured role is greater than that in the disfavoured role by an amount that depends on the effect of sperm limitation (i.e. the probability that there is insufficient sperm to ensure full fertility). If the latter is negligible, or the fertilization raffle fair, this difference is zero, as Parker found originally. In the constant roles situation (where males of a particular type always occupy the same role) the predictions differ: the additive model has the same predictions as in the random roles case, but the multiplicative model predicts that males of the type occupying the favoured role ejaculate less than males of the type occupying the disfavoured role, in accord with Parker's original conclusion. The fitnesses of the two types of male can be calculated in the multiplicative model: the fitness of the favoured male is usually higher, even if he has to expend more energy in "finding" a female, e.g. through fighting, etc. These conclusions relate to inter-male behaviour (i.e. of different male types), as distinct from intra-male behaviour (i.e. of a given male when in different roles). We analyse situations in which one male type has some probability of acting in its less usual role: calculations with varying amounts of sperm limitation are presented. It is found that the presence of a male of a different type has an effect on intra-male ejaculate behaviour, which also depends critically on the role usually occupied. We conclude that the multiplicative model is the more accurate model and provides more information. Some experimental data on sperm numbers are used to find the effects of sperm limitation. For species which conform to the loaded raffle model, sperm limitation typically has small or negligible effects: in this case, we argue that empiricists should look for equal ejaculates in the two roles when studying random role situations; when roles are occupied non-randomly average sperm expenditure should be greater by male types typically occupying the disfavoured role, but within a male type, expenditure should be greater in the role it typically occupies.