Sperm competition games: Sperm size (mass) and number under raffle and displacement, and the evolution of P2

We examine models for evolution of sperm size (i.e. mass m) and number (s) under three mechanisms of sperm competition at low 'risk' levels: (i) raffle with no constraint on space available for competing sperm, (ii) direct displacement mainly by seminal fluid, and (iii) direct displacement mainly by sperm mass. Increasing sperm mass increases a sperm's 'competitive weight' against rival sperm through a diminishing returns function, r(m). ESS total ejaculate expenditure (the product m^*s^*) increases in all three models with sperm competition risk, q. If r(m), or ratio r'(m)/r(m), is independent of ESS sperm numbers, ESS sperm mass remains constant, and the sperm mass/number ratio (m^*/s^*) therefore decreases with risk. Dependency of sperm mass on risk can arise if r(m) depends on competing sperm density (sperm number / space available for sperm competition). Such dependencies generate complex relationships between sperm mass and number with risk, depending both on the mechanism and how sperm density affects r(m). While numbers always increase with risk, mass can either increase or decrease, but m^*/s^* typically decreases with risk unless sperm density strongly influences r(m). Where there is no extrinsic loading due to mating order, ESS paternity of the second (i.e. last) male to mate (P₂) under displacement always exceeds 0.5, and increases with risk (in the raffle P₂=0.5). Caution is needed when seeking evidence for a sperm size-number trade off. Although size and number trade-off independently against effort spent on acquiring matings, their product, m^*s^*, is invariant or fixed at a given risk level, effectively generating a size-number trade off. However, unless controlled for the effects of risk, the relation between m^* and s^* can be either positive or negative (a positive relation is usually taken as evidence against a size-number trade off).