Sperm competition I: basic model, ESS and dynamics

We examine models of sperm competition to determine which strategies are evolutionarily stable according to game theory. Games are considered in which the males of a species must divide a fixed amount of sperm between a fixed number of rounds in competition over fertilization of a given set of eggs. Sperm success with a single female is allocated using the raffle principle". A two round model is formulated and we show that the evolutionarily stable strategy (ESS) is a pure strategy in which a male should use at least half his sperm in the first round if given the opportunity to mate. The ESS is unique and globally stable, in contrast to most classical ESSs which are only locally stable. The model is extended to include the effects of sperm replenishment and egg oviposition between rounds. Both serve to increase the ESS amount of sperm inseminated in round one.