Sperm-competition games: energy dependence and competitor numbers in the continuous-external-fertilization model

This paper is concerned with an externally fertilizing species where several males compete during spawning. We suppose that the sperm compete continuously until all the eggs are fertilized or all the sperm have died. Our aim is to derive the average number 〈N〉 of competitors, the sperm mass m* and the sperm number sI* in terms of external parameters, for example, the total energy R available to a male during the mating period for species with such periods. We derive an equation relating n*, the number of matings for a male, 〈N〉, the average number of males at a spawning, and σ, the sex ratio. This is used to relate R to 〈N〉, through the energy constraint. We extend further the continuous-fertilization model of Ball & Parker (1996, J. Theor. Biol. 180, 141–150; 1997, J. Theor. Biol. 186, 459–466). This examines evolutionarily stable strategies (ESSs) for the sperm mass and sperm number in the ejaculate. We consider two models. In model 1, the males cannot assess the number of competitors, and their ejaculate effort is shaped by the average number 〈N〉 of competitors. In model 2, males can assess exactly the number Ni of competitors at each spawning, and they can regulate the number si of sperm, but not the mass m of their sperm at a given spawning. The numerical calculations of Ball & Parker (1997) are re-expressed in terms of R. The values of 〈N〉 and the fertility are always greater in model 2 than in model 1 for a given R, but the reverse is true for the average sperm number (si), showing that the use of sperm is more efficient in model 2 than in model 1.