Sperm competition games: external fertilization and "adaptive"' infertility.

We develop a model of a continuous fertilization process in which eggs and sperm are shed simultaneously, and in which the eggs are fertilized at a rate proportional to sperm density surrounding the egg mass. The model derives the ESS size and number of sperm in an ejaculate of an externally-fertilizing animal such as a fish species, in which the probability or intensity of sperm competition varies. It also predicts the ESS level of infertility (eggs remaining unfertilized after all sperm have died). Sperm size is assumed to increase sperm competitive ability (e.g. by increasing speed) and is also assumed to affect sperm longevity (either positively or negatively). Ejaculate expenditure is traded off against expenditure on obtaining further spawnings, and size and number of sperm can vary independently. The model predicts that the ESS ejaculate expenditure (product of sperm size and number) should increase, and that the ESS infertility should decrease with sperm competition intensity measured across species. Other results depend on the way that sperm size affects longevity. The available biological evidence suggests that longevity decreases with sperm size, probably because the main increase is in tail length which increases sperm energy expenditure. In this case, sperm size should increase with sperm competition intensity from an optimum at zero competition which maximizes the total distance travelled by the entire ejaculate in its lifetime, to an optimum for maximum sperm competition which maximizes the product of speed and sperm number. However, if longevity increases with sperm size, then the non-competitive optimal sperm size is greater than that for maximum competition, so that sperm size decreases with sperm competition intensity. Sperm numbers typically increase with sperm competition intensity, and always so if sperm competition is high enough, though decreases are possible over a range of low sperm competition intensity if (i) sperm longevity decreases with sperm size, and (ii) infertility is high enough.