Ball M A, Parker G A
Department of Applied Mathematics and Theoretical Physics, University of Liverpool.
J Theor Biol. 1996 May 21;180(2):141-50. doi: 10.1006/jtbi.1996.0090.
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.
我们构建了一个连续受精过程的模型，在这个模型中，卵子和精子同时排出，并且卵子的受精率与围绕卵团的精子密度成正比。该模型推导了体外受精动物（如鱼类）射精中精子的ESS大小和数量，其中精子竞争的概率或强度会有所不同。它还预测了不育的ESS水平（所有精子死亡后仍未受精的卵子）。假设精子大小会提高精子的竞争能力（例如通过提高速度），并且还假设其会影响精子寿命（无论是正向还是负向）。射精支出与获得更多产卵机会的支出相互权衡，精子的大小和数量可以独立变化。该模型预测，ESS射精支出（精子大小与数量的乘积）应该增加，并且随着跨物种测量的精子竞争强度增加，ESS不育水平应该降低。其他结果取决于精子大小影响寿命的方式。现有的生物学证据表明，寿命会随着精子大小的增加而降低，可能是因为主要增加的是尾巴长度，这增加了精子的能量消耗。在这种情况下，精子大小应该随着精子竞争强度的增加而增加，从竞争为零时的最优值（此时整个射精在其生命周期内行进的总距离最大化），到精子竞争最大时的最优值（此时速度与精子数量的乘积最大化）。然而，如果寿命随着精子大小的增加而增加，那么非竞争性的最优精子大小大于最大竞争时的最优值，因此精子大小会随着精子竞争强度的增加而减小。精子数量通常会随着精子竞争强度的增加而增加，如果精子竞争足够激烈则总是如此，不过在精子竞争强度较低的范围内，如果（i）精子寿命随着精子大小的增加而降低，并且（ii）不育程度足够高，精子数量也可能会减少。
