We consider the dynamics imposed by natural selection on the populations of two competing, sexually reproducing, haploid species. In this setting, the fitness of any genome varies over time due to the changing population mix of the competing species; crucially, this fitness variation arises naturally from the model itself, without the need for imposing it exogenously as is typically the case. Previous work on this model [14] showed that, in the special case where each of the two species exhibits just two phenotypes, genetic diversity is maintained at all times. This finding supported the tenet that sexual reproduction is advantageous because it promotes diversity, which increases the survivability of a species. In the present paper we consider the more realistic case where there are more than two phenotypes available to each species. The conclusions about diversity in general turn out to be very different from the two-phenotype case. Our first result is negative: namely, we show that sexual reproduction does not guarantee the maintenance of diversity at all times, i.e., the result of [14] does not generalize. Our counterexample consists of two competing species with just three phenotypes each. We show that, for any time~$t_0$ and any $\varepsilon>0$, there is a time $t\ge t_0$ at which the combined diversity of both species is smaller than~$\varepsilon$. Our main result is a complementary positive statement, which says that in any non-degenerate example, diversity is maintained in a weaker, ``infinitely often'' sense. Thus, our results refute the supposition that sexual reproduction ensures diversity at all times, but affirm a weaker assertion that extended periods of high diversity are necessarily a recurrent event.

翻译：我们研究了自然选择对两个竞争的有性繁殖单倍体物种群体施加的动力学过程。在此设定下，任意基因组的适应度会因竞争物种群体组成的动态变化而随时间波动；关键在于，这种适应度变化源于模型本身，无需像通常做法那样人为外生施加。该模型的先前研究[14]表明，在两种物种各仅呈现两种表型的特殊情形下，遗传多样性始终得以维持。这一发现支持了有性繁殖因促进多样性而增强物种生存能力的论点。本文研究了更符合现实的情形：每个物种可具有两种以上表型。结果表明，关于多样性的结论总体上与双表型情形截然不同。我们的首个结果为否定性结论：即证明有性繁殖不能保证多样性始终维持，换言之，[14]的结论不具备普适性。反例由仅含三种表型的两个竞争物种构成。我们证明，对任意时刻$t_0$和任意$\varepsilon>0$，存在某个$t\ge t_0$使得两个物种的总多样性小于$\varepsilon$。主要结果是互补的肯定性陈述：任何非退化案例中，多样性能在较弱的"无限频繁"意义上得以维持。因此，我们的结果反驳了有性繁殖保证持续多样性的假设，但确认了较弱的论断——高多样性时期必然会反复出现。