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性、有性繁殖、单倍体物种种群施加的动力学。在此设定下，任何基因组的适应度会随着竞争物种种群构成的变化而随时间改变；关键在于，这种适应度变化是模型本身自然产生的，无需像通常情况那样外生施加。先前针对该模型的研究[14]表明，在两个物种各自仅呈现两种表型的特殊情况下，遗传多样性在任何时刻都得以维持。这一发现支持了以下观点：有性繁殖之所以具有优势，是因为它能促进多样性，从而提高物种的生存能力。在本文中，我们考虑了更现实的情况，即每个物种可用的表型超过两种。关于多样性的一般性结论与双表型情况存在显著差异。我们的第一个结果是否定性的：即我们证明有性繁殖并不能保证多样性在所有时刻都得以维持，亦即[14]的结果无法推广。我们的反例由两个竞争物种构成，每个物种仅具有三种表型。我们证明，对于任意时刻~$t_0$ 和任意 $\varepsilon>0$，都存在一个时刻 $t\ge t_0$，使得两个物种的综合多样性小于~$\varepsilon$。我们的主要结果是一个互补的肯定性陈述，即在任何非退化示例中，多样性以一种较弱的“无限频繁”意义得以维持。因此，我们的结果反驳了有性繁殖始终确保多样性的假设，但肯定了一个较弱的断言：高度多样性的延长期必然是循环出现的事件。