  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.
