Diversity in evolutionary dynamics

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 genotype varies over time due to the changing population mix of the competing species; crucially, unlike other approaches to ensuring time-varying fitnesses, in our model, this fitness variation arises intrinsically from fixed-fitness interactions between the species themselves. Previous work on this model 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 article, 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 above two-phenotype result does not generalize. Our counterexample consists of two competing species with just three phenotypes each. We show that, for any time t0 and any ε > 0, there is a time t ≥ t0 at which the combined diversity of both species is smaller than ε. Our main result is a complementary positive statement, which says that in any non-degenerate system, diversity is maintained in a weaker, 'infinitely often' sense. Here, non-degeneracy is the condition that the game possesses no strict pure Nash equilibria. 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.