Diversity in Evolutionary Dynamics

TL;DR

This paper investigates the dynamics of genetic diversity in competing species with multiple phenotypes, revealing that while diversity isn't maintained at all times, it recurs infinitely often due to natural selection.

Contribution

It extends previous models by analyzing multiple phenotypes, showing that sexual reproduction doesn't guarantee constant diversity but ensures its recurrent appearance over time.

Findings

Diversity is not maintained at all times in multi-phenotype models.

Counterexample with three phenotypes shows diversity can drop arbitrarily low.

Diversity recurs infinitely often, indicating a weaker but persistent form of diversity.

Abstract

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.