One of the best-known phenomena in the insect world is the unusual recurrence of various populations of cicada. There aren't any cicadas out here on the West Coast, where I live, but they are endemic to the Northeast. The periodical cicadas (there are non-periodical cicadas, apparently) are notorious for having life cycles that are synchronized to one of two (relatively) large primes: 13 years and 17 years. The big question, of course, is why: Why do cicadas have life cycles that are synchronized in this fashion?
One could divide the 13-year cicadas into 13 distinct subgroups, depending on which year they emerged, and divide the 17-year cicadas into 17 subgroups along the same principle. Physical observation of cicadas, as shown in the Wikipedia plot summary, reveals that only about half of the 13+17 = 30 subgroups actually manifest in the United States (where the cicada is native), however, with two subgroups becoming extinct within the last century or two. Nonetheless, the periodicity is well enough established that there should be a rational explanation of this phenomenon.
One historically proposed reason for the synchronization has been that the long recurrence time limits exposure of the species above ground to predators, and that when they are exposed, there are so many of them that predators cannot possibly decimate them (a fact well attested by the unfortunate farmers who have to deal with them), thereby ensuring the continued existence of the population. Although this is surely part of the answer, it only explains why the period is long; it doesn't explain why the period isn't 12 or 15, for instance, rather than 13 or 17. These latter periods would only provide additional benefit if the likely predators of the cicada likewise had a life cycle punctuated by years of inactivity, which turns out not to be so.
A more successful explanation involves hybridization. It is hypothesized that whatever mechanism governs the return of the population after however many years is based on a biological clock that is adjusted to activate periodically, and that if a 13-year cicada were to mate with a 17-year cicada, the result would be a substantial number of cicadas with unpredictable, but likely shorter, periods. (Too long, and the individuals would die of old age, anyway.) Such offspring would be more vulnerable to predation, so there is an evolutionary premium placed against hybridization. Computer simulation studies show, however, that if we assume an initial species-wide distribution of a variety of periods—some prime-numbered, some composite—the prime-numbered periods remain, but so do some of the composite periods.
This 2009 paper, by Tanaka et al., explains away the remaining composite periods by means of something called the Allee effect. In many population dynamics analyses, it is assumed that the fewer instances of a species exist, the more likely any instance is to survive—it being presumed that there is no disadvantage owing to an excess of resources. There may be no such disadvantage, but it is nonetheless the case that there are situations where the reverse is true, for small populations: the greater the population, the more likely any individual is to survive to reproduce, because it benefits from the increased support and robustness of the larger population, up until the point where that larger population represents more competition than cooperation. This reverse but very natural-seeming tendency constitutes the Allee effect.
Tanaka and company simulated the cicada species under a very simple hybridization model, both with and without the Allee effect, starting with subgroups with a range of periods varying from 10 through 20 years. They found that without the Allee effect, there was broad survival of all of the cicada subgroups, with the 16-year subgroup thriving the best. But with the Allee effect, the result was startlingly different: Only those cicada subgroups with periods of 13, 17, or 19 years survived, depending on some of the initial parameters.
Since the actual mechanism of the periodicity is not well understood yet, this study is more suggestive than dispositive, but the results are provocative.