Permanence induced by life-cycle resonances: the periodical cicada problem.

Periodical cicadas are known for their unusually long life cycle for insects and their prime periodicity of either 13 or 17 years. One of the explanations for the prime periodicity is that the prime periods are selected to prevent cicadas from resonating with predators with submultiple periods. This paper considers this hypothesis by investigating a population model for periodical predator and prey. The study shows that if the periods of the two periodical species are not coprime, then the predator cannot resist the invasion of the prey. On the other hand, if the periods are coprime, then the predator can resist the invasion of the prey. It is also shown that if the periods are not coprime, then the life-cycle resonance can induce a permanent system, in which no cohorts are missing in both populations. On the other hand, if the periods are coprime, then the system cannot be permanent.