Selection for prime-number intervals in a numerical model of periodical cicada evolution

Periodical cicadas are known for unusually long and prime-numbered life cycles (13 and 17 years) for insects. To explain the evolution of prime-numbered reproductive intervals (life cycles), the hybridization hypothesis claims that prime numbers greatly reduce the chance of hybridization with other life cycles. We investigate the hybridization hypothesis using a simulation model. This model is a deterministic, discrete population model with three parameters: larval survival per year, clutch size, and emergence success. Reproductive intervals from 10 years to 20 years compete for survival in the simulations. The model makes three key assumptions: a Mendelian genetic system, random mating among broods of different life-cycle lengths, and integer population sizes. Longer life cycles have larger clutch sizes but suffer higher total mortality than shorter life cycles. Our results show that (1) nonprime-numbered reproductive intervals disappear rapidly in comparison to the selection among the various prime-numbered life cycles, (2) the selection of prime-numbered intervals happens only when populations are at the verge of extinction, and (3) the 13- and 17-year prime phenotypes evolve under certain conditions of the model and may coexist. The hybridization hypothesis is discussed in light of other hypotheses for the evolution of periodical cicada life cycles.