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In addition to their unusually long life cycle, periodical cicadas, {\itMagicicada} spp., provide an exceptional example of spatially synchronized lifestage phenology in nature. Within regions ("broods") spanning 50,000 to 500,000km$^2$, adults emerge synchronously every 13 or 17 years. While satiation ofavian predators is believed to be a key component of the ability of thesepopulations to reach high densities, it is not clear why populations at asingle location remain entirely synchronized. We develop nonlinear Lesliematrix-type models of periodical cicadas that include predation-driven Alleeeffects and competition in addition to reproduction and survival. Using bothanalytical and numerical techniques, we demonstrate the observed presence of asingle brood critically depends on the relationship between fecundity,competition, and predation. We analyze the single-brood, two-brood andall-brood equilibria in the large life-span limit using a tractable hybridapproximation to the Leslie matrix model with continuous time competition inbetween discrete reproduction events. Within the hybrid model we prove that thesingle-brood equilibrium is the only stable equilibrium. This hybrid modelallows us to quantitatively predict population sizes and the range ofparameters for which the stable single-brood and unstable two-brood andall-brood equilibria exist. The hybrid model yields a good approximation to thenumerical results for the Leslie matrix model for the biologically relevantcase of a 17-year lifespan.

A hybrid model for the population dynamics of periodical cicadas