A Hybrid Model for the Population Dynamics of Periodical Cicadas

Abstract

In addition to their unusually long life cycle, periodical cicadas, Magicicada spp., provide an exceptional example of spatially synchronized life stage phenology in nature. Within regions (“broods”) spanning 50,000–500,000 km 2 , adults emerge synchronously every 13 or 17 years. While satiation of avian predators is believed to be a key component of the ability of these populations to reach high densities, it is not clear why populations at a single location remain entirely synchronized. We develop nonlinear Leslie matrix-type models of periodical cicadas that include predation-driven Allee effects and competition in addition to reproduction and survival. Using both analytical and numerical techniques, we demonstrate the observed presence of a single brood critically depends on the relationship between fecundity, competition and predation. We analyze the single-brood, two-brood and all-brood equilibria in the large life span limit using a tractable hybrid approximation to the Leslie matrix model with continuous time competition in between discrete reproduction events. Within the hybrid model, we prove that the single-brood equilibrium is the only stable equilibrium. This hybrid model allows us to quantitatively predict population sizes and the range of parameters for which the stable single-brood and unstable two-brood and all-brood equilibria exist. The hybrid model yields a good approximation to the numerical results for the Leslie matrix model for the biologically relevant case of a 17-year life span.