  In addition to their unusually long life cycle, periodical cicadas, {\it
Magicicada} spp., provide an exceptional example of spatially synchronized life
stage phenology in nature. Within regions ("broods") spanning 50,000 to 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 lifespan.
