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arXiv · 1612.01429

A hybrid model for the population dynamics of periodical cicadas

Jon Machta·Julie Blackwood·Andrew Noble·Andrew Liebhold·Alan Hastings
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Abstract

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.

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Jon Machta, Julie Blackwood, Andrew Noble, Andrew Liebhold, Alan Hastings. 2016-12-05. A hybrid model for the population dynamics of periodical cicadas. https://doi.org/10.1007/s11538-018-00554-0

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