A hybrid model for the population dynamics of periodical cicadas

TL;DR

This paper introduces a hybrid mathematical model combining Leslie matrices and continuous dynamics to explain the synchronized emergence and population stability of periodical cicadas, emphasizing predation, competition, and fecundity effects.

Contribution

It develops a novel hybrid Leslie matrix model that predicts conditions for stable single-brood populations and explains synchronization in periodical cicadas.

Findings

Single-brood equilibrium is the only stable state.

Model accurately predicts population sizes for 17-year cicadas.

Existence of multiple equilibria depends on fecundity, competition, and predation.

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.