Journal
BULLETIN OF MATHEMATICAL BIOLOGY
Volume 81, Issue 4, Pages 1122-1142Publisher
SPRINGER
DOI: 10.1007/s11538-018-00554-0
Keywords
Periodical cicada; Allee effects; Leslie matrix
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Funding
- Santa Fe Institute
- National Science Foundation under INSPIRE Grant [1344187]
- Direct For Biological Sciences [1344187] Funding Source: National Science Foundation
- Division Of Environmental Biology [1344187] Funding Source: National Science Foundation
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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 km2, 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.
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