A continuum approximation to the population dynamics of periodical cicadas

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 to 500,000 km, 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 non-linear Leslie matrixtype models of periodical cicadas that include predation-driven Allee effects and competition in addition to reproduction and survival. Using both numerical and analytical 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 steady state in the large life-span limit using a tractable continuum approximation and show that it yields a good approximation to the numerical results for the biologically relevant case of a 17-year lifespan. J. Machta Department of Physics, University of Massachusetts, Amherst, MA 01003, USA and Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, New Mexico 87501, USA E-mail: [email protected] J.C. Blackwood Department of Mathematics and Statistics, Williams College, Williamstown, MA 01267, USA A.E. Noble Department of Environmental Science and Policy, University of California, Davis, CA 95616, USA A.M. Liebhold Northern Research Station, U.S. Forest Service, Morgantown, WV, 26505, USA A. Hastings Department of Environmental Science and Policy, University of California, Davis, CA 95616, USA ar X iv :1 61 2. 01 42 9v 1 [ qbi o. PE ] 5 D ec 2 01 6 2 Jonathan Machta et al.