A Hybrid Model for the Population Dynamics of Periodical Cicadas

  • Jonathan MachtaEmail author
  • Julie C. Blackwood
  • Andrew Noble
  • Andrew M. Liebhold
  • Alan Hastings


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 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 life span.


Periodical cicada Allee effects Leslie matrix 



The authors thank the Santa Fe Institute for sponsoring three working groups during which much of this work was carried out. JM, AH and AN acknowledge support from the National Science Foundation under INSPIRE Grant No. 1344187. We are grateful to Prof. Odo Diekmann for providing key insights that motivated Theorems 1 and 2.

Supplementary material

11538_2018_554_MOESM1_ESM.nb (53 kb)
Supplementary material 1 (nb 53 KB)


  1. Behncke H (2000) Periodical cicadas. J Math Biol 40(5):413–431MathSciNetCrossRefzbMATHGoogle Scholar
  2. Blackwood J, Meyer A, Noble A, Machta J, Hastings A, Liebhold A (2018) Competition and stragglers as mediators of developmental synchrony in periodical cicadas. Am Nat 192(4):479–489CrossRefGoogle Scholar
  3. Briggs C, Sait S, Begon M, Thompson D, Godfray HCJ (2000) What causes generation cycles in populations of stored-product moths? J Anim Ecol 69:352–366CrossRefGoogle Scholar
  4. Bulmer MG (1977) Periodical insects. Am Nat 111(982):1099–1117CrossRefGoogle Scholar
  5. Cushing JM, Henson SM (2012) Stable bifurcations in semelparous Leslie models. J Biol Dyn 6(sup2):80–102MathSciNetCrossRefGoogle Scholar
  6. Davydova N, Diekmann O, van Gils S (2005) On circulant populations. I. The algebra of semelparity. Linear Algebra Appl 398:185–243 special Issue on Matrices and Mathematical BiologyMathSciNetCrossRefzbMATHGoogle Scholar
  7. de Roos AM, Persson L (2013) Population and community ecology of ontogenetic development. Princeton University Press, PrincetonCrossRefGoogle Scholar
  8. Diekmann O, Planque R (2018) The winner takes it all: how semelparous insects can become periodical. bioRxiv
  9. Dybas HS, Lloyd M (1974) The habitats of 17-year periodical cicadas (homoptera: Cicadidae: Magicicada spp.). Ecol Monogr 44(3):279–324CrossRefGoogle Scholar
  10. Gasciogne JC, Lipcius RN (2004) Allee effects driven by predation. J Appl Ecol 41:801–810CrossRefGoogle Scholar
  11. Gurney W, Nisbet R, Lawton J (1983) The systematic formulation of tractable single-species population models incorporating age structure. J Anim Ecol 52(2):479–495CrossRefGoogle Scholar
  12. Hastings A (1987) Cycles in cannibalistic egg-larval interactions. J Math Biol 24(6):651–666MathSciNetCrossRefzbMATHGoogle Scholar
  13. Hastings A, Costantino RF (1987) Cannibalistic egg-larva interactions in tribolium: an explanation for the oscillations in population numbers. Am Nat 130(1):36–52CrossRefGoogle Scholar
  14. Hastings A, Costantino RF (1991) Oscillations in population numbers: age-dependent cannibalism. J Anim Ecol 60(2):471–482CrossRefGoogle Scholar
  15. Heath JE (1968) Thermal synchronization of emergence in periodical “17-year” cicadas (homoptera, cicadidae, magicicada). Am Midland Nat 80(2):440–448CrossRefGoogle Scholar
  16. Heliövaara K, Väisänen R, Simon C (1994) Evolutionary ecology of periodical insects. Trends Ecol Evol 9(12):475–480CrossRefGoogle Scholar
  17. Hoppensteadt FC, Keller JB (1976) Synchronization of periodical cicada emergences. Science 194(4262):335–337CrossRefGoogle Scholar
  18. Karban R (1982) Increased reproductive success at high densities and predator satiation for periodical cicadas. Ecology 63(2):321–328CrossRefGoogle Scholar
  19. Karban R (1984) Opposite density effects of nymphal and adult mortality for periodical cicadas. Ecology 65(5):1656–1661CrossRefGoogle Scholar
  20. Karban R (1997) Evolution of prolonged development: a life table analysis for periodical cicadas. Am Nat 150(4):446–461CrossRefGoogle Scholar
  21. Koenig WD, Liebhold AM (2013) Avian predation pressure as a potential driver of periodical cicada cycle length. Am Nat 181(1):145–149CrossRefGoogle Scholar
  22. Leonard DE (1964) Biology and ecology of magicicada septendecim (l.) (hemiptera: Cicadidae). J N Y Entomol Soc 72(1):19–23Google Scholar
  23. Lloyd M, Dybas HS (1966) The periodical cicada problem II. Evolution. Evolution 20:466–505CrossRefGoogle Scholar
  24. Lloyd M, White JA (1976) Sympatry of periodical cicada broods and the hypothetical four-year acceleration. Evolution 30(4):786–801CrossRefGoogle Scholar
  25. Mjølhus E, Wikan A, Solberg T (2005) On synchronization in semelparous populations. J Math Biol 50(1):1–21MathSciNetCrossRefzbMATHGoogle Scholar
  26. Tanaka Y, Yoshimura J, Simon C, Cooley JR, Ki Tainaka (2009) Allee effect in the selection for prime-numbered cycles in periodical cicadas. Proc Nat Acad Sci 106(22):8975–8979CrossRefGoogle Scholar
  27. Webb G (2001) The prime number periodical cicada problem. Discrete Contin Dyn Syst Ser B 1(3):387–399MathSciNetCrossRefzbMATHGoogle Scholar
  28. White J, Lloyd M (1979) 17-year cicadas emerging after 18 years: a new brood? Evolution 33(4):1193–1199CrossRefGoogle Scholar
  29. White J, Lloyd M, Zar JH (1979) Faulty eclosion in crowded suburban periodical cicadas: populations out of control. Ecology 60(2):305–315CrossRefGoogle Scholar
  30. White JA, Lloyd M (1975) Growth rates of 17 and 13-year periodical cicadas. Am Midland Nat 94(1):127–143CrossRefGoogle Scholar
  31. Wikan A (2012) On nonlinear age-and stage-structured population models. J Math Stat 8(2):311–322CrossRefGoogle Scholar
  32. Williams KS, Simon C (1995) The ecology, behavior, and evolution of periodical cicadas. Ann Rev Entomol 40:269–295CrossRefGoogle Scholar
  33. Williams KS, Smith KG, Stephen FM (1993) Emergence of 13-yr periodical cicadas (cicadidae: Magicicada): phenology, mortality, and predator satiation. Ecology 74(4):1143–1152CrossRefGoogle Scholar
  34. Yamanaka T, Nelson W, Uchimura K, Bjørnstad O (2012) Generation separation in simple structured life cycles: models and 48 years of field data on a tea tortrix moth. Am Nat 179(1):95–109CrossRefGoogle Scholar
  35. Yoshimura J, Hayashi T, Tanaka Y, Tainaka K, Simon C (2009) Selection for prime-number intervals in a numerical model of periodical cicada evolution. Evolution 63(1):288–294CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2019

Authors and Affiliations

  1. 1.Department of PhysicsUniversity of MassachusettsAmherstUSA
  2. 2.Santa Fe InstituteSanta FeUSA
  3. 3.Department of Mathematics and StatisticsWilliams CollegeWilliamstownUSA
  4. 4.Department of Environmental Science and PolicyUniversity of CaliforniaDavisUSA
  5. 5.Northern Research Station, U.S. Forest ServiceMorgantownUSA

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