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.2009 Jun 2;106(22):8975-9.
doi: 10.1073/pnas.0900215106. Epub 2009 May 18.

Allee effect in the selection for prime-numbered cycles in periodical cicadas

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Allee effect in the selection for prime-numbered cycles in periodical cicadas

Yumi Tanaka et al. Proc Natl Acad Sci U S A..

Abstract

Periodical cicadas are well known for their prime-numbered life cycles (17 and 13 years) and their mass periodical emergences. The origination and persistence of prime-numbered cycles are explained by the hybridization hypothesis on the basis of their lower likelihood of hybridization with other cycles. Recently, we showed by using an integer-based numerical model that prime-numbered cycles are indeed selected for among 10- to 20-year cycles. Here, we develop a real-number-based model to investigate the factors affecting the selection of prime-numbered cycles. We include an Allee effect in our model, such that a critical population size is set as an extinction threshold. We compare the real-number models with and without the Allee effect. The results show that in the presence of an Allee effect, prime-numbered life cycles are most likely to persist and to be selected under a wide range of extinction thresholds.

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Conflict of interest statement

The authors declare no conflict of interest.

Figures

Fig. 1.
Fig. 1.
Temporal dynamics of periodical cicadas starting with 11 pure broods at 10- to 20-year cycles. (A) Real-number model with the Allee effect (Nc = 100). All but 13-, 17-, and 19-year cycles are extinct at an early stage. The order of population sizes at the end (100 years) is: 17 > 13 > 19. (B) Control: real-number model without the Allee effect (Nc = 0). Parameters areS = 0.95 andE = 0.5. All survive until the end (100 years), and the order of population sizes is 16 > 15 > 18 > 17 > 19 > 14 > 20 @k 13 > 12 > 11 > 10.
Fig. 2.
Fig. 2.
Phase planes of 1,000-year survivals for juvenile survival rateS and adult emergence rateE among 11 pure broods of 10- to 20-year cycles with the Allee effect. The extinction thresholds are set as:Nc = 100 (A) andNc = 300 (B). One or two cycles only (red: 11 only; green: 13 only; black: 14 only; blue: 17 only; pink: 19 only; beige: 11 and 13; brown: 13 and 17; yellow: 17 and 19), three or more cycles (gray) and all extinction (white) are shown. The various cycles (gray) include hybrids.
Fig. 3.
Fig. 3.
Phase planes of population sizes (relative to the initial population sizes) of 1,000-year survivals for juvenile survival ratesS and adult emergence rateE among 11 pure broods of 10- to 20-year cycles without the Allee effect (control;Nc = 0). The population sizes are categorized as increasing or decreasing compared with the corresponding initial population sizes (N0 = 1,000). The populations are shown as: three or more cycles increasing (dark gray); shown prime numbers only increasing with the rest decreasing (green: 13 only; blue: 17 only; pink: 19 only; yellow: 17 and 19); and all decreasing (light gray).
Fig. 4.
Fig. 4.
Individual phase planes of 1,000-year survivals (Fig. 2A) for each of 11 pure broods of 10- to 20-year cycles (numbers shown) with the Allee effect. The extinction thresholds are set asNc = 100. The two parameters are juvenile survival ratesS (y axis: range = 0.91–0.97) and adult emergence rateE (x axis: range: 0.13–0.35). The survival areas are larger in 13- and 17-year cycles, followed by 11- and 19-year cycles. Those of non-prime-numbered cycles are all smaller.
Fig. 5.
Fig. 5.
Surviving cycles among 11 pure broods at 10- to 20-year cycles with different levels of the Allee effect, where the extinction threshold is varied, such thatNc = 0 (no Allee effect), 10, 20, … , 300 (step = 10). The gray cells are surviving cycles.E = 0.24 (A), 0.26 (B), 0.27 (C), and 0.3 (D). Other parameters areS = 0.944, andC = 25.A–D correspond with the points along lineS = 0.944 in the phase plane of Fig. 2A.
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References

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