Speaker
Description
Toivonen and Fromhage (2018) proposed a competition model between size-based and age-based populations to study the evolution of periodicity in periodical cicadas. They demonstrated that the effect of predator satiation triggers the replacement of a size-based population by an age-based one, even when the former possesses a higher reproductive number. This suggests that the perfect periodicity observed in periodical cicadas could have evolved from proto-periodicity. In this talk, we modify the model of Toivonen and Fromhage (2018) to derive a piece-wise linear model. This serves as a generalization of the periodical cicada model proposed by May (1979), which was itself a modification of the Hoppensteadt and Keller (1976) model. Using this modified framework, we show that the proto-periodicity in the size-based population observed by Toivonen and Fromhage (2018) is realized by a specific periodic orbit, which we term an "essentially single-class orbit". We further establish the stability conditions for this orbit, demonstrating that an essentially single-class orbit is more likely to be stabilized than an equilibrium solution if reproduction occurs in later age-classes. Finally, we mathematically prove that a size-based population is invaded and eventually eradicated by an age-based population. These results mathematically complement the results by Toivonen and Fromhage (2018).
Bibliography
@article{1,
title = {Synchronization of {Periodical} {Cicada} {Emergences}},
volume = {194},
issn = {0036-8075, 1095-9203},
url = {https://www.science.org/doi/10.1126/science.987617},
doi = {10.1126/science.987617},
language = {en},
number = {4262},
urldate = {2026-05-10},
journal = {Science},
author = {Hoppensteadt, Frank C. and Keller, Joseph B.},
month = oct,
year = {1976},
pages = {335--337},
}
@article{2,
title = {Periodical cicadas},
volume = {277},
copyright = {http://www.springer.com/tdm},
issn = {0028-0836, 1476-4687},
url = {https://www.nature.com/articles/277347a0},
doi = {10.1038/277347a0},
language = {en},
number = {5695},
urldate = {2026-05-10},
journal = {Nature},
author = {May, Robert M.},
month = feb,
year = {1979},
pages = {347--349},
}
@article{ToivonenFromhage2018,
author = {Toivonen, Jaakko and Fromhage, Lutz},
title = {Modelling the Evolution of Periodicity in the Periodical Cicadas},
journal = {Evolutionary Ecology Research},
year = {2018},
volume = {19},
pages = {107--125},
url = {http://evolutionary-ecology.com/abstracts/v19/3125.html}
}
