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
Hoppensteadt and Keller (1976), Synchronization of periodical cicada emergences, Science 194 (4262), 335--337
May (1979), Periodical cicadas, Nature, 277, 347--349
Toivonen and Fromhage (2018). Modelling the evolution of periodicity in the periodical cicadas. Evolutionary Ecology Research, 19, 107--125
