A nonlinear continuous-time model for a semelparous species

Periodical semelparous insects such as cicadas and May beetles exhibit synchronization in age classes such that only one age class is present at any point of time. This leads to outbreaks of adults as they all reach maturity around the same time. Discrete-time models of semelparous species have shown that this type of synchronous cycling can occur as a result of greater between-class competition relative to within-class competition. However, relatively few studies have examined continuous-time models of semelparous species. Here we develop a continuous-time model for a semelparous species using a technique called the linear chain trick to convert a non-linear McKendrick partial differential equation into a finite system of ordinary differential equations. We represent semelparity by a birth function whose age distribution can be made arbitrarily narrow. We show that a Hopf bifurcation may occur in this model as a result of competition between reproducing and non-reproducing classes. This bifurcation leads to stable cycles in which the two classes are out of phase, thus providing a continuous-time counterpart to the synchronous cycles that occur in discrete-time models.