Department of Mathematics, University of Louisiana at Lafayette, Lafayette, LA 70504-1010, USA.
Math Biosci. 2018 Mar;297:1-11. doi: 10.1016/j.mbs.2018.01.003. Epub 2018 Jan 9.
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.
周期性的孤雌生殖昆虫，如蝉和鳃金龟，在龄期上表现出同步性，即在任何时间点只有一个龄期存在。这导致成虫大量出现，因为它们都在同一时间达到成熟。孤雌生殖物种的离散时间模型表明，这种同步循环是由于类间竞争相对于类内竞争更大而产生的。然而，相对较少的研究检验了孤雌生殖物种的连续时间模型。在这里，我们使用一种称为线性链技巧的技术为一种孤雌生殖物种开发了一个连续时间模型，将非线性 McKendrick 偏微分方程转换为有限的常微分方程组。我们通过一个出生函数来表示孤雌生殖，该函数的年龄分布可以任意变窄。我们表明，由于繁殖和非繁殖类之间的竞争，这个模型可能会发生 Hopf 分岔。这个分岔导致稳定的循环，其中两个类相位不同，从而为离散时间模型中发生的同步循环提供了一个连续时间的对应物。
