Exploring cyclic net reproduction

This paper advances a new approach that provides closed form expressions for the birth trajectory produced by a regime of changing vital rates. An exponentiated sinusoidal net maternity function is considered in detail, as populations with cyclically varying net maternity are of particular interest because of their connection to the Easterlin hypothesis. The dynamics of the model are largely determined by the ratio of the population's generation length (A) to the period of cyclicity (T), and relatively simple expressions are found for the phase difference and relative amplification of the birth and net reproduction functions. More generally, an analytical expression for a population's birth trajectory is derived that applies whenever net reproductivity can be written as an exponentiated Fourier series. In the cyclic model, Easterlin's inverse relationship between cohort size and cohort fertility holds whenever the phase difference is zero. At other phase differences, the birth-reproduction equations have the form of predator-prey equations. The present analytical approach may thus be relevant to analyses of interacting populations.