A population model with birth pulses, age structure, and non-overlapping generations

Abstract

We propose a single-species population model based on these assumptions: (i) individuals are sexually immature at birth and are classed as juveniles; (ii) juveniles become sexually mature and are classed as adults if they survive to age τ, where τ is a fixed positive constant called the maturation age; (iii) reproduction occurs in brief periodic episodes called birth pulses; (iv) if an adult is alive at the time of a birth pulse, then it dies immediately afterward. These assumptions may reasonably approximate the life cycles of certain types of insect or fish, in which reproduction occurs at a single particular time of year and adults die shortly after reproducing. Assumptions (i) and (ii) are a simple representation of age structure, and assumption (iv) ensures that the population has non-overlapping generations.In our model, we represent birth pulses (assumption (iii)) as impulsive events, to capture their brevity. The number of offspring at each birth pulse is equal to a “birth function” of the adult population. We consider three such functions - linear, Beverton–Holt, and an extension of the Beverton–Holt function that we call “extended Beverton–Holt”. The per capita mortality rates for juveniles or adults are allowed to be either density-dependent or density-independent. Twelve special cases arise from combining our options for the birth function and per capita mortality rates. We analyse all twelve special cases. In each case, we find: (i) that the model dynamics are driven by a one-dimensional map that arises from the periodic nature of the birth pulses; (ii) that the population may die out or behave periodically. In those special cases with an extended Beverton–Holt birth function, we additionally find that chaos may occur, but only when the per capita mortality rates are not strongly density-dependent.