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

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.