Further analysis of Clark's delayed recruitment model

Abstract

I explore the nonlinear behavior of a model in which the number of adults in each year is the sum of recruitment (which depends nonlinearly on adult abundanceT years in the past) and a constant fraction (survival) of adults in the previous year. Adding even a small amount of age structure to the semelparous version of this model (by increasing adult survival from zero) is stabilizing in that: (1) it shifts the value of slope at which the linearized model becomes unstable about the equilibrium to a lower value; and (2) it stretches the pattern of period doubling bifurcations so that bifurcations occur at much lower values of the slope. For the iteroparous case with a maturation and survival pattern reasonably typical of long-lived organisms, the period of cycles or quasi-cycles produced increases continuously as the slope of the stock-recruitment function decreases. The possibility of arbitrarily long cycles is not predicted by the linear theory, and has important practical implictions for analyses of cyclic populations. Both truncation of the age structure and an upper limit on recruitment seem to remove this gradual increase in period. However, the former can give rise to doubling of the period. Although the nonlinear behavior is not analysed in detail, a qualitative interpretation of the behavior of this model in terms of population inertia seems to explain the behavior observed in these numerical simulations.