PERIODICITY OF 4 IN AGE-STRUCTURED POPULATION-MODELS WITH DENSITY-DEPENDENCE

The instabilities of the equilibria of a class of time discrete popula tion growth models are studied. The models describe the growth of one species with age structure, where the survival probabilities are assum ed to depend on the total population.: Instability occurs when the fec undities or the survival probabilities supersede a certain threshold. The nonlinear development of the instability has, for a large range of parameters, the character of 4-periodic cycles. In the parameter rang e following immediately after threshold, the cycles are only roughly 4 -periodic, such that the long-term dynamics fill a closed curve in pha se space; thus the instability is a supercritical Hopf bifurcation. In creasing the parameter further from threshold, these roughly periodic cycles become frequency-locked into an exact 4-periodic cycle. As the parameter is even further increased, we also find period doubling of t he Feigenbaum type into 8-, 16- and 32-cycles and so on, and finally t he dynamics becomes chaotic. Even in the 4 x 2(k)-periodic, and also i n part of the chaotic regime, a qualitative character of 4-cycles is p reserved.