Periodicity of 4 in age-structured population models with density dependence

The instabilities of the equilibria of a class of time discrete population growth models are studied. The models describe the growth of one species with age structure, where the survival probabilities are assumed to depend on the total population. Instability occurs when the fecundities 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 range following immediately after threshold, the cycles are only roughly 4-periodic, such that the long-term dynamics fill a closed curve in phase space; thus the instability is a supercritical Hopf bifurcation. Increasing 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 the Feigenbaum type into 8-, 16- and 32-cycles and so on, and finally the dynamics becomes chaotic. Even in the 4 × 2k-periodic, and also in part of the chaotic regime, a qualitative character of 4-cycles is preserved.