A. Wikan et E. Mjolhus, PERIODICITY OF 4 IN AGE-STRUCTURED POPULATION-MODELS WITH DENSITY-DEPENDENCE, Journal of theoretical biology, 173(2), 1995, pp. 109-119
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.