Abstract
The scope of this review article is to present and discuss various aspects of discrete nonlinear age- and stage-structured population models. We show that such models cover species which exhibit a wide range of different life histories and that one may use them in order to deduce fairly general ecological principles with respect to stability and dynamical outcomes. From a mathematical point of view, we give several examples of the fact that the nonstationary dynamics generated by such maps is indeed rich as a result of different types of bifurcations of various nature as well as other mechanisms like frequency locking and crises that may occur.
Highlights
In an influential review article “Simple and Gurney et al (1990)
(1977); Feigenbaum (1978) and Singer (1978) it was demonstrated that simple one-dimensional nonlinear maps of biological relevance could exhibit an extraordinary rich dynamical behaviour ranging from stable fixed points, periodic points to chaotic behaviour
From a biological point of view there is a variety of cases where one-dimensional population models are not sufficient modelling tools
Summary
Population models by Patten (1976); Boling (1973); see Metz and Diekmann (1986); Hallam et al (1990). If a fixed point undergoes a supercritical bifurcation when the parameter μ is increased to a value μ0 there is established an attracting orbit Example 4 (Precocious and delayed semelparity): First we consider the precocious case where both fecundity and survival probability is density dependent, i.e.: Assuming n even, as x* becomes larger successive flip bifurcations create stable SYC cycles of period 2kn. Since (14a) is associated with the possibility that (x1*,x2*) shall undergo a flip bifurcation at instability threshold it is natural to seek for a stable 2-cycle in case of x* small Such a stable 2-cycle exists and the points in the cycle are on the form (x1, x 2 ) = (A,0) or (0,B) which implies that only one age class is populated at each time.
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