Nonstationary and Chaotic Dynamics in Age-Structured Population Models

Arild Wikan,Ørjan Kristensen

doi:10.1155/2017/1964286

Abstract

The dynamics from nonlinear discrete age-structured population models is under consideration. Focus is on bifurcations, as well as nonstationary and chaotic dynamics. We also explore different mechanisms which may lead to periodic phenomena. Some new results are also presented, in particular from models where both fecundity and survival terms contain nonlinear elements.

Highlights

As it is well known, simple one-dimensional maps of biological relevance can exhibit an extraordinary rich dynamical behaviour ranging from stable fixed points to chaotic oscillations
As it is clear from the examples, nonlinear age-structured population models are excellent tools in order to study the dynamical outcomes of biological populations
When n is large, x∗(P) is an increasing function which clearly suggests that an increase of n acts stabilizing to the dynamics

Summary

Introduction

As it is well known (cf. [1,2,3,4]), simple one-dimensional maps of biological relevance can exhibit an extraordinary rich dynamical behaviour ranging from stable fixed points to chaotic oscillations. Whenever the number of age classes n is small, we classify the species as having a precocious semelparous life history. Considering the multidimensional model (2), the eigenvalues of the linearization may leave the unit circle through λ = 1, λ = −1, or λ = ± exp(iθ), where θ = 2πk/n, k = 0, 1, . Assuming that the bifurcation is of supercritical nature, an attracting period 2 orbit is established when the fixed point goes unstable. When a pair of complex modulus 1 eigenvalues leave the unit circle, the fixed point will go through a Neimark Sacker (Hopf) bifurcation. Bifurcations of subcritical nature appear to be rare events in population models like (2), but an excellent example may be obtained in the prey-predator model discussed in [26]

Examples

Summary