Implications of semi-stable attractors for ecological modelling

Dynamical systems frequently are attracted to fixed points and invariant, chaotic sets which are not stable but which can dominate the dynamics of a system for an extended period of time. These sets are attractors which have been destabilized by a change in one or more model parameters. We call these sets semi-stable attractors since they remain attractive, with phase volumes decreasing around them, while being unstable in at least one direction, that in which distances expand. Simulations used to illustrate the idea of the semi-stable attractor based on a non-linear, biological age-structured population model typically remained near the semi-stable attractor for approximately 500 iterations (or five centuries if the time unit of one year is used). We present an empirical frequency distribution of residence times for our age-structured model, and the empirical scaling relation of mean residence time versus our control parameter: T similar to (q - 0.984)(-1.07). Knowledge of this scaling function allows assessment of the stability of the invariant sets for the region of parameter space of interest.