IMPLICATIONS OF SEMI-STABLE ATTRACTORS FOR ECOLOGICAL MODELING

Dynamical systems frequently are attracted to fixed points and invaria nt, chaotic sets which are not stable but which can dominate the dynam ics of a system for an extended period of time. These sets are attract ors which have been destabilized by a change in one or more model para meters. We call these sets semi-stable attractors since they remain at tractive, with phase volumes decreasing around them, while being unsta ble in at least one direction, that in which distances expand. Simulat ions used to illustrate the idea of the semi-stable attractor based on a non-linear, biological age-structured population model typically re mained 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-str uctured model, and the empirical scaling relation of mean residence ti me versus our control parameter: T similar to (q - 0.984)(-1.07). Know ledge of this scaling function allows assessment of the stability of t he invariant sets for the region of parameter space of interest.