Moving toward an unstable equilibrium: Saddle nodes in population systems

Abstract

1. We identify an unstable equilibrium with a two-dimensional stable manifold and a one-dimensional unstable manifold in a three-state variable (larva, pupa, adult) insect population growth model. 2. The saddle node forecasts that the time series of some initial numbers of larvae, pupae and adults are drawn closely to the unstable equilibrium before approaching the asymptotic stable attractor (a two-cycle), while the time series of other initial points are not. 3. Using two quantitative indices, we examine time series from a Tribolium experiment for evidence of the predicted saddle node. We conclude that a saddle node accounts for the transient dynamics in these data and for the differences between the transient behaviour of different replicates of the same experiment.