Transitions in systems with High-Dimensional Stochastic Complex Dynamics: Monitoring and Forecasting

We analyst in detail a new approach to the monitoring and forecasting of the onset of transitions in high dimensional complex systems (see Phys. Rev. Lett . vol. 113, 264102 (2014)) by application to the Tangled Nature Model of evolutionary ecology and high dimensional replicator systems with a stochastic element. A high dimensional stability matrix is derived for the mean field approximation to the stochastic dynamics. This allows us to determine the stability spectrum about the observed quasi-stable configurations. From overlap of the instantaneous configuration vector of the full stochastic system with the eigenvectors of the unstable directions of the deterministic mean field approximation we are able to construct a good early-warning indicator of the transitions occurring intermittently. Inspired by these findings we are able to suggest an alternative simplified applicable forecasting procedure which only makes use of observable data streams.