Critical speeding up as an early warning signal of stochastic regime shifts

Abstract

The use of critical slowing down as an early warning indicator for regime switching in observations from noisy dynamical systems and models has been widely studied and implemented in recent years. Some systems, however, have been shown to avoid critical slowing down prior to a transition between equilibria (Ditlevsen and Johnsen, Geophysical Research Letters, 37(19), 2010; Hastings and Wysham, Ecol Lett 13(4):464–472, 2010). Possible explanations include a non-smooth potential driving the dynamic (Hastings and Wysham, Ecol Lett 13(4):464–472, 2010) or large perturbations driving the system out of the initial basin of attraction (Boettiger and Batt 2020). In this paper, we discuss a phenomenon analogous to critical slowing down, where a slow parameter change leads to a high likelihood of a regime shift and creates signature warning signs in the statistics of the process’s sample paths. This effect, which we dub “critical speeding up,” is demonstrated using a simple population model exhibiting an Allee effect. In short, if a basin of attraction is compressed under a parameter change then the potential well steepens, leading to a drop in the time series’ variance and autocorrelation; precisely the opposite warning signs exhibited by critical slowing down. The fact that either falling or rising variance / autocorrelation can indicate imminent state change should underline the need for reliable modeling of any empirical system where one desires to forecast regime change.