Critical speeding up as an early warning signal of stochastic regime shifts
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ORIGINAL PAPER
Critical speeding up as an early warning signal of stochastic regime shifts Mathew Titus1,2
· James Watson1
Received: 8 August 2019 / Accepted: 3 February 2020 © Springer Nature B.V. 2020
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 2010; Hastings and Wysham Ecology Letters 13(4):464–472 2010). Possible explanations include a non-smooth potential driving the dynamic (Hastings and Wysham Ecology Letters 13(4):464–472 2010) or large perturbations driving the system out of the initial basin of attraction (Boettiger and Batt 2018). 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. Keywords Early warning signals · Regime change · Critical slowing down · Critical speeding up · Complex systems · Stability
Introduction When studying time series data for dynamical systems which exhibit critical transitions—that is, sudden changes in equilibrium behavior—a widely used early warning signal for an oncoming transition is critical slowing down (Dakos et al. 2008, 2012; Scheffer et al. 2009). This signature for the system being at risk of a large transition is based on the theory of stochastic dynamical systems (Hastings and Wysham 2010; Drake 2013; Kuehn 2013) and has been observed in a variety of empirical tests, both in nature and in the laboratory (e.g., Carpenter and Brock 2006; Scheffer et al. 2012; van Belzen et al. 2017; Wen et al. 2018). At its core, critical slowing down (CSD) assumes that the stochastic process X experiences a smooth Mathew Titus
[email protected] 1
College of Earth, Ocean and Atmospheric Sciences, Oregon State University, Corvallis, OR, USA
2
The Prediction Lab, Corvallis, OR, USA
potential Vt which is varying slowly in time, dXt = −∇Vt (Xt )dt + ξt where ξ is some random process. If the potential nears a bifurcation point, such as a pitchfork or fold bifurcation, then the shape of V around its equilibrium necessarily flattens out as its minimum (the stable point) becomes a degenerate critical point. The lesseni
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