Detecting and distinguishing tipping points using spectral early warning signals

Abstract

Theory and observation tell us that many complex systems exhibit tipping points—thresholds involving an abrupt and irreversible transition to a contrasting dynamical regime. Such events are commonly referred to as critical transitions. Current research seeks to develop early warning signals (EWS) of critical transitions that could help prevent undesirable events such as ecosystem collapse. However, conventional EWS do not indicate the type of transition, since they are based on the generic phenomena of critical slowing down. For instance, they may fail to distinguish the onset of oscillations (e.g. Hopf bifurcation) from a transition to a distant attractor (e.g. Fold bifurcation). Moreover, conventional EWS are less reliable in systems with density-dependent noise. Other EWS based on the power spectrum (spectral EWS) have been proposed, but they rely upon spectral reddening, which does not occur prior to critical transitions with an oscillatory component. Here, we use Ornstein–Uhlenbeck theory to derive analytic approximations for EWS prior to each type of local bifurcation, thereby creating new spectral EWS that provide greater sensitivity to transition proximity; higher robustness to density-dependent noise and bifurcation type; and clues to the type of approaching transition. We demonstrate the advantage of applying these spectral EWS in concert with conventional EWS using a population model, and show that they provide a characteristic signal prior to two different Hopf bifurcations in data from a predator–prey chemostat experiment. The ability to better infer and differentiate the nature of upcoming transitions in complex systems will help humanity manage critical transitions in the Anthropocene Era.

Deep learning for early warning signals of regime shifts

“…The early warning signals provided by lag-1 AC, variance, and the DL algorithm can be compared as progressively more of the time series leading up to the bifurcation is made available for their computation, as might occur in real-world settings where a variable is monitored over time. For the two ecological models exhibiting the fold, Hopf and transcritical bifurcations (Figure 1a-c), the lag-1 AC and variance increase progressively before all three transition types except for the Hopf bifurcation, where lag-1 AC decreases due to the presence of an oscillatory component to the motion 18 ) (Figure 1d-i). Hence the trends in these two indicators suggest that a transition will occur.…”

“…Other early warning approaches have been developed to predict the type of regime shift 18,20,30 . However, these approaches tend to be system-specific.…”

“…Generic early warning indicators are computed using the Python package ewstools 18 which implements established methods 42 . This first involves detrending the time series to obtain residual dynamics.…”

“…Higher-order approximations can distinguish between different types of bifurcations. But they are not often used to develop early warning indicators because (1) the first-order approximation dominates dynamics sufficiently close to the equilibrium, causing critical slowing down to generate the strongest signal, and (2) the first-order approximation is more tractable to mathematical analysis of stochastic systems than the higher-order approximations 18 . However, as a system gets closer to a bifurcation it can drift further from equilibrium due to critical slowing down.…”

“…As a consequence, the higher-order terms become significant and may be large enough to provide clues about the type of transition that will occur. Statistical measures such as skew and kurtosis reflect the influence of these highest order terms, for instance [18][19][20] . Higher-order terms could be associated with features in time series data that are subtle but detectable, if we knew what to look for.…”