Tipping points -- thresholds separating two contrasting system regimes -- exist across a wide variety of ecological systems due to underlying positive feedbacks. The consequences of crossing a tipping point are often detrimental to the functioning of an ecosystem and the services it provides, and can be challenging if not impossible to reverse. The development of early warning indicators (EWIs) for approaching tipping points has thus become an area of intense research and currently revolves around the dynamic phenomena of critical slowing down -- the tendency of a system to respond more ‘sluggishly’ to disturbances as a tipping point is neared. However, critical slowing down precedes all local bifurcations, not all of which correspond to tipping points. We therefore need EWIs that are specific to the type of upcoming bifurcation to avoid false alarms. We use theory from stochastic processes to derive analytic expressions for EWIs in the small-noise approximation to gain insight into the behaviour of these EWIs in generic pre-bifurcation regimes. Simple stochastic ecological models exhibiting a variety of bifurcations are simulated to demonstrate how these insights can help us determine characteristics of an approaching bifurcation.
Results/Conclusions
We find that autocorrelation, a commonly used EWI, displays qualitatively different behavior depending on the relative magnitude of the lag time and the intrinsic frequency of system oscillations upon disturbance. Contrary to the notion that autocorrelation always increases with critical slowing down, lag times of a given range result in decreasing autocorrelation for certain bifurcation types -- a behavior observed in a recent empirical study and elucidated by our theoretical analysis. Uncertainty in choice of lag time can be circumvented by using the power spectral density, a metric which captures equivalent information to the autocorrelation at every lag time. We find that the power spectral density captures properties of the system fluctuations that can distinguish between the onset of collapse (fold bifurcation) from the onset of oscillations (Hopf bifurcation) in ecosystems. These results suggest that the use of power spectral density should become standard protocol when investigating the resilience of ecological systems to global change.