Complex systems

Foreseeing tipping points

Theory suggests that the risk of critical transitions in complex systems can be revealed by generic indicators. A lab study of extinction in plankton populations provides experimental support for that principle. See Letter p. 456

On page 456 of this issue, Drake and Griffen1 show that subtle changes in the pattern of fluctuations in a population can indicate whether that population is close to extinction. This is a step forward for conservation biology, but the wider implications are even more profound. The symptoms detected belong to a family of generic leading indicators that may help to determine whether a complex system is on the brink of collapse.

Although mathematical models often predict gradual trends of change quite well, we are still badly equipped when it comes to foreseeing radical transitions such as the crash of financial markets, the onset of severe droughts, epileptic seizures or the collapse of coral ecosystems. However, a new development in our ability to predict such events2 stems from the insight that some dramatic shifts in complex systems may be related to the existence of tipping points (or 'catastrophic bifurcations'). As a system comes close to such a critical point, even small perturbations can trigger a massive shift, much as capsizing becomes increasingly likely as more cargo is loaded onto the deck of a ship. It is notoriously hard to know if a system is close to a tipping point. We simply do not see the 'brittleness' of the situation unless the transition happens.

The new approaches for probing the vicinity of a tipping point are based on the idea that, whereas the equilibrium state reveals little at all, non-equilibrium dynamics should change in universal ways in the vicinity of tipping points2,3,4,5. Thus, rather than looking at the state itself, we may have to look at its fluctuations if we want to know how vulnerable a system is.

The main principle behind this theory (reviewed in ref. 2) is the fact that systems that are close to a tipping point become very slow in recovering from perturbations (Fig. 1), a phenomenon known as 'critical slowing down'. The most straightforward implication is that system fragility can in principle be probed by studying its recovery rate following experimental perturbations6. As the test perturbations can be tiny, this may be done with little risk of causing the actual transition. For large, complex systems, it will often be difficult to systematically test recovery rates. But there is a way around that problem. Virtually all systems are permanently subject to natural perturbations. In such situations, it can be shown that, as a critical point is approached, critical slowing down will be reflected in characteristic changes in the frequency spectrum and variance of the fluctuations in the system (Fig. 1).

Figure 1: Tipping points and leading indicators.
figure1

The loss of resilience in the vicinity of tipping points can be understood from stability landscapes, shown here. a, Under conditions far from tipping points, a system is resilient: the basin of attraction is large, and perturbations will not easily drive the system towards an alternative state. b, If a system is close to a tipping point, the basin of attraction will be small, and a perturbation may easily push the system into an alternative basin. The state of the system by itself does not reveal such 'brittleness', but the system dynamics around the equilibrium differ in characteristic ways from those seen when the basin of attraction is large (as in a). In the risky state (b), the recovery rate from a small perturbation is slower (arrow), and the fluctuations in a stochastic environment will tend to be larger and more time-correlated, as shown in the insets. Such changes in dynamics are generic indicators for the proximity of tipping points, including those of the Daphnia populations investigated by Drake and Griffen1. (Modified from ref. 2.)

One such change is an increase in autocorrelation: subsequent states in a time series will become more alike. This phenomenon has been found, for example, in models of the collapse of ocean thermohaline circulation — the ocean's 'conveyor belt'3,7. But a systematic rise in autocorrelation has also been found before eight major climate transitions in the past5. Critical slowing down will also cause the correlation between linked units to rise as a tipping point is approached8. For instance, the correlation between financial markets may increase before a collapse, and the synchrony between neurons in the brain rises before an epileptic seizure2. Other leading indicators that may signal an upcoming transition are the rising variance4 and skewness9 of fluctuations.

The contribution of Drake and Griffen1 is a landmark, as it reveals theoretically predicted signals in a controlled, living system. In a large replicated experiment, they exposed populations of zooplankton (Daphnia) to slowly deteriorating conditions in the form of declining food provision. The population displayed early warning signals, such as a rise in the coefficient of variation, skewness and autocorrelation, as much as eight generations before extinction occurred. Drake and Griffen's observations point to novel ways to judge population viability. Moreover, their demonstration of the practical detectability of generic indicators suggests that this approach might help to assess the risk of transitions in systems ranging from the brain to ecosystems, climate and society.

But we are not there yet. Plankton populations that are gently pushed to extinction under controlled conditions represent a particular case (a transcritical bifurcation). The critical point of no return in this instance is probably simply at a population size of one. In nature, populations often have a higher critical density that represents a tipping point (a fold bifurcation) beyond which numbers will enter free fall owing to the so-called Allee effect10. Theory predicts the same critical slowing-down symptoms in these situations. However, follow-up studies on a variety of controlled systems would help to give a broader view of how the critical phenomena will show up in practice.

At the same time, we may bridge the gap between the laboratory and the real world by mining data from the wide range of systems that occasionally go through sharp transitions. Detection of leading indicators is a notoriously data-hungry problem, and good, long time-series are difficult to obtain. But smart data-processing techniques4,5,7 and analyses of spatial patterns8,11 may help to reduce the lead time for warning. Still, reducing the chances of false negatives or false positives will be a challenge, and the risk of some kinds of transitions will simply remain impossible to assess.

From the perspective of managing single large systems such as climate, there is yet another issue12: even if we detect good early-warning indications, could we expect society to take timely action to turn a system back from the brink in the face of large uncertainty and the high costs involved? In other situations, the barrier to practical application may be lower. For example, if we are concerned about a particular endangered species, we may scan data on dynamics and spatial patterns for leading indicators to find out which populations seem closest to a critical point for extinction. Subsequently, conservation efforts could be targeted specifically at those populations.

Clearly, we have just scratched the surface in exploring the possibilities and limitations of this emerging field. However, the prospect of identifying generic indicators is exciting, as the approach may provide an independent way to assess the risk of critical transitions. Expansion of our toolbox for prediction is especially welcome given the almost insurmountable problem of modelling complex systems in a quantitatively accurate way.

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Scheffer, M. Foreseeing tipping points. Nature 467, 411–412 (2010). https://doi.org/10.1038/467411a

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