Exit time as a measure of ecological resilience

INTRODUCTION

Financial markets may collapse, rainforest can shift to savanna, a person can become trapped in a depression, and the Gulf Stream can come to a standstill. Such critical transitions may happen in complex systems when resilience is low, allowing a perturbation to trigger self-propelled change toward the contrasting state. An intuitive way of visualizing this is to depict the system as a marble in a cup. If the cup is shallower, resilience is lower, and it becomes easier to flip the marble out. Resilience may sometimes be inferred from autocorrelation and variance of time series because they carry information about recovery rates from small perturbations, thus reflecting the slope of the attraction basin. However, although differences in such indicators may reflect differences in resilience, they cannot be interpreted in an absolute sense. The alternative—measuring resilience as the maximum perturbation that a system can take—may seem more attractive. However, this classical ecological definition of resilience assumes that the system is affected by distinct, isolated perturbations. In reality, most systems are perturbed instead by a never-ending natural regime of shocks and fluctuations. Thus, they rarely recover from a perturbation before the next one comes. As a result, the imaginary marble in the cup keeps wandering around, and occasionally a sequence of small but synergistic “perturbations” will push the system across the border of the basin of attraction. How then can we characterize the resilience of such systems in a useful and practical way?

RATIONALE

We propose to use “life expectancy” as a measure of resilience, which can be formalized as the mean exit time from an attraction basin (i.e., the expected time required to cross the border of the basin). This approach has the advantage of taking the natural variability of real complex systems explicitly into account. If we have many observations of shifts, we can calculate the mean exit time simply as the average time the system spends in each given state. However, such data are rare. Moreover, the permanent fluctuations also contain information about the system, which would be lost if one considers only the rare occasions at which a shift occurs. In the approach we outline, this information is used as we infer the deterministic and stochastic components of the underlying dynamical system from observed fluctuations. Applying techniques from statistical mechanics, we show how one can subsequently use the reconstructed empirical model to compute the expected mean exit time for each basin of attraction.

RESULTS

After using model-generated data to demonstrate the method, we applied it to two examples spanning very different time scales: the rapid dynamics of Cyanobacteria in a lake, and the much slower alternations between cold glacial and warmer interstadial regimes in the climate (the so-called Dansgaard-Oeschger events). For both time series, we show that one can reconstruct a model that has two alternative attractors. We estimated the mean exit time for each of these attractors, and we show how one can analyze the uncertainty of this estimate.

A major challenge is that the approach requires high-resolution time series that cover dynamics across both basins of attraction. Such information can be obtained from a single long time series if it includes enough shifts between attraction basins. Alternatively, one may piece the required data together using shorter time series from sets of similar systems.

CONCLUSION

Characterizing resilience as the estimated life expectancy—of a rainforest, a coral reef, or the thermohaline circulation—is a natural and intuitively straightforward way to go. The high-resolution time series required for the approach we outline are still relatively rare. However, possibilities for automated sensing are rapidly expanding in fields as diverse as biomedicine, climate science, ecology, and financial markets. Against this background, the technique for estimating mean exit times is an exciting additional tool to anticipate critical transitions in the many complex systems on which humanity depends.