Understanding the stability metrics
In order to explain the resilience metrics more thoroughly, example time series of a stable and
unstable ecosystem, shown respectively as dashed lines in figure S1a,b are used. The time series
represent the response of an ecosystem variable, e.g. biomass or vegetation greenness. For both
time series, the variable shows a repeated seasonal pattern, presented by a black line. The latter is
further referred to as the seasonal climatology (tsC) of the ecosystem and is characteristic for
many time series of vegetative systems. For example, in temperate climates more biomass is
produced during summer months compared to winter months.
The stability of an ecosystem can be derived using the anomalies (tsA) relative to the seasonal
climatology. For example, consider an ecosystem which is not subjected to disturbances, nor
internal changes such as a shift in species composition. In this case the environmental conditions
remain on their average climatological state, thus the ecosystem is also assumed to follow the
seasonal climatology. However, in reality, perturbations such as droughts and wildfires occur. As
a response to these changing environmental conditions, the ecosystem's variable deviates from
the seasonal climatological pattern (tsC), as illustrated by the coloured line. The properties of the
anomalies caused by perturbations, such as magnitude and behaviour over time, represent the
ecosystem stability.
Resilience is defined as the rate of return to its equilibrium state after a perturbation. When an
ecosystem has a low resilience, it will return slowly to its equilibrium state, while highly resilient
ecosystems have a fast return. If an ecosystem with low resilience would be subjected to a
perturbation, a negative anomaly might occur while the following observations will remain
negative, although slowly return to zero. This would result in a high correlation of subsequent
observations. In contrast, the anomalies of a highly resilient ecosystem would return quickly to
zero, giving a nearly random distribution of the anomalies. The temporal relation between
observations thus serves as a metric of resilience and can be expressed by the autocorrelation at
lag one (ρ1). The higher ρ1, the more similar to subsequent anomalies are and thus the slower the
return to equilibrium. Therefore, resilience might be expressed as 1-ρ1. For example, time series
Y shows longer periods below or above the seasonal climatology compared to time series X and
therefore has a lower resilience (see figure S1a,b). This is confirmed by the autocorrelation at lag
1, i.e. ρ1 of time series X = 0.71 while ρ1 of Y = 0.34 (shown in figure S1d).
Another group of resilience metrics is based on the distribution of the variance of the anomaly
time series tsA over the frequencies, i.e. the spectrum of the anomalies. When the variance is
equal for all frequencies of the spectrum, the anomalies are independent of each other and the
ecosystem is highly resilient, whereas high power for low frequencies
suggests persistent
behaviour (i.e. positive/negative anomalies are likely followed by positive/negative anomalies).
This is the case when an ecosystem slowly returns to its equilibrium state after a perturbation. In
reverse,
anti-persistent
behaviour
(i.e.
positive/negative
anomalies
are
followed
by
negative/positive anomalies) occurs when the power is large for high frequencies. To summarize
these possible differences in variance distribution two metrics have been developed, i.e. the
normalised spectral entropy and the spectral scaling component (α). The normalised spectral
entropy expresses the evenness of the distribution of variance. With equal variance for all
frequencies, the normalised spectral entropy reaches a maximum of one. When one frequency
contains a high amount of variance, while the other frequencies contain little variance, the
normalised spectral entropy will be lower. Therefore a higher entropy indicates a more resilient
ecosystem. The spectral scaling component is given by the slope of the logarithm of the spectrum
upon the logarithm of the reverse frequency. An α equal to, larger and smaller than zero can
respectively be interpreted as an indication of a white noise process, persistent behaviour and
anti-persistent behaviour. Therefore, highly resilient ecosystems are expected to have an α
approaching zero.
The distribution of the variance of the anomalies for ecosystem X and Y, i.e. their spectra, are
shown in figure S1e. The anomaly spectrum of ecosystem Y is less flat than the spectrum of X.
Ecosystem X is therefore more resilient than Y. Similar conclusions can be drawn for the spectral
scaling coefficient, where both the anomalies of time series X and Y are persistent, but with
lower spectral scaling coefficients for X (0.51) than for Y (1.27) (lower graph of figure S1e). The
persistence of the anomalies of time series X is smaller, i.e. they tend to return more quickly to
equilibrium.
Figure S1: Example of the stability metrics: a) The time series (ts) of a stable ecosystem X (red
dashed line), the climatology (tsC, black) and the anomalies (tsA, red); b) The time series of a
stable ecosystem Y (red dashed line), the climatology (black) and the anomalies (blue); c) The
histogram of the anomalies of time series X (red) and Y (blue); d) The sample autocorrelation of
time series X (red) and Y (blue), e) The spectrum upon the frequency and the logarithmic
relationship between the spectrum and 1/frequency for time series X (red) and Y (blue).