Predicting temporal stability and resilience from resistance and recovery

Abstract

Stability can be desirable for many natural and social systems. Temporal stability, the invariability of a system over time, can be enhanced by resisting displacement during perturbations, accelerating recovery after them, or both^1–4. Likewise, resilience (sensu proximity to unperturbed levels after a perturbation^5–10) also has components of withstanding (resistance) and recovering after perturbations^11,12. Here we develop and test new predictions for how temporal stability and resilience depend on their resistance and recovery components. We find that temporal stability could often be predicted from resistance, even without information about how quickly the system recovers. By contrast, resilience is predicted to depend at least as much on recovery as on resistance, as in earlier theory^11,12. Using plant productivity data from the world’s longest-running biodiversity experiment, we find that long-term temporal stability, quantified over a quarter century at the ecosystem or species level, is predicted with moderate accuracy from single-year estimates of resistance alone, with only slight improvement by also considering recovery. Resilience was predicted with moderate accuracy by a combination of resistance and recovery at the ecosystem level. We also find that ecosystem drought resistance can be forecasted by monitoring temporal stability before the drought. Our results reveal that long-term temporal stability and short-term resistance may often be predicted from one another and clarify how resistance and recovery can be leveraged to enhance the stability of both natural and managed systems.

New predictions for how temporal stability and resilience depend on their resistance and recovery components are explored.

Main

Temporal stability and resilience can be desirable properties for many dynamical systems in nature and society, such as ecosystems, the climate, agriculture and economic markets. For instance, ecosystem stability, often quantified as the temporal stability of plant community productivity, underpins the reliable supply of many ecosystem services, including the production of crops, livestock and timber. Species stability, the temporal stability of the abundance or biomass of a species, can affect whether a species persists or is lost. Resilience, the proximity to unperturbed levels shortly after a perturbation^5–10, indicates, for example, whether and how quickly a harvested system returns to a profitable level or a species returns to a viable population size. Global environmental changes, such as climate change, pollution and land use changes, are now altering the temporal stability and resilience of ecosystems and species^13–16. New approaches are needed to predict and enhance the temporal stability and resilience of both natural and managed systems, and to forecast the sensitivity of these systems to perturbations, such as extreme climate events that are becoming increasingly frequent and intense¹⁶.

Intuitively, temporal stability^1–4 and resilience^5–10 can be enhanced by features that lead to resisting displacement during perturbations, accelerating recovery after them, or both (Fig. 1). Although many previous studies have considered multiple types (sometimes referred to as dimensions) of stability^{1–3,5,7,17–23}, the expected relationships among stability measures remain unclear. For example, if temporal stability depends equally on resistance and recovery, then the two components may be substitutable. Alternatively, if temporal stability depends mostly on either resistance or recovery, then enhancing temporal stability may require focusing on its most critical stability component. Formalizing these expected relationships may help to predict and enhance temporal stability or resilience by focusing on their most critical component (or components).

Fig. 1. Theoretical predictions for the dependence of temporal stability and resilience on their resistance and recovery components.

a, Hypothetical time series of a system property (that is, state variable) with negative and positive perturbations occurring respectively at time steps one and six, and a period of recovery following each perturbation. Example time series are shown for contrasting combinations of high or low levels of resistance and recovery. In the examples shown, P = 2 perturbations, T = 10 time steps, ${\mathrm{\Omega }}_2=0.75$ for high resistance or ${\mathrm{\Omega }}_2=0.25$ for low resistance, ${\mathrm{\Delta }}_2=0.75$ for rapid recovery or ${\mathrm{\Delta }}_2=0.25$ for slow recovery. Horizontal dashed line shows average unperturbed level. b,c, Predicted dependence of temporal stability (b) and resilience (c) on their resistance and recovery components. The lines are isocline contours; the points and symbols correspond to examples in panel a; and the temporal stability and resilience values are, respectively, predicted from E2 and E4 in Table 1 (b,c). Table 1 provides additional definitions of measures and parameters. Extended Data Fig. 1 shows how these relationships depend on perturbation frequency, time steps since perturbation and stability measures.

Here we developed and tested a new theory showing how these four types of stability (temporal stability, resilience, resistance and recovery) can be predicted from the others. We first show mathematically how temporal stability and resilience can be predicted from their resistance and recovery components. Given that expected relationships among stability components will depend on the measures used, we considered two sets of stability measures (Table 1). We then tested these theoretical predictions with data from the world’s longest-running biodiversity experiment^24–30. Applying our novel theory to data from this experiment allowed us to consider stability components not only at the aggregated ecosystem level but also for individual species in monocultures, where some species may exhibit more resistance or recovery than others.

Table 1.

Definitions and predicted relationships for two sets of stability measures

Stability measures	Definition		Inverse measures of stability		Complement measures of stability
Integrated measures of stability
Temporal stability	Invariability of the system over time, relative to the long-term average level	D1	$I_1\equiv \frac{\mu }{\sigma }=\frac{1}{\mathrm{CV}}$	D5	$I_2\equiv 1-\frac{\sigma }{\mu }=1-\mathrm{CV}$
		E1	$I_1=\frac{{\mathrm{\Omega }}_1}{{\mathrm{\Delta }}_1\sqrt{\frac{P\left(1-{{\mathrm{\Delta }}_1}^{-2T/P}\right)}{\left(T-1\right)\left({{\mathrm{\Delta }}_1}^2-1\right)}}}$	E2	$I_2=1+\left({\mathrm{\Omega }}_2-1\right)\sqrt{\frac{P\left[(1-{{\mathrm{\Delta }}_2)}^{2T/P}-1\right]}{\left(T-1\right){\mathrm{\Delta }}_2\left({\mathrm{\Delta }}_2-2\right)}}$
		A1	$I_1\approx \frac{{\mathrm{\Omega }}_1}{\sqrt{\frac{P}{T-1}}}$	A2	$I_2\approx 1+\left({\mathrm{\Omega }}_2-1\right)\sqrt{\frac{P}{T-1}}$
Resilience	Proximity to normal levels after a perturbation, relative to the normal level	D4	${\mathrm{\Phi }}_{1,\mathit{x}}\equiv \frac{\overline{Y_n}}{|\overline{Y_n}-Y_{e+x}|}$	D8	${\mathrm{\Phi }}_{2,\mathit{x}}\equiv 1-\frac{|\overline{Y_n}-Y_{e+\mathit{x}}|}{\overline{Y_n}}$
		E3	${\mathrm{\Phi }}_{1,\mathit{x}}={\mathrm{\Omega }}_1{\mathrm{\Delta }}_1^x$	E4	${\mathrm{\Phi }}_{2,\mathit{x}}=1-{\left(1-{\mathrm{\Delta }}_2\right)}^x+{\mathrm{\Omega }}_2{\left(1-{\mathrm{\Delta }}_2\right)}^x$
Component measures of stability
Resistance	Lack of displacement during a perturbation, relative to the normal level	D2	${\mathrm{\Omega }}_1\equiv \frac{\overline{Y_n}}{|\overline{Y_n}-Y_e|}$	D6	${\mathrm{\Omega }}_2\equiv 1-\frac{|\overline{Y_n}-Y_e|}{\overline{Y_n}}$
		A3	${\mathrm{\Omega }}_1\approx I_1\sqrt{\frac{P}{T-1}}$	A4	${\mathrm{\Omega }}_2\approx 1+\frac{{\mathrm{I}}_2-1}{\sqrt{\frac{P}{T-1}}}$
Recovery	Rate of recovery just after a perturbation, relative to the displacement	D3	${\mathrm{\Delta }}_1\equiv |\frac{\overline{Y_n}-Y_e}{\overline{Y_n}-Y_{e+1}}|$	D7	${\mathrm{\Delta }}_2\equiv 1-|\frac{\overline{Y_n}-Y_{e+1}}{\overline{Y_n}-Y_e}|$

$\overline{Y_n}$, Y_e, Y_e + 1 and Y_e + x are, respectively, values (for example, ecosystem productivity or other measures of a system level) during normal times (mean across unperturbed times), during a perturbation event, one time unit after a perturbation, and x time units after a perturbation (Fig. 1a). P and T are the number of perturbations and the length of the time series, respectively. μ and σ are the mean and standard deviation of the time series, respectively. See Methods for additional information for definitions (D1 to D8), equations (E1 to E4) and approximations (A1 to A4). CV, coefficient of variation.

On the basis of a mathematical analysis of the relationships among stability components (Methods), we found that temporal stability is predicted to depend mostly on resistance, and only strongly depends on recovery when both perturbations are infrequent and recovery is slow (E1 and E2 in Table 1 and Fig. 1b). For example, perturbations may be considered relatively infrequent when they occur only 20% or less of the time (P = 2 in Extended Data Fig. 1b,d) and recovery may be considered relatively slow when the system recovers less than halfway to unperturbed levels each time step (${∆}_2<0.5$ in Fig. 1b). Long-term temporal stability may therefore often be predicted from short-term estimates of resistance alone using approximations A1 and A2 in Table 1. By contrast, resilience is predicted to depend equally on recovery and resistance initially, just after a perturbation (that is, when x = 1) and, unsurprisingly^11,12, to increasingly depend mostly on recovery when it is quantified longer after a perturbation (that is, with increasing x; E3 and E4 in Table 1; Fig. 1c and Extended Data Fig. 1c,e). These predictions apply to both sets of stability measures that we considered (Extended Data Fig. 1b–e). We also found some additional predictions that differ between these two sets of stability measures (Methods), although here we focused on testing these more general predictions. Given that these hypotheses arise from simplifying assumptions, empirical tests of these new theoretical predictions and the model assumptions are needed (Methods).

Extended Data Fig. 1. Theoretical predictions for the dependence of temporal stability and resilience on their resistance and recovery components.

(a) Hypothetical time series of a system property (i.e., state variable) with negative and positive perturbations occurring respectively at time steps one and six, and a period of recovery following each perturbation. Example time series are shown for contrasting combinations of high or low levels of resistance and recovery. In the examples shown, P = 2 perturbations and T = 10 time steps, ${\mathrm{\Omega }}_1=4$ and ${\mathrm{\Omega }}_2=0.75$ for high resistance or ${\mathrm{\Omega }}_1=1.33$ and ${\mathrm{\Omega }}_2=0.25$ for low resistance, ${\mathrm{\Delta }}_1=4$ and ${\mathrm{\Delta }}_2=0.75$ for rapid recovery or ${\mathrm{\Delta }}_1=1.33$ and ${\mathrm{\Delta }}_2=0.25$ for slow recovery. (b-e) Predicted dependence of temporal stability and resilience on resistance and recovery for two sets of stability measures: (b,c) inverse measures and (d,e) complement measures. Lines are isocline contours; points and symbols correspond to examples in (a); and temporal stability and resilience values in (b-e) are respectively predicted from E1, E3, E2, and E4 in Table 1. See Table 1 for additional definitions of measures and parameters. The upper panel in (d), where P = 2, and the middle panel in (e), where x = 2, are also included in Fig. 1. Here, as in Table 1, x is the number of time steps after a perturbation. P = 2 except where otherwise specified in the middle and lower panels of (b,d). Note that no points are shown where P = 5 and P = 8 because this perturbation frequency does not correspond to the time series shown in (a).

Next, we empirically test these new hypotheses at the ecosystem and species levels over a quarter century in the world’s longest-running biodiversity experiment. Since 1994, the BioDIV experiment in central Minnesota, USA has manipulated grassland plant species richness in plots planted with 1, 2, 4, 8 or 16 plant species^24–30. In this temperate grassland, measurements of peak aboveground biomass, collected annually since 1996, serve as a proxy for annual aboveground net primary productivity (henceforth productivity) because all aboveground plant biomass dies during winter and is removed by prescribed burning each spring. We considered the resistance and recovery of plant aboveground productivity to climate events that occurred during our study (Extended Data Fig. 2) where, as in many other ecosystems^31–34, variation in plant productivity over time is partly explained by changes in water availability^3,7,35 (Extended Data Fig. 3). First, we considered resistance to an extreme wet event that occurred in 2002, given that this was the most extreme climate event during the study (Extended Data Fig. 2), as well as recovery the following year in 2003 and resilience by 2004. Second, we forecast resistance to an extreme drought that occurred in 2021, given that this was the first extreme drought during the study and we wanted to test whether drought responses could be predicted using data before that year (Extended Data Fig. 2). We considered these events because they are the most extreme climate events observed thus far during this experiment and there was a normal climate year before and after these events, making the dynamics during and after these events less likely to be influenced by other events (Extended Data Fig. 2). We fit diversity–interactions models^36–38, which are regression models that jointly estimate the effects of species composition, diversity and relative abundances on the response variable, and we used predictions from these models as estimates of our stability measures, which are the inputs to our subsequent stability analyses. This approach helps to improve estimates of stability measures by accounting for uncertainty (for example, due to outliers) and allowing estimation of stability for each species (for example, when data for a species in monoculture is limited; Methods).

Extended Data Fig. 2. Standardized Precipitation-Evapotranspiration Index (SPEI) during the four months (May, June, July, August) of the growing seasons and peak aboveground plant biomass production at our field site, Cedar Creek Ecosystem Science Reserve in Minnesota, USA.

(a) SPEI water balances are normalized differences between precipitation and potential evapotranspiration (PET). Values show the SPEIbase version 2.9 of SPEI, which quantifies PET by the FAO-56 Penman-Monteith model. Positive (or negative) SPEI values indicate water balances that are wetter (or drier) than those observed near the study site throughout the reference period, which was January 1901 to December 2022. When estimating resistance and recovery to the observed extreme wet event, Y_e, Y_e+1, and Y_e+2 were respectively based on the average productivity levels in 2002, 2003, and 2004. This was the most extreme event of the time series and had normal years just before and after it. When estimating resistance to the observed extreme drought in 2021, Y_e was based on productivity in 2021. In all cases, $\overline{Y_n}$ was based on productivity averaged across all normal years that did not follow an extreme event: 1996, 2001, 2005, 2007, 2009, 2012, 2013, and 2018. (b) Mean peak aboveground plant biomass production, averaged on log-transformed biomass across all BioDIV plots within each year, by year. The BioDIV experiment was planted in 1994 and 1995. Since 1996, plant biomass production has been measured annually. See Extended Data Fig. 3 for the relationship between these two variables.

Extended Data Fig. 3. Mean peak aboveground plant biomass production, averaged across all BioDIV plots within each year, increased with SPEI-04, growing season water balance.

Note the contrasting positions of 2002, the year of the extreme wet event (upper right corner), and 2021, the year of the recent drought (lower left corner). Line shows a linear fit and band shows 95% CI.

Consistent with our theoretical predictions (Table 1 and Fig. 1), we found that temporal stability was positively correlated with resistance, but only marginally significantly correlated with recovery, at both the species and the ecosystem levels (Fig. 2a). That is, species and experimental communities that exhibited a small percentage change in productivity during an extreme wet event in 2002 (Extended Data Fig. 2) were those that exhibited the greatest ecosystem stability from 1996 to 2022 (Fig. 2a). By contrast, species and experimental communities that most rapidly recovered to normal levels of productivity in 2003, the year after this wet event, were not necessarily those that exhibited the greatest ecosystem stability over this quarter century (Fig. 2a). On the basis of our theoretical results (Fig. 1b), we might expect such a correlation between temporal stability and recovery to be more common in other dynamical systems that have infrequent perturbations and relatively slow recovery. By contrast, most of our data were at relatively high resistance and rapid recovery (Fig. 2a), in which temporal stability is predicted to be relatively insensitive to changes in recovery (Fig. 1b).

Fig. 2. Correlations between temporal stability and its resistance and recovery components, and resilience and its resistance and recovery components.

a, Consistent with our theoretical predictions (Table 1 and Fig. 1), at both the ecosystem (top) and the species (bottom) levels, temporal stability is positively correlated with resistance, but less so with recovery. b, Consistent with our theoretical predictions (Table 1 and Fig. 1), resilience increases with both resistance and recovery at the ecosystem level. At the species level, resilience increased with resistance. Note that recovery may still contribute to temporal stability and resilience, even when it is not correlated with them, because the predicted relationships (Table 1 and Fig. 1) are more complex than these linear bivariate relationships. Points are ecosystem-level (that is, plot) or species-level estimates from diversity–interactions models. The solid lines show mean linear fits and the shaded bands show 95% CIs. Coefficients of determination and P values are shown for each panel.

Also consistent with our theoretical predictions (Table 1 and Fig. 1c), we found that resilience was positively correlated with both resistance and recovery at the ecosystem level (Fig. 2b). That is, experimental communities that exhibited a small percentage increase in productivity during an extreme wet event in 2002 or rapid recovery to normal levels of productivity in 2003 tended to be those that were resilient (sensu close to normal levels) in 2004 (Fig. 2b). At the species level, resilience was positively correlated with resistance, as predicted, but not significantly with recovery (Fig. 2b). There is a relatively wide range of conditions over which resilience is predicted to depend on both resistance and recovery (Fig. 1c), and thus recovery may more often be correlated with resilience than with temporal stability, as we found here at the ecosystem level (Fig. 2b).

Although bivariate correlations are a natural place to begin exploring these relationships, the theoretical predictions in Table 1 and Fig. 1 indicate that they are an oversimplification of more complex relationships. Even when uncorrelated, recovery may still contribute to temporal stability or resilience, albeit non-linearly and under constrained conditions (that is, a subset of possible combinations of resistance, recovery, perturbation frequency and time since perturbation). Next, we more formally predicted temporal stability and resilience from their resistance and recovery components at the ecosystem and species levels using the equations and approximations in Table 1.

We found that long-term temporal stability, observed over a quarter century at either the ecosystem or the species level, was predicted with moderate accuracy from single-year estimates for resistance and recovery (Fig. 3). First, we predicted ecosystem stability from E2 (Table 1 and Fig. 3a), which accounts for both resistance to the extreme wet event in 2002 and recovery the year after it in 2003 (Extended Data Fig. 2). On average, temporal stability was predicted within 1.1% or 2.3% of its observed value, as estimated from diversity–interactions models at the ecosystem or species levels, respectively, when it was predicted from both resistance and recovery (Fig. 3a). Second, we predicted temporal stability from approximation A2 (Table 1), which accounts for resistance but not recovery. Predictions were nearly as accurate when based only on resistance (Fig. 3b): on average, temporal stability was predicted within 1.6% or 2.8% of its observed value, estimated at the ecosystem or species level, respectively (Fig. 3b). Temporal stability tended to be underestimated when it was low and overestimated when it was high (Fig. 3a,b). Although more work will be needed to improve the accuracy and precision of these estimates, this is, to our knowledge, the first approach for quantitatively predicting long-term multi-decadal temporal stability from short-term single-year estimates of its resistance and recovery components.

Fig. 3. Accuracy for predicting temporal stability and resilience from resistance and recovery for ecosystems and species.

a, Long-term temporal stability, observed over 27 years (1996–2022), was predicted from single-year estimates of the resistance of productivity to an extreme wet climate event in 2002 and its recovery in 2003 using E2 in Table 1. Resilience, observed in 2004, the second year after the event, was predicted from resistance in 2002 and recovery in 2003 using E4 in Table 1. b, Predictions of temporal stability were nearly as accurate when based only on resistance in 2002 using approximation A2 in Table 1. The dashed lines show the 1:1 relationship. The solid lines indicate the mean linear model fits to data and the shaded bands show 95% CI. Points are ecosystem-level (that is, plot) or species-level estimates from diversity–interactions models. Relationships are shown for complement measures of stability (Table 1). MAPE, mean absolute percentage error.

Resilience in 2004, 2 years after the extreme climate event, was predicted with moderate accuracy from resistance in 2002 and recovery in 2003 (Fig. 3a). On average, resilience was predicted within 3.0% or 4.7% of its observed value, estimated at the ecosystem or species levels, respectively (Fig. 3a). However, at the species level, there was a non-significant relationship between the observed and predicted values for resilience (Fig. 3a; P > 0.1). Resilience tended to be underestimated when it was low and overestimated when it was high (Fig. 3a). We suspect that it will often be more challenging to predict resilience, which is based on only 1 year of data (for example, 2004 here), and thus is more susceptible to observation error than temporal stability, which is based on the full time series (for example, 1996–2022 here).

Next, we tested whether drought resistance can be forecasted from monitoring the temporal stability of productivity before the drought. Note that the equations and approximations in Table 1 can be algebraically rearranged to solve for other stability measures of interest. For example, rearranging approximation A2 to solve for resistance provides a way to forecast resistance to a perturbation, before it occurs, by monitoring temporal stability (Methods). We used this approach to forecast resistance to a recently observed extreme drought that occurred in 2021 (Extended Data Fig. 2), using approximation A4 (Table 1). Before 2021, there were no extreme droughts (defined here as a 1-in-10-year event) during our experiment (Extended Data Fig. 2).

We found that resistance to an observed extreme drought, which occurred in 2021, could be forecasted based on temporal stability observed from 1996 to 2020 (Fig. 4). Specifically, we found that the observed level of drought resistance was forecastable with moderate accuracy, on average within 3.1% or 7.0%, when estimated at the ecosystem and species levels, respectively (Fig. 4). However, at the species level, there was a non-significant relationship between the forecasted and observed values (Fig. 4b; P > 0.1). Drought resistance was not systematically overestimated or underestimated (Fig. 4). The error on these predictions decreased with the number of years of data and at least 17 years of data were needed before the relationship became statistically significant at the ecosystem level (Extended Data Fig. 4). These results suggest that it may be possible, in some cases, to forecast drought resistance at the ecosystem level from monitoring productivity and its temporal stability, especially from relatively long time series.

Fig. 4. Forecasting drought resistance at the ecosystem and species levels by monitoring temporal stability before the drought.

a,b, Resistance to a drought that occurred in 2021 was forecasted from temporal stability observed before the drought (1996–2020) using approximation A4 in Table 1. The dashed lines show the 1:1 relationship. The solid lines indicate the mean linear model fits to data and the shaded bands show 95% CI. Points are ecosystem-level (that is, plot; a) or species-level (b) estimates from diversity–interactions models (Extended Data Table 1). Relationships are shown for complement measures of stability (Table 1).

Extended Data Fig. 4. Drought resistance at the ecosystem level was better forecasted when temporal stability was monitored for more years.

Resistance to a drought that occurred in 2021 was forecasted from temporal stability using approximation A2 in Table 1, with temporal stability observed over a time window that included the first three, four, five, and so on years of data. MAPE is the mean absolute percentage error of the forecast. The relationship between the observed and forecasted values became statistically significant when temporal stability was based on at least 17 years of data.

Our finding that temporal stability and resistance can be predicted from one another is intuitive, but not inevitable. It is intuitive that systems better able to resist displacement during a perturbation also exhibit greater long-term temporal stability. Indeed, in our mathematical predictions, the deviation during a perturbation is a component of both the long-term standard deviation and resistance. However, we have also shown that predictions can be made out of sample. For example, when forecasting resistance to the 2021 drought, we used temporal stability observed before that year, from 1996 to 2020 (Fig. 4). Furthermore, resistance to the 2002 wet event is not only correlated with temporal stability across all years (Fig. 2a) but also when 2002 is excluded from temporal stability (that is, correlation between temporal stability, $I_2$, and resistance, ${\mathrm{\Omega }}_2$) at the ecosystem level (Pearson’s r = 0.84, P < 2.2 × 10⁻¹⁶) and at the species level (r = 0.76, P = 6.7 × 10⁻⁴). In addition, as would be expected, predictions are less accurate when resistance is quantified during normal climate years, rather than during extreme climate events. For example, at the ecosystem level, errors for approximations of temporal stability (using A2 in Table 1) were nearly twice as large when resistance was quantified for normal years (average mean absolute percentage error = 4.1% for 1996, 2001, 2005, 2007, 2009, 2012, 2013 and 2018), rather than years with extreme climate events (average mean absolute percentage error = 2.3% for 2002 and 2021). This underscores the importance of identifying the relevant perturbations that drive temporal variation when using our approach to predict stability.

Indeed, applying our approach requires knowledge of the relevant perturbations and their frequency. For example, results of previous studies informed our decisions to consider interannual changes in water balance as a relevant perturbation and to choose our parameter estimate for P, the number of years in which there was a perturbation. At our study site, as in many other ecosystems^31–34, variation in plant productivity over time is partly explained by changes in water availability^3,7,35 (Extended Data Fig. 3). Furthermore, productivity responds not only to extreme and infrequent events, but also to more moderate and frequent wet and dry events³. Thus, we defined P accordingly. For simplicity when introducing these ideas, here we have focused on climate events as a single type of perturbation. However, we acknowledge that dynamical systems often have multiple types of perturbations and systems may, for example, be resistant to one but not another type of perturbation, and different types of perturbations may reinforce or counterbalance one another. In addition, although our approach can be applied to both pulse and press perturbations (Methods), detrending or other approaches may be required to account for directionally shifting systems. We encourage future studies that explore these complexities.

Our finding that recovery often contributes relatively little to temporal stability is interpretable, even if unexpected. Although temporal stability can, under some conditions, depend on recovery^1–4 (Fig. 1b), here we found that recovery only contributes substantially to temporal stability when perturbations are relatively infrequent and recovery is relatively slow (Fig. 1b and Extended Data Fig. 1b,d). In other words, recovery contributes to temporal stability especially when a system spends much of its time recovering. When perturbations are frequent, the system has little time to recover between perturbations and thus recovery contributes relatively little to temporal stability. When perturbations are infrequent and recovery is sufficiently rapid, the system spends most of its time essentially recovered and thus recovery again contributes little to temporal stability. Our findings help to clarify the conditions under which recovery rates may be ignored when predicting temporal stability from its resistance and recovery components.

To our knowledge, our work is the first framework for predicting long-term temporal stability from a small subset of that information: that is, single-year estimates of resistance and recovery, or just resistance. Other approaches for predicting ecosystem stability, such as predicting mixtures from monocultures³⁹ or partitioning ecosystem stability into its species stability and species asynchrony components⁴⁰, require the full time series for the predictors. Furthermore, although some other variables, such as species richness, can explain much of the variation in temporal stability^3,25, they cannot quantitatively predict the level of temporal stability or explain why some species exhibit greater temporal stability than others in monocultures, within a level of species richness. For example, in the biodiversity experiment considered here, nearly half (R² = 0.48) of the variation in ecosystem stability is explained by the species richness treatment, whereas nearly three-quarters of its variation is explained by the predictions based on either resistance (R² = 0.73) or both resistance and recovery (R² = 0.74). In these ways, our approach substantially extends previous related work.

Our theoretical and empirical results reveal that resistance and recovery are not substitutable for providing temporal stability or resilience. This finding may be especially important in systems that exhibit a tradeoff between resistance and recovery. In such systems, a shift towards more rapid recovery and less resistance could change how integrated measures of stability arise (that is, increasingly from recovery rather than from resistance), while also systematically decreasing temporal stability and increasing resilience. For example, plants exhibit a tradeoff along a fast–slow continuum of resource acquisition traits and growth^41,42, which is hypothesized to correspond to a tradeoff along a recovery–resistance continuum^4,21. If, for instance, global changes favour fast plants over slow plants²¹, then integrated measures of stability may depend increasingly on the recovery provided by fast plant species and decreasingly on the resistance provided by slow plant species, and temporal stability may decrease while resilience may increase (for x > 1). An important next step will be to investigate potential tradeoffs between resistance and recovery, and how shifts towards one component or the other systematically alter temporal stability and resilience. This may be especially important in other dynamical systems that exhibit a wider range of resistance than the plant communities considered here.

Our predicted relationships may also be applicable in systems with multiple stable states (or other dynamic attractors)⁴³, including undesirable states. Stability is not always desirable. When a system is stuck in an undesirable state, making the system less resistant, for example, may allow it to cross a critical threshold and transition to an alternative, more desirable state. Note that our approach focuses on system dynamics and relationships among stability components within a basin of attraction. Our predicted relationships could be separately applied to multiple (desirable or undesirable) basins of attraction before and after a critical transition⁴³. To do so, a time series could be subdivided into separate periods of time when the system is thought to be fluctuating within each basin of attraction, excluding time periods when the system undergoes a critical transition between basins of attraction. New studies will be needed to predict relationships among stability components during critical transitions (for example, by bridging our work with studies of critical slowing and other early warning signals^12,44).

In conclusion, our results help to move from a longstanding recognition that there are many types or dimensions of stability^{1–3,5,7,17–23} to a hierarchical organization for four types of stability, with two integrated measures (that is, temporal stability and resilience) each having two components (that is, resistance and recovery), which vary in relative importance according to predictable relationships for sets of measures (Table 1). Although systems can be unstable in some ways and stable in others (for example, highly resistant but slowly recovering), some integrated types of stability nevertheless predictably arise from other component types of stability. We found that temporal stability may often be predicted from resistance, without needing to know how quickly the system recovers. By contrast, resilience is predicted to depend at least as much on recovery as on resistance. Our results show that resistance and recovery are not substitutable for providing temporal stability and resilience, instead focusing attention on their most critical components. Furthermore, our results provide a way to forecast resistance to future perturbations from monitoring temporal stability. Together, these findings may help to predict and enhance the temporal stability and resilience of both natural and managed systems, and forecast the sensitivity of these systems to perturbations, such as extreme climate events that are becoming increasingly frequent and intense¹⁶.

Methods

Here we develop an approach for predicting temporal stability and resilience from their resistance and recovery components. We began by considering a common measure of temporal stability, the inverse of the coefficient of variation, and defined similar measures of resistance, recovery and resilience that are also the inverse of a deviation. Given that the predicted relationships between integrated measures of stability (that is, temporal stability and resilience) and their component measures of stability (that is, resistance and recovery) will depend on the measures used, we also defined a second set of stability measures as the complement, rather than the inverse, of a deviation. We then showed how temporal stability and resilience can be predicted from their resistance and recovery components, and how resistance can be forecasted from monitoring temporal stability. After generating these new theoretical predictions, we empirically tested them with data from the world’s longest-running biodiversity experiment, allowing us to consider stability at both the ecosystem and the species levels.

Defining two sets of stability measures

We defined temporal stability (also known as invariability) with a common measure, the inverse of the coefficient of variation (CV; D1 in Table 1):

$$I_1\equiv \frac{\mu }{\sigma }=\frac{1}{\mathrm{CV}},$$

and we similarly defined resistance (D2 in Table 1):

$${\mathrm{\Omega }}_1\equiv \frac{\overline{Y_n}}{|\overline{Y_n}-Y_e|},$$

and recovery (D3 in Table 1):

$${\mathrm{\Delta }}_1\equiv \left|\frac{\overline{Y_n}-Y_e}{\overline{Y_n}-Y_{e+1}}\right|,$$

with measures from our previous related work³. We similarly defined resilience (sensu proximity to unperturbed levels after a perturbation^5–10) as (D4 in Table 1):

$${\mathrm{\Phi }}_{1,x}\equiv \frac{\overline{Y_n}}{|\overline{Y_n}-Y_{e+x}|}.$$

Here $\mu$ and $\sigma$ are the mean and standard deviation, respectively, of the time series and $\overline{Y_n}$, Y_e, Y_e + 1 and Y_e + x are respectively values (for example, ecosystem productivity or other measures of a system level) during normal times (mean across all non-perturbation times), during a perturbation, and during one (or x) time unit (or units) after a perturbation. We refer to these four (D1 to D4) as inverse measures of stability, given that they are all the inverse of a deviation. We used a subscript of one to denote inverse measures of stability.

Given that the predicted relationships between an integrated measure of stability and its components will probably depend on the measures used, we additionally considered a second set of stability measures that define temporal stability (D5 in Table 1):

$$I_2\equiv 1-\frac{\sigma }{\mu }=1-\mathrm{CV},$$

resistance (D6 in Table 1):

$${\mathrm{\Omega }}_2\equiv 1-\frac{|\overline{Y_n}-Y_e|}{\overline{Y_n}},$$

recovery (D7 in Table 1):

$${\mathrm{\Delta }}_2\equiv 1-|\frac{\overline{Y_n}-Y_{e+1}}{\overline{Y_n}-Y_e}|,$$

and resilience (D8 in Table 1):

$${\mathrm{\Phi }}_{2,x}\equiv 1-\frac{|\overline{Y_n}-Y_{e+\mathit{x}}|}{\overline{Y_n}}.$$

We refer to these as complement measures of stability, given that they are the complement (rather than the inverse) of a deviation. We used a subscript of two to denote complement measures of stability.

All of these stability measures (definitions D1 to D8) are dimensionless, and thus can be directly compared among studies and communities with different normal levels, such as different levels of normal productivity, or among data with different units from different dynamical systems. They are also symmetric, and thus can be directly compared between positive and negative perturbations (Fig. 1a), such as wet and dry climate events. Furthermore, these stability measures can be applied to dynamical systems that exhibit either monotonic recovery or damped oscillations after a perturbation.

For both sets of measures, resistance indicates a lack of displacement during a perturbation, quantified relative to the normal level. Inverse resistance (${\mathrm{\Omega }}_1$) is quantified as the inverse of a proportional displacement from normal levels during a perturbation. For example, if the system is reduced by one-quarter of its normal level during a perturbation, then ${\mathrm{\Omega }}_1$ = 4 (that is, the inverse of one-quarter; Fig. 1a, high-resistance case). Complement resistance (${\mathrm{\Omega }}_2$) is quantified as the complement of a proportional displacement from normal levels during a perturbation. For example, if the system is reduced by one-quarter of its normal level during a perturbation, then ${\mathrm{\Omega }}_2$ = 0.75 (that is, the complement of one-quarter; Fig. 1a, high-resistance case).

For both sets of measures, recovery indicates the rate of return to normal levels just after a perturbation, relative to the magnitude of displacement. Inverse recovery (${\mathrm{\Delta }}_1$) is quantified as the inverse of a proportional lack of recovery just after a perturbation. For example, if the system recovers all but one-quarter of the way from perturbed to normal levels during the time step following a perturbation, then ${\mathrm{\Delta }}_1$ = 4 (that is, the inverse of the one-quarter proportion of displacement not yet recovered; Fig. 1a, high-recovery case). Complement recovery (${\mathrm{\Delta }}_2$) is quantified as the complement of a proportional lack of recovery after a perturbation. For example, if the system recovers all but one-quarter of the way from perturbed to normal levels during the time step following a perturbation, then ${\mathrm{\Delta }}_2$ = 0.75 (that is, the complement of the one-quarter proportion of displacement not yet recovered; Fig. 1a, high-recovery case). Here we focused on the case in which the system returns to normal levels following a perturbation (${\mathrm{\Delta }}_1$ > 1, ${\mathrm{\Delta }}_2$ > 0); however, future studies could additionally consider cases in which a system fails to recover and instead moves away from a repeller or has multiple attractors (for example, ${\mathrm{\Delta }}_1$ < 1, ${\mathrm{\Delta }}_2$ < 0).

For both sets of measures, resilience indicates the proximity to normal levels after a perturbation, relative to the normal level. We recognize that the concept of resilience is multifaceted and has multiple definitions^45,46. In fact, we were unable to find any definition of resilience that all co-authors of this paper agreed with. In previous work³, some of us have used a narrower definition of resilience that is mathematically independent of resistance and equivalent to how recovery is defined here (D3). Here we instead used the term resilience consistent with several other previous studies^5–10 where resilience includes both resistance and recovery components. Inverse resilience (${\mathrm{\Phi }}_{1,x}$) is quantified as the inverse of a proportional deviation from normal levels at some time, x, after a perturbation. For example, if the system is reduced by one-quarter of its normal level during a perturbation (${\mathrm{\Omega }}_1$ = 4, ${\mathrm{\Omega }}_2$ = 0.75, $Y_e=75$) and if the system then recovers all but one-quarter of the way from perturbed to normal levels during the first time step following a perturbation (${\mathrm{\Delta }}_1$ = 4, ${\mathrm{\Delta }}_2$ = 0.75, $Y_{e+1}=93.75$), and then recovers all but one-quarter of the remaining distance to normal levels during the second time step following a perturbation ($Y_{e+2}=98.44$), then ${\mathrm{\Phi }}_{1,2}=64$ (Fig. 1a, high-resistance and high-recovery case). Complement resilience (${\mathrm{\Phi }}_{2,x}$) is quantified as the complement of a proportional deviation from normal levels at some time, x, after a perturbation. For example, in the case of high-resistance and high-recovery shown in Fig. 1a, ${\mathrm{\Phi }}_{2,2}=0.984$. Our consideration of resilience is a simple discrete-time extension of previous theoretical work in ecology, which considered continuous-time and small perturbations from an equilibrium value^11,12. Although less novel, we included resilience in our framework to relate our novel findings for temporal stability and its resistance and recovery components to this earlier work considering resilience.

Complement measures of stability have two advantages over inverse measures: (1) all four complement measures will often be in the range from zero to one with intuitive interpretations (see below), and (2) three of the four complement measures (that is, $I_2,{\mathrm{\Omega }}_2,{\mathrm{\Phi }}_{2,x}$) avoid running off to infinity by avoiding having a difference that can approach zero in their denominator (Extended Data Fig. 1).

Complement measures of stability have intuitive interpretations. The complement measure of resistance indicates the proportion of the normal level retained during the perturbation. For example, ${\mathrm{\Omega }}_2$ = 0.75 indicates that the system retained 75% of its normal level during the perturbation. The complement measure of recovery indicates that the proportion of the displacement recovered in the time step after the perturbation. For example, ${\mathrm{\Delta }}_2$ = 0.75 indicates the system recovered 75% of the way back to normal levels just after the perturbation. The complement measure of resilience indicates that the proportion of the normal level at some specified time after the perturbation. For example, ${\mathrm{\Phi }}_{2,2}=0.984$ indicates that the system is at 98.4% of its normal level two time steps after the perturbation.

One potential weakness of complement measures of stability is that they can be negative. For complement measures of resilience and recovery, this simply means that the system is moving farther from, rather than closer to, unperturbed levels, or possibly that observation error is overwhelming any recovery. Complement measures of resistance will be negative when the system is displaced by more than 100%, and temporal stability will be negative when the standard deviation is larger than the mean. In the data considered here, only recovery was observed to be negative in some cases (Fig. 2).

Predicting temporal stability from resistance and recovery

Consider a discrete time series of system levels (for example, ecosystem productivities), periodically perturbed (for example, by climate events), with recovery back towards an attractor or away from a repeller after perturbations. If we assume that, during a perturbation, the system level depends only on resistance, then solving for Y_e in the resistance formula gives the system level during a negatively perturbed time (for example, productivity during drought), $\overline{Y_n}$ – $\overline{Y_n}$/${\mathrm{\Omega }}_1$, or a positively perturbed time, $\overline{Y_n}$ + $\overline{Y_n}$/${\mathrm{\Omega }}_1$ (Fig. 1a). If we assume recovery occurs at a constant proportional rate, then solving for Y_e + 1 in the recovery formula gives the system level during recovery times t either for monotonic recovery, Y_t = $\overline{Y_n}$ − ($\overline{Y_n}$ − Y_t − 1)/${\mathrm{\Delta }}_1$ (Fig. 1a), or for damped oscillations, Y_t = $\overline{Y_n}$ + ($\overline{Y_n}$ − Y_t − 1)/${\mathrm{\Delta }}_1$. Without loss of generality, consider the case in which there is a negative perturbation during time one and recovery via damped oscillations during subsequent times. The first four values of the time series would be given by:

$$Y_1=\overline{Y_n}-\frac{\overline{Y_n}}{{\mathrm{\Omega }}_1},$$ M1

$$Y_2=\overline{Y_n}+\frac{\overline{Y_n}-Y_1}{{\mathrm{\Delta }}_1}=\overline{Y_n}+\frac{\overline{Y_n}-(\overline{Y_n}-\overline{Y_n}/{\mathrm{\Omega }}_1)}{{\mathrm{\Delta }}_1}=\overline{Y_n}+\frac{\overline{Y_n}}{{\mathrm{\Omega }}_1{\mathrm{\Delta }}_1},$$ M2

$$Y_3=\overline{Y_n}+\frac{\overline{Y_n}-Y_2}{{\mathrm{\Delta }}_1}=\overline{Y_n}+\frac{\overline{Y_n}-\left(\overline{Y_n}+\frac{\overline{Y_n}}{{\mathrm{\Omega }}_1{\mathrm{\Delta }}_1}\right)}{{\mathrm{\Delta }}_1}=\overline{Y_n}-\frac{\overline{Y_n}}{{\mathrm{\Omega }}_1{∆}_1^2},$$ M3

$$Y_4=\overline{Y_n}+\frac{\overline{Y_n}-Y_3}{{\mathrm{\Delta }}_1}=\overline{Y_n}+\frac{\overline{Y_n}-\left(\overline{Y_n}-\frac{\overline{Y_n}}{{\mathrm{\Omega }}_1{∆}_1^2}\right)}{{\mathrm{\Delta }}_1}=\overline{Y_n}+\frac{\overline{Y_n}}{{\mathrm{\Omega }}_1{∆}_1^3},$$ M4

and therefore, by induction, values at time t in this time series can be given by:

$$Y_t=\overline{Y_n}+s_t\frac{\overline{Y_n}}{{\mathrm{\Omega }}_1{∆}_1^{t-1}},$$ M5

where $s_t$ is a sign factor indicating whether the system is below or above $\overline{Y_n}$ at time t:

$$s_t=\left\{\begin{array}{cc}-1,& Y_t<\overline{Y_n}\\ +1,& Y_t>\overline{Y_n}\end{array}\right).$$

For monotonic recovery, we take $s_t=s$ for all t, with $s\in \{-1,+1\}$ fixed for a given trajectory ($s=-1$ for a negative perturbation and $s=+1$ for a positive perturbation). For damped oscillations, we instead set $s_t=s{\left(-1\right)}^{t-1}$ so that the deviation from $\overline{Y_n}$ alternates in sign over time. For the case considered above, which has a negative perturbation followed by recovery via damped oscillations, this gives $s_t=-{\left(-1\right)}^{t-1}={\left(-1\right)}^t$. M5 also applies to positive (rather than negative) perturbations and to monotonic (rather than oscillatory) recovery.

Next, if we further assumed that positive and negative perturbations are symmetric such that the time series mean equals the unperturbed mean, μ = $\overline{Y_n}$, where μ is the mean value across all times, and that perturbations are equally spaced over time with T/P time units between perturbations, where T is the length of the time series and P is the number of perturbations, then the variance of the time series can be given as a function of resistance and recovery:

$${\sigma }^2=\frac{1}{T-1}\sum _{t=1}^T{\left(Y_t-\overline{Y_n}\right)}^2=\frac{P}{T-1}\sum _{t=1}^{T/P}{\left(Y_t-\overline{Y_n}\right)}^2=\frac{P}{T-1}\sum _{t=1}^{T/P}{\left(\frac{\overline{Y_n}}{{\mathrm{\Omega }}_1{∆}_1^{t-1}}\right)}^2,$$ M6

where the second form groups time steps by perturbation cycles, and temporal stability can then be specified as a function of its resistance and recovery components:

$$I_1=\frac{\mu }{\sigma }=\frac{\overline{Y_n}}{\sqrt{\frac{P}{T-1}{\sum }_{t=1}^{T/P}{\left(\frac{\overline{Y_n}}{{\mathrm{\Omega }}_1{∆}_1^{t-1}}\right)}^2}}.$$ M7

Examples of dynamics that meet all these assumptions for contrasting levels of resistance and recovery are shown in Fig. 1a. For cases where ${\mathrm{\Omega }}_1$ > 0, ${\mathrm{\Delta }}_1$ > 0 and $\overline{Y_n}$ > 0, and given that for any geometric series ${\sum }_{k=1}^n{\mathrm{c}}^k=c\frac{c^n-1}{c-1}$, M7 can be further simplified to (E1 in Table 1):

$$I_1=\frac{{\mathrm{\Omega }}_1}{{\mathrm{\Delta }}_1\sqrt{\frac{P\left(1-{∆}_1^{-2T/P}\right)}{\left(T-1\right)\left({∆}_1^2-1\right)}}}.$$

E1 in Table 1 specifies how increasing either resistance or recovery could increase temporal stability, as shown in Extended Data Fig. 1b.

E1 in Table 1 also shows that temporal stability may often depend much more on resistance than on recovery. First, note that recovery contributes very little to temporal stability as the frequency of perturbations increases. That is, as P approaches T (with 0 ≤ P ≤ T), E1 in Table 1 simplifies to the following approximation (A1 in Table 1):

$$I_1\approx \frac{{\mathrm{\Omega }}_1}{\sqrt{\frac{P}{\left(T-1\right)}}},$$

and temporal stability depends only on resistance. To see this simplification, recall that $1-{\mathrm{\Delta }}_1^{-2}$ can be rewritten as $\left({\mathrm{\Delta }}_1^2-1\right)/{\mathrm{\Delta }}_1^2$ and thus ${\mathrm{\Delta }}_1\sqrt{\left(({\mathrm{\Delta }}_1^2-1)/{\mathrm{\Delta }}_1^2\right)/({\mathrm{\Delta }}_1^2-1)}={\mathrm{\Delta }}_1\sqrt{1/{\mathrm{\Delta }}_1^2}=1$. Furthermore, for frequent perturbations and sufficiently large T (that is, in the limit as $P\to T$ and $T\to \infty$), A1 becomes $I_1\approx {\mathrm{\Omega }}_1$, and thus temporal stability becomes nearly equivalent in magnitude to resistance. When perturbations are frequent, recovery contributes little to temporal stability because the period of recovery between perturbations is brief. Second, note that recovery also contributes very little to temporal stability when recovery is relatively rapid, even if perturbations are infrequent. When recovery is rapid or perturbations are infrequent, $1-{∆}_1^{-2T/P}$ approaches one; and when recovery is rapid, ${∆}_1\sqrt{1/\left({∆}_1^2-1\right)}$ approaches one. In these cases, E1 in Table 1 again simplifies to approximation A1. When recovery is rapid, even if perturbations are infrequent, recovery contributes little to temporal stability because the system is essentially fully recovered during much of the period between perturbations. Therefore, recovery will only substantially contribute to temporal stability when the system spends much of its time recovering, which happens when two conditions are met: (1) perturbations are infrequent, and (2) recovery is slow.

Consequently, temporal stability can often be approximated by the resistance of the system to perturbations and frequency of perturbations, irrespective of its rate of recovery following perturbations. For relatively long time series, $T-1\approx T$, and approximation A1 can be further simplified to:

$$I_1\approx {\mathrm{\Omega }}_1/\sqrt{p},$$

where p is the proportion of times in which there is a perturbation. This simple approximation may be useful for many dynamical systems.

Similar predictions can be made for complement measures of stability. Using the relationship between the two sets of measures, $I_2=1-\frac{1}{{\mathrm{I}}_1}$, M7 can be rewritten as:

$$I_2=1-\frac{1}{I_1}=1-\frac{\sqrt{\frac{P}{T-1}{\sum }_{t=1}^{T/P}{\left(\frac{\overline{Y_n}}{{\mathrm{\Omega }}_1{∆}_1^{t-1}}\right)}^2}}{\overline{Y_n}}.$$ M8

Substituting ${\mathrm{\Omega }}_2=1-\frac{1}{{\mathrm{\Omega }}_1}$ and ${\mathrm{\Delta }}_2=1-\frac{1}{{\mathrm{\Delta }}_1}$ gives the complement measure of temporal stability as a function of its resistance and recovery components:

$$I_2=1-\frac{\sqrt{\frac{P}{T-1}{\sum }_{t=1}^{T/P}{\left(\overline{Y_n}\right(1-{\mathrm{\Omega }}_2\left){(1-{\mathrm{\Delta }}_2)}^{t-1}\right)}^2}}{\overline{Y_n}}=1-(1-{\mathrm{\Omega }}_2)\sqrt{\frac{P}{T-1}{\sum }_{t=1}^{T/P}{(1-{\mathrm{\Delta }}_2)}^{2t-2}}.$$ M9

For cases where ${\mathrm{\Omega }}_2$ > 0, ${\mathrm{\Delta }}_2$ > 0 and $\overline{Y_n}$ > 0, and given that for any geometric series ${\sum }_{k=1}^n{\mathrm{c}}^k=c\frac{c^n-1}{c-1}$, M9 can be further simplified to (E2 in Table 1):

$$I_2=1+\left({\mathrm{\Omega }}_2-1\right)\sqrt{\frac{P\left[(1-{{\mathrm{\Delta }}_2)}^{2T/P}-1\right]}{\left(T-1\right){\mathrm{\Delta }}_2\left({\mathrm{\Delta }}_2-2\right)}}.$$

E2 in Table 1 also shows that temporal stability may often depend much more on resistance than on recovery. First, note that recovery contributes very little to temporal stability as the frequency of perturbations increases. For example, as P approaches T, E2 in Table 1 simplifies to (A2 in Table 1):

$$I_2\approx 1+\left({\mathrm{\Omega }}_2-1\right)\sqrt{\frac{P}{T-1}}$$

and for sufficiently large T, this becomes $I_2\approx {\mathrm{\Omega }}_2$. Second, note that recovery also contributes very little to temporal stability when recovery is relatively rapid. That is, as ${\mathrm{\Delta }}_2$ approaches one, E2 in Table 1 again simplifies to approximation A2.

For relatively long time series, $T-1\approx T$, and approximation A2 can be further simplified to:

$$I_2\approx 1+\left({\mathrm{\Omega }}_2-1\right)\sqrt{p},$$

These equations and approximations (E1, E2, A1 and A2 in Table 1) can be applied to both pulse and press perturbations because: (1) the proportion of times in which a perturbation occurs can range from 0 to 1, and (2) the time steps can be defined over very brief (for example, daily measurements of soil moisture) or long (for example, multiyear droughts) periods. Note that there may be a similar level of temporal stability resulting from low resistance to an infrequent event (for example, p = 0.01 for a 1 in 100 year drought) or high resistance to a more frequent event (for example, p = 0.1 for a 1 in 10 year drought).

Also, note that these simple approximations (A1 and A2) can be used to estimate temporal stability over any time interval at least as long as the time interval over which the system recovers relatively quickly. For example, temporal stability is often insensitive to recovery rates when ${\mathrm{\Delta }}_1>2$ (Extended Data Fig. 1b). Thus, if it takes 3 years for the system to recover halfway back to normal levels, then it may be possible to approximate temporal stability across intervals of 3 or more years.

In addition, to avoid using a long time series for estimating the unperturbed system level, $\overline{Y_n}$, one can use other data (for example, unperturbed controls in an experiment) or the value observed one time step before the perturbation event, $Y_{e-1}$, assuming the system was unperturbed at that time.

Predicting resilience from resistance and recovery

Next, we considered how resilience depends on its resistance and recovery components. M5 can be rearranged to solve for the deviation between normal levels and the level at a specified time of interest (that is, where $Y_t=Y_{e+x}$):

$$|\overline{Y_n}-Y_{e+x}|=\frac{\overline{Y_n}}{{\mathrm{\Omega }}_1{∆}_1^{t-1}}.$$ M10

Then, given definition D4 above and that x = t − 1 because the perturbation occurred at time one and x counts the time steps after the perturbation (E3 in Table 1):

$${\mathrm{\Phi }}_{1,x}={\mathrm{\Omega }}_1{\mathrm{\Delta }}_1^x.$$

This shows that resilience depends on its resistance and recovery components. Initially, one time step after a perturbation (x = 1), resilience depends equally on resistance and recovery. Longer after a perturbation, resilience depends increasingly and eventually almost exclusively on recovery.

Similar predictions can be made for complement measures of stability. Given that ${\mathrm{\Phi }}_{2,x}=1-\frac{1}{{\mathrm{\Phi }}_{1,x}}$, E3 in Table 1 can be rewritten as:

$${\mathrm{\Phi }}_{2,x}=1-\frac{1}{{\mathrm{\Omega }}_1{∆}_1^x}.$$ M11

Substituting ${\mathrm{\Omega }}_2=1-\frac{1}{{\mathrm{\Omega }}_1}$ and ${\mathrm{\Delta }}_2=1-\frac{1}{{\mathrm{\Delta }}_1}$ gives the complement measure of resilience as a function of its resistance and recovery components (E4 in Table 1):

$${\mathrm{\Phi }}_{2,x}=1-{\left(1-{\mathrm{\Delta }}_2\right)}^x+{\mathrm{\Omega }}_2{\left(1-{\mathrm{\Delta }}_2\right)}^x.$$

Initially, just after a perturbation (when x = 1), E4 simplifies to ${\mathrm{\Phi }}_{2,1}={\mathrm{\Omega }}_2+{\mathrm{\Delta }}_2-{{\mathrm{\Omega }}_2\mathrm{\Delta }}_2$. Thus, as shown above for the inverse measures of stability, resilience initially depends equally on resistance and recovery, but increasingly depends on its recovery component after there has been more time for recovery following a perturbation.

Forecasting resistance by monitoring temporal stability

Next, we show an example of how the above equations and approximations can be rearranged to solve for other variables of interest. It may be useful to forecast resistance to a perturbation, before it occurs, by monitoring temporal stability. To do so, one could simply rearrange the above approximations A1 and A2, solving for resistance, to get (A3 in Table 1):

$${\mathrm{\Omega }}_1\approx I_1\sqrt{\frac{P}{T-1}},$$

and (A4 in Table 1):

$${\mathrm{\Omega }}_2\approx 1+\frac{I_2-1}{\sqrt{\frac{P}{T-1}}},$$

and then monitor temporal stability to forecast resistance, as empirically demonstrated below.

Predictions that differ between the two sets of stability measures

In the main text, we have emphasized and tested general predictions that are consistent between the two sets of stability measures. Here we note some differences between inverse and complement measures of stability and encourage future studies to test these as well.

For inverse measures of stability, temporal stability is predicted to depend on recovery more at high than at low resistance (Extended Data Fig. 1b). For complement measures of stability, temporal stability is predicted to depend on recovery more at low than at high resistance (Extended Data Fig. 1d). For inverse measures of stability, initial resilience is predicted to increase especially when both recovery and resistance are increased (concave up relationship in Extended Data Fig. 1c). By contrast, for complement measures of stability, initial resilience is predicted to increase when either recovery or resistance are increased (concave down relationship in Extended Data Fig. 1e).

Considering an alternative definition of resistance

Here and in our previous work³, we defined resistance relative to the unperturbed system level, $\overline{Y_n}$, the mean value observed across the unperturbed times. Some previous studies have instead quantified resistance relative to the pre-perturbed system level, Y_e − 1 (for example, replacing $\overline{Y_n}$ with Y_e − 1 in D2 or D6). The two references will be similar ($Y_{e-1}\approx \overline{Y_n}$) when the system has had sufficient time to nearly recover between perturbations. By contrast, when there are multiple sequential perturbations, the two references make very different assumptions about how the system responds. Our definitions above consider how far the perturbation displaces the system from the unperturbed level, regardless of its pre-perturbed level. Alternatively, perturbations may displace the system some distance from its pre-perturbed level, regardless of its unperturbed level. For example, consider what might occur if a slowly recovering system has no recovery time between two consecutive climate events. If the system starts at its unperturbed level during year 1, then decreases to 50% below its unperturbed level during a negative perturbation in year 2, then a positive perturbation in year 3 might increase the system to:

1. 150% of its unperturbed level, decreasing and increasing by the same percentage (that is, 50%) above and below the unperturbed system level (as assumed in D2 and D6);

2. 100% of its unperturbed level, decreasing and increasing by the same absolute amount (that is, 50% of the unperturbed level) during the dry and wet year; or

3. 75% of its unperturbed level, decreasing and increasing by the same percentage (50%) during the dry and wet year, relative to the pre-perturbed level, which is 100% for the dry event and 50% for the wet event.

Our definitions above (D2 and D6) assume the first of these three possibilities. Below, we empirically tested this assumption and found some support for it (see ‘Testing assumptions’ below). Next, we also considered mathematically the third possibility: the system changes by a constant percentage relative to the pre-perturbed system level.

To consider how our main results might differ had we used a different definition of resistance, we determined how temporal stability depends on resistance and recovery when resistance is defined in reference to the pre-perturbation, rather than the unperturbed, system level. To do so, we replaced $\overline{Y_n}$ with Y_e − 1 in D2, providing an alternative definition of resistance as:

$$\widehat{{\mathrm{\Omega }}_1}\equiv \frac{Y_{e-1}}{|Y_{e-1}-Y_e|},$$

M1–M5 remain unchanged, simply with Y_e − 1 in place of $\overline{Y_n}$, and thus M5 becomes:

$$Y_t=Y_{e-1}+s_t\frac{Y_{e-1}}{{\mathrm{\Omega }}_1{∆}_1^{t-1}},$$ M12

When resistance is defined with respect to the pre-perturbation system level, rather than the unperturbed system level, M6 becomes:

$${\sigma }^2=\frac{P}{T-1}\sum _{t=1}^{T/P}{\left(Y_{e-1}-\overline{Y_n}+s_t\frac{Y_{e-1}}{\widehat{{\mathrm{\Omega }}_1}{∆}_1^{t-1}}\right)}^2,$$ M13

and M7 becomes:

$$\widehat{I_1}=\frac{\mu }{\sigma }=\frac{\overline{Y_n}}{\sqrt{\frac{P}{T-1}\sum _{t=1}^{T/P}{\left(Y_{e-1}-\overline{Y_n}+s_t\frac{Y_{e-1}}{\widehat{{\mathrm{\Omega }}_1}{∆}_1^{t-1}}\right)}^2}}.$$ M14

When the pre-perturbation system level is near the unperturbed system level, $Y_{e-1}-\overline{Y_n}\approx 0$, the two definitions for resistance are equivalent (${\mathrm{\Omega }}_1=\widehat{{\mathrm{\Omega }}_1}$) and the predictions and approximations above hold (that is, M14 simplifies to E1 and A1 in Table 1).

However, when the pre-perturbation system level is far from unperturbed levels, as might be the case when there are multiple sequential perturbations, this difference cannot be ignored. Furthermore, in this case, the sign factor does not immediately cancel and thus, for monotonic recovery, E1 in Table 1 becomes:

$$\begin{array}{c}\widehat{I_{1,m}}=\hfill \\ \frac{\overline{Y_n}}{\sqrt{\frac{P}{T-1}\left[\frac{T}{P}{\left(Y_{e-1}-\overline{Y_n}\right)}^2+\frac{2s{Y}_{e-1}\left(Y_{e-1}-\overline{Y_n}\right)}{\widehat{{\mathrm{\Omega }}_1}}\frac{{∆}_1\left(1-{∆}_1^{-T/P}\right)}{{∆}_1-1}+\frac{{Y_{e-1}}^2}{{\widehat{{\mathrm{\Omega }}_1}}^2}\frac{{{∆}_1}^2\left(1-{{∆}_1}^{-2T/P}\right)}{{{∆}_1}^2-1}\right]}}.\hfill \end{array}$$ M15

and, for damped oscillations, E1 in Table 1 becomes:

$$\begin{array}{c}\widehat{I_{1,d}}=\hfill \\ \frac{\overline{Y_n}}{\sqrt{\frac{P}{T-1}\left[\frac{T}{P}{\left(Y_{e-1}-\overline{Y_n}\right)}^2+\frac{2s{Y}_{e-1}\left(Y_{e-1}-\overline{Y_n}\right)}{\widehat{{\mathrm{\Omega }}_1}}\frac{{∆}_1\left(1-{{(-∆}_1^{-1})}^{T/P}\right)}{{∆}_1+1}+\frac{{Y_{e-1}}^2}{{\widehat{{\mathrm{\Omega }}_1}}^2}\frac{{{∆}_1}^2\left(1-{{∆}_1}^{-2T/P}\right)}{{{∆}_1}^2-1}\right]}}.\hfill \end{array}$$ M16

assuming in both cases positive parameters and ${∆}_1\ne 1$. To obtain M15 and M16, we expanded the squared term in M14 and then summed over all time steps t between perturbations, grouping terms for clarity. This produces three components: (1) the deviation of the pre-perturbation system level from the unperturbed mean, (2) a cross-term representing the interaction between this deviation and subsequent recovery dynamics, and (3) a term describing the recovery trajectory. That is, summing each component separately gives:

$$\sum _{t=1}^{T/P}{\left(Y_{e-1}-\overline{Y_n}+s_t\frac{Y_{e-1}}{\widehat{{\mathrm{\Omega }}_1}{∆}_1^{t-1}}\right)}^2=\frac{T}{P}{\left(Y_{e-1}-\overline{Y_n}\right)}^2+2s{Y}_{e-1}\left(Y_{e-1}-\overline{Y_n}\right)\frac{1}{\widehat{{\mathrm{\Omega }}_1}}\sum _{t=1}^{T/P}{s}_t{∆}_1^{-(t-1)}+\frac{Y_{e-1}^2}{{\hat{\mathrm{\Omega }}}_1^2}\sum _{t=1}^{T/P}{∆}_1^{-2(t-1)}.$$

For monotonic recovery, $s_t=s$ for all t, giving the geometric sum:

$$\sum _{t=1}^{T/P}{∆}_1^{-(t-1)}={∆}_1\left(1-{∆}_1^{-T/P}\right)/({∆}_1-1).$$

For damped oscillations, $s_t=s{(-1)}^{t-1}$, giving an alternating geometric series:

$$\sum _{t=1}^{T/P}{(-1)}^{t-1}{∆}_1^{-(t-1)}={∆}_1\left(1-{{(-∆}_1^{-1})}^{T/P}\right)/({∆}_1+1).$$

Substituting these into M14 gives the two closed forms M15 and M16, respectively.

If perturbations are symmetric, such that positive and negative deviations are equally frequent and the time-series mean equals the unperturbed mean (μ = $\overline{Y_n}$), the contributions of the sign factor cancel when summed over all perturbations. We note that, over infinite time, $\mu \ne \overline{Y_n}$ with this definition of resistance ($\widehat{{\mathrm{\Omega }}_1}$) and these dynamics (M12). However, for finite time, μ can approximately equal $\overline{Y_n}$, especially when the system nearly recovers between perturbations (that is, rapid recovery, infrequent perturbations, or both). With the symmetry assumption, the cross-term vanishes, and both M15 and M16 simplify to the following general expression:

$$\widehat{I_1}=\frac{\overline{Y_n}}{\sqrt{\frac{T}{T-1}{\left(Y_{e-1}-\overline{Y_n}\right)}^2+\frac{P}{T-1}\frac{Y_{e-1}^2}{{\hat{\mathrm{\Omega }}}_1^2}\frac{{∆}_1^2\left(1-{∆}_1^{-2T/P}\right)}{{∆}_1^2-1}}}.$$ M17

Thus, M17 describes both monotonic and oscillatory recovery dynamics under symmetric perturbations with resistance defined by the pre-perturbation, rather than the unperturbed, system level.

This general expression can be further simplified as recovery becomes rapid (${∆}_1\to \infty$), perturbations become frequent ($P\to T$), or both:

$$\frac{{∆}_1^2\left(1-{∆}_1^{-2T/P}\right)}{{∆}_1^2-1}\to 1,$$

the recovery term vanishes, and M17 simplifies to the following approximation (in place of A1 in Table 1):

$$\widehat{I_1}\approx \frac{\overline{Y_n}}{\sqrt{\frac{T}{T-1}{\left(Y_{e-1}-\overline{Y_n}\right)}^2+\frac{P}{T-1}\frac{Y_{e-1}^2}{{\hat{\mathrm{\Omega }}}_1^2}}}.$$

Therefore, considering this alternative definition of resistance provides further support that temporal stability can be approximated by resistance alone when recovery is rapid, perturbations are frequent, or both. Only when perturbations are infrequent and recovery is slow does recovery substantially influence temporal stability. These results parallel those derived above for M7, and E1 and A1 in Table 1, confirming that they hold even when resistance is defined relative to the pre-perturbation system level, rather than the unperturbed level.

Study site

Our field experiment is located at Cedar Creek Ecosystem Science Reserve, near East Bethel, MN, USA. The climate of the study site is continental, with mean January temperature of −11 °C, mean July temperature of 22 °C, and mean annual precipitation of 660 mm per year⁴⁷. The field site is located on a glacial outwash plain and has sandy and nutrient-poor soils. Before European colonization, the experimental site was an oak savanna, with wolves (Canis lupus), bison (Bison bison), elk (Cervus elaphus) and other animals and plants that later became extirpated. Subsequently, the site was used for various row crops for an unknown number of years^48,49. Between 1956 and 1968, fields on this part of the Reserve were abandoned from agriculture and became old fields undergoing succession.

BioDIV experiment

We tested these new theoretical predictions with data from the world’s longest-running biodiversity experiment: the BioDIV experiment (E120 of the Cedar Creek Long-Term Ecological Research program)^{24,26–30,50}. In 1993, in a 7-ha area: (1) glyphosate herbicide was applied; (2) the dead vegetation was burned; (3) the top 6–8 cm of soil was removed by a bulldozer to reduce the seed bank, which also reduced soil carbon and nutrients; and (4) the remaining soil was ploughed and harrowed. This site management homogenized the 7-ha area, in preparation for planting the BioDIV experiment. In 1994 and 1995, different numbers and combinations of grassland plant species were randomized to replicate plots sown with 1, 2, 4, 8 or 16 plant species from a pool of 18 species, including 16 herbaceous species and 2 oak (Quercus) species. The main experiment consists of 154 plots, each 9 × 9 m, that have been burned each spring and hand-weeded each summer to remove non-sown species. Additional experimental design details are available online (see E120 on Cedar Creek’s Long-Term Ecological Research website: https://www.cedarcreek.umn.edu).

Since 1996, peak aboveground biomass has been clipped in two strips, each 10 × 600 cm, in each of the 154 plots that have been maintained. Clipped biomass was dried to a constant mass and weighed. In these temperate grasslands, aboveground peak biomass is a proxy for annual aboveground net primary productivity because all aboveground tissues die in the winter season and are removed by annual spring burning. Plant aboveground biomass data and metadata for this experiment are available from the Environmental Data Initiative⁵¹.

Identifying wet and dry climate events

We defined all wet and dry events based on the standardized precipitation–evapotranspiration index (SPEI)³¹. Drought occurs when water availability remains below normal levels over some period of time³¹. Several drought indices consider water balance, accounting for both water inputs (precipitation) and water losses (potential evapotranspiration (PET)). Warming temperatures can contribute to drought by increasing PET. Here we used SPEI to identify and quantify the magnitude of wet and dry climate events (Extended Data Fig. 2). SPEI is a standard normal variable for water balances aggregated over a given number of months at a particular location. SPEI values are based on month-by-month variations in climate observed since 1901, based on monthly means of measurements made at more than 4,000 weather stations worldwide, and provided on 0.5 × 0.5 degree grids globally. For example, our study site experienced a value of SPEI-04 = 1.9 for August of 2002 (Extended Data Fig. 2), which corresponds to a 1-in-35-year wet event for the 4 months (thus the 04 in SPEI-04) of the growing season before the biomass harvest (that is, May, June, July and August), relative to the water balances observed for those 4 months at our site since 1901.

We obtained SPEI values from SPEIbase (v2.9)⁵² for our study site. We used SPEI-04 throughout the article because we found stronger relationships with growing season, rather than annual, water balances at our study site, probably partly because there is a short growing season and sandy, well-drained soils, making the system most sensitive to precipitation that occurs during the summer months while the plants are actively growing.

We classified experiment years as extremely dry, moderately dry, normal, moderately wet and extremely wet, as in our previous work³ (Extended Data Fig. 2). Extreme events (extremely dry or extremely wet) were defined as those that historically occurred less frequently than once per decade. Moderate events were defined as those that historically occurred between once in 4 years and once per decade. Normal years were defined as those within the interquartile range of historical water balances that did not follow an extreme event (Extended Data Fig. 2).

Statistical analyses

All analyses were conducted in R (v4.4.3)⁵³ using the tidyverse (v2.0.0)⁵⁴, DImodels (v1.3.2)³⁸ and ggpubr (v0.6.0)⁵⁵ packages.

For each BioDIV plot, we log transformed plot-level productivity and then calculated D1 to D8 in Table 1. That is, to help meet the assumption of the theoretical predictions that positive and negative perturbations are symmetric such that μ = $\overline{Y_n}$, we first log transformed productivity. Temporal stability measures were calculated for two different time intervals: 1996–2022 and 1996–2020, with the latter used for forecasting. For all measures, $\overline{Y_n}$ is the mean plot-level productivity, averaged across all normal years that did not follow an extreme event (Extended Data Fig. 2). Resistance measures were quantified for the 2002 wet event and the 2021 drought (that is, Y_e is the plot-level value observed in 2002 or 2021), with the latter used for forecasting. Recovery measures were quantified for 2003, the year after the wet event (that is, Y_e + 1 is the plot-level value observed in 2003). Resilience measures were quantified for 2004, 2 years after the wet event (that is, Y_e + 2 is the plot-level value observed in 2004).

Measures of stability that have a difference in their denominator tend to run off to infinity as this difference approaches zero, creating outliers. This issue arises for all the inverse measures of stability, but only for the complement measure of recovery (Table 1). Thus, here we largely avoid the issues of outliers by considering complement, rather than inverse, measures of stability. Although there are no perfect solutions, the influence of outliers can be assessed by censoring or transforming the data, excluding outliers, or through other regression approaches (for example, generalized linear models, beta regression and generalized additive models). Here we censored the complement measure of recovery to a minimum value of zero, which assumes that individual plots are not moving farther from normal the year after a perturbation. Our main results were qualitatively similar when we instead skipped this censoring step (although, in this case, parameter estimates are influenced by outliers), excluded these negative outliers (although, in this case, sample size is reduced to n = 99 plots), or used minimum–maximum normalization (although, in this case and with other transformations, there is an altered interpretation of parameter estimates). When considering inverse measures of stability, it might be helpful to assume that the system moves at least 1% from normal levels during a perturbation (${\mathrm{\Omega }}_1\le 100$) and that the system recovers no more than 99% of the way back to normal levels just after a perturbation (${\mathrm{\Delta }}_1\le 100$).

Next, to account for observation error when estimating stability components, and to get species-level estimates for all measures, we fit diversity–interactions (DI) models^36,37 using least squares estimation via the autoDI function in the DImodels package³⁸. DI models estimate species-specific effects, known as species identity effects in the modelling framework, and species interaction effects.

DI models have flexibility in the specification of species interactions. For example, the autoDI function compares the fits of models that assume: (1) there are no species interactions; (2) all pairs of species interact with the same strength (for example, interspecific interactions differ from intraspecific interactions); (3) each pair of functional groups has a unique pairwise interaction; or (4) each pair of species has a unique pairwise interaction. The second option was the most parsimonious for most of our measures of stability (Extended Data Table 1). For this case, where interspecific interactions differ from intraspecific interactions and are similar between all pairs of species, the DI model is:

$$y=\sum _{i=1}^S{\mathit{\beta }}_i{p}_i+\delta \sum _{1\le i<j\le \mathit{S}}{\left(p_i{p}_j\right)}^{\theta }+\mathit{ϵ}$$

where y is the community-level response (for example, biomass for a monoculture or mixture plot), $S$ is richness (the number of species), $p_i$ and ${\mathit{\beta }}_i$ are the initial proportion and identity effect of species i, respectively, $\delta$ is the interaction effect for all pairs of species, $\theta$ determines the shape of the relationship between biodiversity and ecosystem functioning or stability (for example, linear, decelerating and saturating), and $\mathit{ϵ}$ is a normally distributed random error term with constant variance.

Extended Data Table 1.

Estimates (standard errors) for complement measures of stability from parsimonious Diversity-Interactions models

*Identity effect parameters are species abbreviations: Achmi = Achillea millefolium (lanulosa); Agrsm = Agropyron smithii; AmoPet = Amorpha canescens, Petalostemum villosum, and Petalostemum candidum; Andge = Andropogon gerardii; Asctu = Asclepias tuberosa; Elyca = Elymus canadensis; Koecr = Koeleria cristata; Lesca = Lespedeza capitata; Liaas = Liatris aspera; Luppe = Lupinus perennis; MonSol = Monarda fistulosa and Solidago rigida; Panvi = Panicum virgatum; Petpu = Petalostemum purpureum; Poapr = Poa pratensis; Schsc = Schizachyrium scoparium; Sornu = Sorghastrum nutans; cases with multiple species listed for a single abbreviation had errors in planting, as described in the metadata⁵⁷.

**AV is the effect of species diversity; the most parsimonious models included an average interaction effect in which all pairs of species had statistically indistinguishable effects, as can occur, for example, when interspecific competition is less than intraspecific competition for all pairs of species.

***theta is an exponent parameter on the species interactions in DI models; theta values less than one indicate stronger than expected contributions of species interactions to the response³⁷; when the model selection process identifies that theta differs from 1.00, a 95% confidence interval is given in parentheses.

Parameter estimates from a DI model can be used to predict the response for any combinations and proportions of species. For monocultures, predictions rely only on the species identity effect in the model (for example, the prediction for species i in monoculture is $\hat{y}=\widehat{{\beta }_i}{p}_i=\widehat{{\beta }_i}\times 1=\widehat{{\beta }_i}$). For mixtures, predictions are determined by both the identity effects of the component species and any interaction effects (for example, for a two-species equi-proportional mixture of species 1 and 2: $\hat{y}=\widehat{{\beta }_1}\times 0.5+\widehat{{\beta }_2}\times 0.5+{\hat{\delta }(0.5\times 0.5)}^{\hat{\theta }}$). Here we used these models to predict species-level estimates of stability measures, which are the estimated values for monocultures of each of the 16 study species, as well as the estimated stability measures for each of the 154 experimental plots. A summary of the most parsimonious models is shown in Extended Data Table 1. Species-level predictions are the identity effects estimated in these models. This gives, for each species, an estimated value in monoculture, accounting for observation error. Ecosystem-level predictions are the plot-level estimates from these models.

Using predictions from DI models to estimate stability measures, instead of raw data, can help to reduce the influence of outliers that may arise from observation error. This is particularly helpful when using stability measures that may otherwise run off to infinity for resistance or recovery, which then leads to unrealistic predictions for temporal stability and resilience.

Note that a DI model may give a negative prediction for a community, if identity effect and/or interaction effect estimates are negative. For instance, for recovery, we found that the most parsimonious model produced negative estimates for the identity effects of two species (Extended Data Table 1). However, note that these estimates were not significantly different from zero (compare these estimates and their standard errors in Extended Data Table 1). Thus, this simply means that, in the time step after a perturbation, there was no evidence that these species significantly recovered (or significantly moved away from normal levels).

Note that our stability predictions that follow can be applied to many types of experimental or observational data, and to other types of perturbations, and thus they do not require the data from biodiversity experiments, the consideration of climate extremes, or the DI modelling framework described above. Here we tested our theoretical stability predictions by using data from a long-term biodiversity experiment because this allows us to consider stability at two levels of biological organization: species and ecosystem. We considered extreme climate events because, in grasslands, these perturbations contribute significantly to the temporal dynamics of species and ecosystem productivity. We used DI models because they are designed for the analysis of data from biodiversity experiments. We encourage future studies to apply our stability predictions below to other observational or experimental data, to carefully consider which types of perturbations are important in those systems, and to use, as a first step, a statistical modelling framework that is suited for the data and study design.

Using these plot-level and species-level estimates of resistance and recovery, we then predicted temporal stability and resilience using E1–E4 and A1–A2 in Table 1. Using plot-level and species-level estimates of temporal stability from 1996 to 2020, we used approximations A3–A4 to forecast resistance to the 2021 drought. In all cases, P = 16 non-normal years and T = 27 total years.

We chose P based on previous results, which have shown that productivity responds not only to extreme and infrequent climate events but also to more moderate and frequent wet and dry events³. Note that E1 and A1 in Table 1 can account for the fact that systems may be less resistant to infrequent than frequent perturbations. For example, the same estimate of temporal stability could arise from an observation of either relatively low resistance to an extreme 1-in-100-year drought (P/T = p = 0.01) or relatively high resistance to a more moderate 1-in-4-year drought (P/T = p = 0.25). Thus, we could have chosen different p estimates based on the extremity of the wet event in 2002 and the drought in 2021. However, updating p in this way would assume that resistance scales with the inverse of the square root of the frequency of perturbations, which may not always hold. In our case, updating p in this way would lead to extreme overestimates of ecosystem stability, probably because we know from previous results that productivity responds not only to these extreme and infrequent events but also to more moderate and frequent wet and dry events³. In other words, in our system, productivity is frequently perturbed by climate events, rather than infrequently perturbed with substantial periods of recovery between perturbations.

Next, we tested for correlations between each integrated measure of stability and its resistance and recovery components. We fit linear models (black lines and confidence intervals in all figures) using the lm function in the stats package via the geom_smooth function in the ggplot2 package and added P values and R² values with the stat_cor function in the ggpubr package⁵⁵.

Finally, we tested for relationships between the observed and predicted or forecasted values. For comparability and consistency across all relationships, here we used sequential (type I) sums of squares for all relationships. For temporal stability, a case may be made for plotting the observed values on the x axis rather than the y axis, given that we expect much greater uncertainty in the predicted values, which are based on single-year estimates, than the observed values, which are based on a quarter century of data. We did not do this here, however, because the same argument would not apply to resilience and we wanted to compare with other predictors, such as species richness. For resilience, future studies may consider using major axis regression (type II sums of squares) to account for substantial uncertainty in both the predicted and the observed variables, which are based on only 2 or 1 year of data, respectively. We did not do this here, however, because this is only recommended for sample sizes much larger than those that we were able to consider for species-level relationships. Although there is no perfect solution for all the relationships that we considered, for comparability and consistency, here we used sequential (type I) sums of squares and observed versus predicted values for all relationships. We did not focus on the estimated slopes and intercepts for these relationships because they depend on the choices described above (that is, type I versus type II sums of squares and observed versus predicted, or vice versa). Instead, we quantified error in the predictions and forecasts by quantifying the mean absolute percentage error and the R² values for each relationship.

Testing assumptions

We tested whether our data met several assumptions of our model. First, consistent with our assumption that there are symmetrical responses to perturbation, we found no significant difference in resistance between positive (wet) and negative (dry) perturbations (Extended Data Fig. 5a). Using the measure of resistance defined here and SPEI, we previously found symmetric responses of grassland plant productivity to observed wet and dry events across 46 experiments³. By contrast, a separate meta-analysis of 83 studies found that productivity often responded more to experimental wet than to experimental drought treatments, with the magnitude of wet or dry events defined by the percentage change in precipitation⁵⁶. These previous results are not necessarily inconsistent, given the many differences between how wet and dry events were defined and whether climate events were observed or experimentally created. Instead, these contrasting results indicate that the symmetry or asymmetry of responses to wet and dry events probably depends on how these events are defined and imposed. We suspect the symmetry assumption of our analysis can often be met by log-transforming productivity and defining wet and dry events based on SPEI, which accounts for water balance, including both water inputs and water losses due to PET^3,56.

Extended Data Fig. 5. Testing whether empirical data meet mathematical model assumptions.

(a) Consistent with our assumption that there are symmetrical responses to perturbation, we find no significant difference in resistance between positive (wet) and negative (dry) perturbations (F_1,14 = 0.11, P = 0.75, df = 14). Bars show mean +/− 95% confidence intervals. n = 16 event years, including 3 dry event years and 13 wet event years. (b) Consistent with our assumption that resistance can be defined with respect to the average unperturbed system level (${\mathrm{\Omega }}_1$), rather than the level at the pre-perturbation time step ($\widehat{{\mathrm{\Omega }}_1}$), we find that resistance does not depend on the level of biomass observed in the year prior to a wet event (F_1,11 = 0.05, P = 0.83, n = 13 wet event years). If instead multiple sequential positive perturbations (e.g., wet events in 2014, 2015, 2016, and 2017, see Extended Data Fig. 2a) displaced the system by the same proportional amount from the previous level of biomass, we would see a negative relationship here (e.g., 2015 would be to the right and below 2014, 2016 would be to the right and below 2015, and so on), in which case we would consider an alternative definition of resistance, such as $\widehat{{\mathrm{\Omega }}_1}$.

Second, consistent with our assumption that resistance can be defined with respect to the average unperturbed system level (${\mathrm{\Omega }}_1$), rather than the level at the pre-perturbation time step ($\widehat{{\mathrm{\Omega }}_1}$), we found that resistance — quantified with D2 — does not depend on the level of biomass observed in the year before the perturbation (Extended Data Fig. 5b). We were only able to test this for wet events, given the small number of dry events observed during the study (Extended Data Fig. 2a). Furthermore, see above (‘Considering an alternative definition of resistance’) for further evidence that the deviation of the state variable from the mean after a perturbation does not depend on the value of the state variable before the perturbation.

Third, consistent with our assumption of a constant recovery rate, we found that resilience two time steps after the perturbation can be predicted from resistance during the perturbation and recovery the year after it at the ecosystem level (Fig. 3a). This prediction assumes that the recovery rate during the second year after the perturbation is the same as was observed during the first year after the perturbation. We are unable to have a longer-term test of the assumption of a constant recovery rate because there are no extended periods of recovery with many years between climate events (Extended Data Fig. 2). For other empirical dynamical systems with extended periods of recovery, we encourage additional tests of the assumption of a constant recovery rate.

Reporting summary

Further information on research design is available in the Nature Portfolio Reporting Summary linked to this article.

Online content

Any methods, additional references, Nature Portfolio reporting summaries, source data, extended data, supplementary information, acknowledgements, peer review information; details of author contributions and competing interests; and statements of data and code availability are available at 10.1038/s41586-026-10498-4.

Author contributions

F.I. conceived the project, with input from A.S.M., M. Loreau and D.T. F.I. completed the mathematical analysis. D.T. designed the biodiversity experiment, with contributions from P.B.R. and S.N., and led data collection, with contributions from F.I., M.I.A., A.T.C., S.H., S.V. and C.R.E.W. F.I. led the data analysis, with input from A.S.M., C.B. and Y.H. F.I. secured funding for the project, in collaboration with P.B.R. and S.E.H. F.I. led the writing, with A.S.M., M. Loreau, P.B.R., D.T., M.I.A., C.B., K.C., Q.C., A.C.C., A.T.C., D.C., N.E., H.C.F., L.A.G., Y.H., M.H., J.-S.H., A.H., S.H., S.E.H., P.H., G.L., M. Liang, S.L., N.M., S.N., P.A.N., X.P., C.P.-R., B.S., H.E.R.S., S.V., M.P.V., S.W., C.R.E.W., Q.X., L.Z. and C.R.Z. contributing substantively to revisions.

Peer review

Peer review information

Nature thanks Sara König and the other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available.

Data availability

The data are included in a Code Ocean capsule (see ‘Code availability’). Biomass data and metadata are publicly available at the Environmental Data Initiative (10.6073/pasta/1f10cb47b9e121f3ea0361eeb1cc7be6). SPEI data and metadata are publicly available (10.20350/digitalCSIC/15470).

Code availability

The R code used to analyse the data and generate the figures and tables is available as a reproducible Code Ocean capsule (https://codeocean.com/capsule/0529317/tree). The capsule contains all analyses and data needed to reproduce all results.

Competing interests

The authors declare no competing interests.

Extended data

is available for this paper at 10.1038/s41586-026-10498-4.

Supplementary information

The online version contains supplementary material available at 10.1038/s41586-026-10498-4.