Rate of forcing and the forecastability of critical transitions

Early warning indicators capture catastrophic transitions driven by explicit rates of environmental change

In response to external changes, ecosystems can undergo catastrophic transitions. Early warning indicators aim to predict such transitions based on the phenomenon of critical slowing down at bifurcation points found under a constant environment. When an explicit rate of environmental change is considered, catastrophic transitions can become distinct phenomena from bifurcations, and result from a delayed response to noncatastrophic bifurcations. We use a trophic metacommunity model where transitions in time series and bifurcations of the system are distinct phenomena. We calculate early warning indicators from the time series of the continually changing system and show that they predict not the bifurcation of the underlying system but the actual catastrophic transition driven by the explicit rate of change. Predictions based on the bifurcation structure could miss catastrophic transitions that can still be captured by early warning signals calculated from time series. Our results expand the repertoire of mechanistic models used to anticipate catastrophic transitions to nonequilibrium ecological systems exposed to a constant rate of environmental change.

Early warnings of tipping in a non-autonomous turbulent reactive flow system: Efficacy, reliability, and warning times

Real-world complex systems such as the earth's climate, ecosystems, stock markets, and combustion engines are prone to dynamical transitions from one state to another, with catastrophic consequences. State variables of such systems often exhibit aperiodic fluctuations, either chaotic or stochastic in nature. Often, the parameters describing a system vary with time, showing time dependency. Constrained by these effects, it becomes difficult to be warned of an impending critical transition, as such effects contaminate the precursory signals of the transition. Therefore, a need for efficient and reliable early-warning signals (EWSs) in such complex systems is in pressing demand. Motivated by this fact, in the present work, we analyze various EWSs in the context of a non-autonomous turbulent thermoacoustic system. In particular, we investigate the efficacy of different EWS in forecasting the onset of thermoacoustic instability (TAI) and their reliability with respect to the rate of change of the control parameter. This is the first experimental study of tipping points in a non-autonomous turbulent thermoacoustic system. We consider the Reynolds number (Re) as the control parameter, which is varied linearly with time at finite rates. The considered EWSs are derived from critical slowing down, spectral properties, and fractal characteristics of the system variables. The state of TAI is associated with large amplitude acoustic pressure oscillations that could lead thermoacoustic systems to break down. We consider acoustic pressure fluctuations as a potential system variable to perform the analysis. Our analysis shows that irrespective of the rate of variation of the control parameter, the Hurst exponent and variance of autocorrelation coefficients warn of an impending transition well in advance and are more reliable than other EWS measures. Additionally, we show the variation in the warning time to an impending TAI with rates of change of the control parameter. We also investigate the variation in amplitudes of the most significant modes of acoustic pressure oscillations with the Hurst exponent. Such variations lead to scaling laws that could be significant in prediction and devising control actions to mitigate TAI.

Predicting discrete-time bifurcations with deep learning

Many natural and man-made systems are prone to critical transitions-abrupt and potentially devastating changes in dynamics. Deep learning classifiers can provide an early warning signal for critical transitions by learning generic features of bifurcations from large simulated training data sets. So far, classifiers have only been trained to predict continuous-time bifurcations, ignoring rich dynamics unique to discrete-time bifurcations. Here, we train a deep learning classifier to provide an early warning signal for the five local discrete-time bifurcations of codimension-one. We test the classifier on simulation data from discrete-time models used in physiology, economics and ecology, as well as experimental data of spontaneously beating chick-heart aggregates that undergo a period-doubling bifurcation. The classifier shows higher sensitivity and specificity than commonly used early warning signals under a wide range of noise intensities and rates of approach to the bifurcation. It also predicts the correct bifurcation in most cases, with particularly high accuracy for the period-doubling, Neimark-Sacker and fold bifurcations. Deep learning as a tool for bifurcation prediction is still in its nascence and has the potential to transform the way we monitor systems for critical transitions.

Early warning indicators of population collapse in a seasonal environment

Recent studies have demonstrated that generic statistical signals derived from time series of population abundance and fitness-related traits of individuals can provide reliable indicators of impending shifts in population dynamics. However, how the seasonal timing of environmental stressors influences these early warning indicators is not well understood. The goal of this study was to experimentally assess whether the timing of stressors influences the production, detection and sensitivity of abundance- and trait-based early warning indicators derived from declining populations. In a multi-generation, season-specific habitat loss experiment, we exposed replicate populations of Drosophila melanogaster to one of two rates of chronic habitat loss (10% or 20% per generation) in either the breeding or the non-breeding period. We counted population abundance at the beginning of each season, and measured body mass and activity levels in a sample of individuals at the end of each generation. When habitat was lost during the breeding period, declining populations produced signals consistent with those documented in previous studies. Inclusion of trait-based indicators generally improved the detection of impending population collapse. However, when habitat was lost during the non-breeding period, the predictive capacity of these indicators was comparatively diminished. Our results have important implications for interpreting signals in the wild because they suggest that the production and detection of early warning indicators depends on the season in which stressors occur, and that this is likely related to the capacity of populations to respond numerically the following season.

Manipulating the strength of organism-environment feedback increases nonlinearity and apparent hysteresis of ecosystem response to environmental change

Theory predicts that organism-environment feedbacks play a central role in how ecological communities respond to environmental change. Strong feedback causes greater nonlinearity between environmental change and ecosystem state, increases the likelihood of hysteresis in response to environmental change, and augments the possibility of alternative stable regimes. To illustrate these predictions and their dependence on a temporal scale, we simulated a minimal ecosystem model. To test the predictions, we manipulated the feedback strength between the metabolism and the dissolved oxygen concentration in an aquatic heterotrophic tri-trophic community in microecosystems. The manipulation consisted of five levels, ranging from low to high feedback strength by altering the oxygen diffusivity: free gas exchange between the microcosm atmosphere and the external air (metabolism not strongly affecting environmental oxygen), with the regular addition of 200, 100, or 50 ml of air and no gas exchange. To test for nonlinearity and hysteresis in response to environmental change, all microecosystems experienced a gradual temperature increase from 15 to 25°C and then back to 15°C. We regularly measured the dissolved oxygen concentration, total biomass, and species abundance. Nonlinearity and hysteresis were higher in treatments with stronger organism-environment feedbacks. There was no evidence that stronger feedback increased the number of observed ecosystem states. These empirical results are in broad agreement with the theory that stronger feedback increases nonlinearity and hysteresis. They therefore represent one of the first direct empirical tests of the importance of feedback strength. However, we discuss several limitations of the study, which weaken confidence in this interpretation. Research demonstrating the causal effects of feedback strength on ecosystem responses to environmental change should be placed at the core of efforts to plan for sustainable ecosystems.

Effect of time series length and resolution on abundance- and trait-based early warning signals of population declines

Natural populations are increasingly threatened with collapse at the hands of anthropogenic effects. Predicting population collapse with the help of generic early warning signals (EWS) may provide a prospective tool for identifying species or populations at highest risk. However, pattern-to-process methods such as EWS have a multitude of challenges to overcome to be useful, including the low signal-to-noise ratio of ecological systems and the need for high quality time series data. The inclusion of trait dynamics with EWS has been proposed as a more robust tool to predict population collapse. However, the length and resolution of available time series are highly variable from one system to another, especially when generation time is considered. As yet, it remains unknown how this variability with regards to generation time will alter the efficacy of EWS. Here we take both a simulation- and experimental-based approach to assess the impacts of relative time series length and resolution on the forecasting ability of EWS. We show that EWS' performance decreases with decreasing time-series length. However, there was no evident decrease in EWS performance as resolution decreased. Our simulations suggest a relative time series length between 10 and five generations as a minimum requirement for accurate forecasting by abundance-based EWS. However, when trait information is included alongside abundance-based EWS, we find positive signals at lengths one-half of what was required without them. We suggest that, in systems where specific traits are known to affect demography, trait data should be monitored and included alongside abundance data to improve forecasting reliability.

Simple signals indicate which period of the annual cycle drives declines in seasonal populations

For declining wild populations, a critical aspect of effective conservation is understanding when and where the causes of decline occur. The primary drivers of decline in migratory and seasonal populations can often be attributed to a specific period of the year. However, generic, broadly applicable indicators of these season-specific drivers of population decline remain elusive. We used a multi-generation experiment to investigate whether habitat loss in either the breeding or non-breeding period generated distinct signatures of population decline. When breeding habitat was reduced, population size remained relatively stable for several generations, before declining precipitously. When non-breeding habitat was reduced, between-season variation in population counts increased relative to control populations, and non-breeding population size declined steadily. Changes in seasonal vital rates and other indicators were predicted by the season in which habitat loss treatment occurred. Per capita reproductive output increased when non-breeding habitat was reduced and decreased with breeding habitat reduction, whereas per capita non-breeding survival showed the opposite trends. Our results reveal how simple signals inherent in counts and demographics of declining populations can indicate which period of the annual cycle is driving declines.

Early warning signals of recovery in complex systems

Early warning signals (EWSs) offer the hope that patterns observed in data can predict the future states of ecological systems. While a large body of research identifies such signals prior to the collapse of populations, the prediction that such signals should also be present before a system’s recovery has thus far been overlooked. We assess whether EWSs are present prior to the recovery of overexploited marine systems using a trait-based ecological model and analysis of real-world fisheries data. We show that both abundance and trait-based signals are independently detectable prior to the recovery of stocks, but that combining these two signals provides the best predictions of recovery. This work suggests that the efficacy of conservation interventions aimed at restoring systems which have collapsed may be predicted prior to the recovery of the system, with direct relevance for conservation planning and policy.

While several studies have documented early warning signals of population collapse, the use of such signals as indicators of population recovery has not been investigated. Here the authors use models and empirical fisheries data to show that there are statistical indicators preceding recovery of cod populations.

Early warning signals for critical transitions in cardiopulmonary health, related to air pollution in an urban Chinese population

Respiratory, and cardio-cerebrovascular health-related diseases significantly threaten human health and together with air pollution form a complex pathophysiological system. Other complex biological systems show that increased variance and autocorrelations in time series may act as valid early warning signals for critical transitions. On population level, we determined the likelihood that increased variance and autocorrelation of hospital visit on cardiopulmonary disease preceded critical transitions in population health by human-pollution interactions. We investigated long-term hospital visits from a hospital in Nanjing City, China during 2006-2016 for the most important cardiopulmonary diseases likely to be influenced by air pollution: cerebrovascular accident disease (CVAD), coronary artery disease (CAD), chronic obstructive pulmonary disease (COPD), lung cancer disease (LCD), and the grouped categories of respiratory system disease (RESD) and cardio-cerebrovascular system disease (CCD). The time series of standard deviations (SDs) and autocorrelation at-lag-1 (AR-1) were studied as potential Early-Warning Indicators (EWIs) of transitions in population health. Elevated SDs provided an early warning for critical transitions in visit for LCD and overall CCD and CVAD, for the period of 2012-2013, after which a real transition of increased visit occurred for these disease categories. Statistical testing showed that these SDs were significantly increased (p < 0.1). The long-term air pollution together with intermittent pollution episodes may have triggered critical transitions in population health for cardiopulmonary disease. It is recommended to consider significant increases in variability in time series of relevant system parameters, such as visit, as early warning signs for future transitions in populations' health states.

Indicators of transitions in biological systems

In the face of global biodiversity declines, predicting the fate of biological systems is a key goal in ecology. One popular approach is the search for early warning signals (EWSs) based on alternative stable states theory. In this review, we cover the theory behind nonlinearity in dynamic systems and techniques to detect the loss of resilience that can indicate state transitions. We describe the research done on generic abundance-based signals of instability that are derived from the phenomenon of critical slowing down, which represent the genesis of EWSs research. We highlight some of the issues facing the detection of such signals in biological systems - which are inherently complex and show low signal-to-noise ratios. We then document research on alternative signals of instability, including measuring shifts in spatial autocorrelation and trait dynamics, and discuss potential future directions for EWSs research based on detailed demographic and phenotypic data. We set EWSs research in the greater field of predictive ecology and weigh up the costs and benefits of simplicity vs. complexity in predictive models, and how the available data should steer the development of future methods. Finally, we identify some key unanswered questions that, if solved, could improve the applicability of these methods.

How one might miss early warning signals of critical transitions in time series data: A systematic study of two major currency pairs

There is growing interest in the use of critical slowing down and critical fluctuations as early warning signals for critical transitions in different complex systems. However, while some studies found them effective, others found the opposite. In this paper, we investigated why this might be so, by testing three commonly used indicators: lag-1 autocorrelation, variance, and low-frequency power spectrum at anticipating critical transitions in the very-high-frequency time series data of the Australian Dollar-Japanese Yen and Swiss Franc-Japanese Yen exchange rates. Besides testing rising trends in these indicators at a strict level of confidence using the Kendall-tau test, we also required statistically significant early warning signals to be concurrent in the three indicators, which must rise to appreciable values. We then found for our data set the optimum parameters for discovering critical transitions, and showed that the set of critical transitions found is generally insensitive to variations in the parameters. Suspecting that negative results in the literature are the results of low data frequencies, we created time series with time intervals over three orders of magnitude from the raw data, and tested them for early warning signals. Early warning signals can be reliably found only if the time interval of the data is shorter than the time scale of critical transitions in our complex system of interest. Finally, we compared the set of time windows with statistically significant early warning signals with the set of time windows followed by large movements, to conclude that the early warning signals indeed provide reliable information on impending critical transitions. This reliability becomes more compelling statistically the more events we test.

Experiments and modelling of rate-dependent transition delay in a stochastic subcritical bifurcation

Complex systems exhibiting critical transitions when one of their governing parameters varies are ubiquitous in nature and in engineering applications. Despite a vast literature focusing on this topic, there are few studies dealing with the effect of the rate of change of the bifurcation parameter on the tipping points. In this work, we consider a subcritical stochastic Hopf bifurcation under two scenarios: the bifurcation parameter is first changed in a quasi-steady manner and then, with a finite ramping rate. In the latter case, a rate-dependent bifurcation delay is observed and exemplified experimentally using a thermoacoustic instability in a combustion chamber. This delay increases with the rate of change. This leads to a state transition of larger amplitude compared with the one that would be experienced by the system with a quasi-steady change of the parameter. We also bring experimental evidence of a dynamic hysteresis caused by the bifurcation delay when the parameter is ramped back. A surrogate model is derived in order to predict the statistic of these delays and to scrutinize the underlying stochastic dynamics. Our study highlights the dramatic influence of a finite rate of change of bifurcation parameters upon tipping points, and it pinpoints the crucial need of considering this effect when investigating critical transitions.

Rate-induced tipping from periodic attractors: Partial tipping and connecting orbits

We consider how breakdown of the quasistatic approximation for attractors can lead to rate-induced tipping, where a qualitative change in tracking/tipping behaviour of trajectories can be characterised in terms of a critical rate. Associated with rate-induced tipping (where tracking of a branch of quasistatic attractors breaks down), we find a new phenomenon for attractors that are not simply equilibria: partial tipping of the pullback attractor where certain phases of the periodic attractor tip and others track the quasistatic attractor. For a specific model system with a parameter shift between two asymptotically autonomous systems with periodic attractors, we characterise thresholds of rate-induced tipping to partial and total tipping. We show these thresholds can be found in terms of certain periodic-to-periodic and periodic-to-equilibrium connections that we determine using Lin's method for an augmented system.

Forecasting infectious disease emergence subject to seasonal forcing

BACKGROUND

Despite high vaccination coverage, many childhood infections pose a growing threat to human populations. Accurate disease forecasting would be of tremendous value to public health. Forecasting disease emergence using early warning signals (EWS) is possible in non-seasonal models of infectious diseases. Here, we assessed whether EWS also anticipate disease emergence in seasonal models.

METHODS

We simulated the dynamics of an immunizing infectious pathogen approaching the tipping point to disease endemicity. To explore the effect of seasonality on the reliability of early warning statistics, we varied the amplitude of fluctuations around the average transmission. We proposed and analyzed two new early warning signals based on the wavelet spectrum. We measured the reliability of the early warning signals depending on the strength of their trend preceding the tipping point and then calculated the Area Under the Curve (AUC) statistic.

RESULTS

Early warning signals were reliable when disease transmission was subject to seasonal forcing. Wavelet-based early warning signals were as reliable as other conventional early warning signals. We found that removing seasonal trends, prior to analysis, did not improve early warning statistics uniformly.

CONCLUSIONS

Early warning signals anticipate the onset of critical transitions for infectious diseases which are subject to seasonal forcing. Wavelet-based early warning statistics can also be used to forecast infectious disease.

Rate of forcing and the forecastability of critical transitions

Critical transitions are qualitative changes of state that occur when a stochastic dynamical system is forced through a critical point. Many critical transitions are preceded by characteristic fluctuations that may serve as model‐independent early warning signals, implying that these events may be predictable in applications ranging from physics to biology. In nonbiological systems, the strength of such early warning signals has been shown partly to be determined by the speed at which the transition occurs. It is currently unknown whether biological systems, which are inherently high dimensional and typically display low signal‐to‐noise ratios, also exhibit this property, which would have important implications for how ecosystems are managed, particularly where the forces exerted on a system are anthropogenic. We examine whether the rate of forcing can alter the strength of early warning signals in (1) a model exhibiting a fold bifurcation where a state shift is driven by the harvesting of individuals, and (2) a model exhibiting a transcritical bifurcation where a state shift is driven by increased grazing pressure. These models predict that the rate of forcing can alter the detectability of early warning signals regardless of the underlying bifurcation the system exhibits, but that this result may be more pronounced in fold bifurcations. These findings have important implications for the management of biological populations, particularly harvested systems such as fisheries, and suggest that knowing the class of bifurcations a system will manifest may help discriminate between true‐positive and false‐positive signals.