Resilience indicators: prospects and limitations for early warnings of regime shifts

In the vicinity of tipping points—or more precisely bifurcation points—ecosystems recover slowly from small perturbations. Such slowness may be interpreted as a sign of low resilience in the sense that the ecosystem could easily be tipped through a critical transition into a contrasting state. Indicators of this phenomenon of ‘critical slowing down (CSD)’ include a rise in temporal correlation and variance. Such indicators of CSD can provide an early warning signal of a nearby tipping point. Or, they may offer a possibility to rank reefs, lakes or other ecosystems according to their resilience. The fact that CSD may happen across a wide range of complex ecosystems close to tipping points implies a powerful generality. However, indicators of CSD are not manifested in all cases where regime shifts occur. This is because not all regime shifts are associated with tipping points. Here, we review the exploding literature about this issue to provide guidance on what to expect and what not to expect when it comes to the CSD-based early warning signals for critical transitions.     

生态系统状态跃变的早期预警信号及检测

生态系统在气候变化和土地利用及人类活动等的影响下其状态会由某一稳态转变到另一稳态。由于环境压力的复杂性、非线性、随机性等特征，往往导致状态转变表现为非线性、突变、跃变等特点。准确界定系统状态跃变的拐点或阈值点存在很大的挑战，而捕捉接近临界拐点前的生态系统结构和属性上的变化特征作为早期预警信号是切实可行的。早期预警信号理论经历理论框架构建、方法确立、机理认知等近半个多世纪的探索，已经由最初的通过仅依赖检测临界点恢复力的速率减慢、方差增加、系统自相关增强等统计学信号过度到更加多样化的检测方法，如检测系统组分属性的变化特征，诊断系统组分各属性之间的关系变化，系统组分的性状变化、系统组分网络结构变化等等，并且试图整合多信号提高预警的精确性。利用来自自然生态系统的长时间高密度数据集和空间代替时间的数据集，基于多度及性状信号的早期预警，结合稳定性、临界恢复力的减速、以及统计参数的指示作用对系统跃变进行早期诊断和预警是预测生态学的主旨。早期预警信号的深入研究不仅能够完善已有理论的不足，同时还能够为生态系统的保护和管理提供切实有效的理论指导。     

陆地生态系统临界转换理论及其生态学机制研究进展

随着气候变化和人类活动对陆地生态系统双重扰动的不断加剧,越来越多的研究已经意识到生态系统结构和功能会发生难以预知的突变,并且恢复起来需要很长时间.开发判别典型生态系统临界转换的早期预警模型及理解其生态学机制成为生态学研究的热点.目前,基于跨越多个时空尺度的理论和实验研究,提出了多种预警陆地生态系统临界转换的理论框架和指...     

Flickering as an early warning signal

Most work on generic early warning signals for critical transitions focuses on indicators of the phenomenon of critical slowing down that precedes a range of catastrophic bifurcation points. However, in highly stochastic environments, systems will tend to shift to alternative basins of attraction already far from such bifurcation points. In fact, strong perturbations (noise) may cause the system to “flicker” between the basins of attraction of the system’s alternative states. As a result, under such noisy conditions, critical slowing down is not relevant, and one would expect its related generic leading indicators to fail, signaling an impending transition. Here, we systematically explore how flickering may be detected and interpreted as a signal of an emerging alternative attractor. We show that—although the two mechanisms differ—flickering may often be reflected in rising variance, lag-1 autocorrelation and skewness in ways that resemble the effects of critical slowing down. In particular, we demonstrate how the probability distribution of a flickering system can be used to map potential alternative attractors and their resilience. Thus, while flickering systems differ in many ways from the classical image of critical transitions, changes in their dynamics may carry valuable information about upcoming major changes.     

Early warning signals: the charted and uncharted territories

The realization that complex systems such as ecological communities can collapse or shift regimes suddenly and without rapid external forcing poses a serious challenge to our understanding and management of the natural world. The potential to identify early warning signals that would allow researchers and managers to predict such events before they happen has therefore been an invaluable discovery that offers a way forward in spite of such seemingly unpredictable behavior. Research into early warning signals has demonstrated that it is possible to define and detect such early warning signals in advance of a transition in certain contexts. Here, we describe the pattern emerging as research continues to explore just how far we can generalize these results. A core of examples emerges that shares three properties: the phenomenon of rapid regime shifts, a pattern of “critical slowing down” that can be used to detect the approaching shift, and a mechanism of bifurcation driving the sudden change. As research has expanded beyond these core examples, it is becoming clear that not all systems that show regime shifts exhibit critical slowing down, or vice versa. Even when systems exhibit critical slowing down, statistical detection is a challenge. We review the literature that explores these edge cases and highlight the need for (a) new early warning behaviors that can be used in cases where rapid shifts do not exhibit critical slowing down; (b) the development of methods to identify which behavior might be an appropriate signal when encountering a novel system, bearing in mind that a positive indication for some systems is a negative indication in others; and (c) statistical methods that can distinguish between signatures of early warning behaviors and noise.     

浅水湖泊稳态转换预警识别方法局限与展望

浅水湖泊水体底泥交换强烈,极易受人类活动干扰,超过一定阈值即可能发生灾难性的稳态转换,对其有效识别有助于湖泊富营养化的及时防控与修复。浅水湖泊稳态转换可通过系统关键变量(叶绿素、溶解氧、浮游动物、鱼类等)的时间序列(判别不同稳态)、预警信号及阈值等进行识别,其中预警识别可为湖泊生态系统稳态转换提供预判信息,有利于早预警早行动。目前,浅水湖泊稳态转换预警识别因子(方差及自相关性等)主要用于"临界慢化"现象,但在强大外力作用、强烈随机扰动及极端事件下,这些"临界慢化"因子则可能出现误用或错用。基于浅水湖泊基本特征,针对稳态转换的不同驱动机制,探讨"临界慢化"因子的适用性与局限性,并展望其未来发展方向,旨在为湖泊生态系统稳态转换预警识别提供科学参考。     

Slow recovery from perturbations as a generic indicator of a nearby catastrophic shift

The size of the basin of attraction in ecosystems with alternative stable states is often referred to as "ecological resilience." Ecosystems with a low ecological resilience may easily be tipped into an alternative basin of attraction by a stochastic event. Unfortunately, it is very difficult to measure ecological resilience in practice. Here we show that the rate of recovery from small perturbations (sometimes called "engineering resilience") is a remarkably good indicator of ecological resilience. Such recovery rates decrease as a catastrophic regime shift is approached, a phenomenon known in physics as "critical slowing down." We demonstrate the robust occurrence of critical slowing down in six ecological models and outline a possible experimental approach to quantify differences in recovery rates. In all the models we analyzed, critical slowing down becomes apparent quite far from a threshold point, suggesting that it may indeed be of practical use as an early warning signal. Despite the fact that critical slowing down could also indicate other critical transitions, such as a stable system becoming oscillatory, the robustness of the phenomenon makes it a promising indicator of loss of resilience and the risk of upcoming regime shifts in a system.     

Early warning signals of simulated Amazon rainforest dieback

We test proposed generic tipping point early warning signals in a complex climate model (HadCM3) which simulates future dieback of the Amazon rainforest. The equation governing tree cover in the model suggests that zero and non-zero stable states of tree cover co-exist, and a transcritical bifurcation is approached as productivity declines. Forest dieback is a non-linear change in the non-zero tree cover state, as productivity declines, which should exhibit critical slowing down. We use an ensemble of versions of HadCM3 to test for the corresponding early warning signals. However, on approaching simulated Amazon dieback, expected early warning signals of critical slowing down are not seen in tree cover, vegetation carbon or net primary productivity. The lack of a convincing trend in autocorrelation appears to be a result of the system being forced rapidly and non-linearly. There is a robust rise in variance with time, but this can be explained by increases in inter-annual temperature and precipitation variability that force the forest. This failure of generic early warning indicators led us to seek more system-specific, observable indicators of changing forest stability in the model. The sensitivity of net ecosystem productivity to temperature anomalies (a negative correlation) generally increases as dieback approaches, which is attributable to a non-linear sensitivity of ecosystem respiration to temperature. As a result, the sensitivity of atmospheric CO₂ anomalies to temperature anomalies (a positive correlation) increases as dieback approaches. This stability indicator has the benefit of being readily observable in the real world.     

Critical Slowing Down Governs the Transition to Neuron Spiking

Many complex systems have been found to exhibit critical transitions, or so-called tipping points, which are sudden changes to a qualitatively different system state. These changes can profoundly impact the functioning of a system ranging from controlled state switching to a catastrophic break-down; signals that predict critical transitions are therefore highly desirable. To this end, research efforts have focused on utilizing qualitative changes in markers related to a system’s tendency to recover more slowly from a perturbation the closer it gets to the transition—a phenomenon called critical slowing down. The recently studied scaling of critical slowing down offers a refined path to understand critical transitions: to identify the transition mechanism and improve transition prediction using scaling laws.Here, we outline and apply this strategy for the first time in a real-world system by studying the transition to spiking in neurons of the mammalian cortex. The dynamical system approach has identified two robust mechanisms for the transition from subthreshold activity to spiking, saddle-node and Hopf bifurcation. Although theory provides precise predictions on signatures of critical slowing down near the bifurcation to spiking, quantitative experimental evidence has been lacking. Using whole-cell patch-clamp recordings from pyramidal neurons and fast-spiking interneurons, we show that 1) the transition to spiking dynamically corresponds to a critical transition exhibiting slowing down, 2) the scaling laws suggest a saddle-node bifurcation governing slowing down, and 3) these precise scaling laws can be used to predict the bifurcation point from a limited window of observation. To our knowledge this is the first report of scaling laws of critical slowing down in an experiment. They present a missing link for a broad class of neuroscience modeling and suggest improved estimation of tipping points by incorporating scaling laws of critical slowing down as a strategy applicable to other complex systems.     

生态系统多稳态研究进展

在多稳态的生态系统中,外力可能导致生态系统状态突然之间发生不可逆转的转变,从而达到一个新的平衡状态。但目前对多稳态理论的系统研究很少,如何使用预警信号来预测生态系统的状态转变依旧是个难题。通过多稳态理论的梳理提出了一个更加综合的多稳态定义,并以放牧模型为例,系统总结了多稳态理论的相关概念,将多稳态理论应用在生态系统演替和扰沌理论的解释中;通过对生态系统稳态转换预警信号的原理、优缺点和应用条件的分析,对不同尺度下多稳态的研究方法进行了归纳;最后提出了目前多稳态领域的研究问题和未来的研究重点。结果表明:(1)将时间和空间预警信号结合在一起,并量化正确预警信号的概率,对错误预警信号的比例进行加权,可能会提供更准确的稳态转换的预报。(2)定量观测试验适用于小尺度的研究,而较大尺度的研究则采用简化的模型来模拟研究,选择正确的尺度极有可能改变预警信号的可靠性。(3)结合多稳态理论研究生态系统临界转换和反馈控制机制,并将基于性状的特征指标和进化动力学纳入其中,是生态系统修复实践的重要研究方向。(4)将多稳态相关理论和生态保护管理政策的实践相结合,是多稳态理论未来应用的前景。本研究为多稳态理论和实践的...     

Catastrophic Collapse Can Occur without Early Warning: Examples of Silent Catastrophes in Structured Ecological Models

Catastrophic and sudden collapses of ecosystems are sometimes preceded by early warning signals that potentially could be used to predict and prevent a forthcoming catastrophe. Universality of these early warning signals has been proposed, but no formal proof has been provided. Here, we show that in relatively simple ecological models the most commonly used early warning signals for a catastrophic collapse can be silent. We underpin the mathematical reason for this phenomenon, which involves the direction of the eigenvectors of the system. Our results demonstrate that claims on the universality of early warning signals are not correct, and that catastrophic collapses can occur without prior warning. In order to correctly predict a collapse and determine whether early warning signals precede the collapse, detailed knowledge of the mathematical structure of the approaching bifurcation is necessary. Unfortunately, such knowledge is often only obtained after the collapse has already occurred.     

Critical slowing down as an indicator of transitions in two-species models

Transitions in ecological systems often occur without apparent warning, and may represent shifts between alternative persistent states. Decreasing ecological resilience (the size of the basin of attraction around a stable state) can signal an impending transition, but this effect is difficult to measure in practice. Recent research has suggested that a decreasing rate of recovery from small perturbations (critical slowing down) is a good indicator of ecological resilience. Here we use analytical techniques to draw general conclusions about the conditions under which critical slowing down provides an early indicator of transitions in two-species predator-prey and competition models. The models exhibit three types of transition: the predator-prey model has a Hopf bifurcation and a transcritical bifurcation, and the competition model has two saddle-node bifurcations (in which case the system exhibits hysteresis) or two transcritical bifurcations, depending on the parameterisation. We find that critical slowing down is an earlier indicator of the Hopf bifurcation in predator-prey models in which prey are regulated by predation rather than by intrinsic density-dependent effects and an earlier indicator of transitions in competition models in which the dynamics of the rare species operate on slower timescales than the dynamics of the common species. These results lead directly to predictions for more complex multi-species systems, which can be tested using simulation models or real ecosystems.     

Lack of Critical Slowing Down Suggests that Financial Meltdowns Are Not Critical Transitions,yet Rising Variability Could Signal Systemic Risk

Complex systems inspired analysis suggests a hypothesis that financial meltdowns are abrupt critical transitions that occur when the system reaches a tipping point. Theoretical and empirical studies on climatic and ecological dynamical systems have shown that approach to tipping points is preceded by a generic phenomenon called critical slowing down, i.e. an increasingly slow response of the system to perturbations. Therefore, it has been suggested that critical slowing down may be used as an early warning signal of imminent critical transitions. Whether financial markets exhibit critical slowing down prior to meltdowns remains unclear. Here, our analysis reveals that three major US (Dow Jones Index, S&P 500 and NASDAQ) and two European markets (DAX and FTSE) did not exhibit critical slowing down prior to major financial crashes over the last century. However, all markets showed strong trends of rising variability, quantified by time series variance and spectral function at low frequencies, prior to crashes. These results suggest that financial crashes are not critical transitions that occur in the vicinity of a tipping point. Using a simple model, we argue that financial crashes are likely to be stochastic transitions which can occur even when the system is far away from the tipping point. Specifically, we show that a gradually increasing strength of stochastic perturbations may have caused to abrupt transitions in the financial markets. Broadly, our results highlight the importance of stochastically driven abrupt transitions in real world scenarios. Our study offers rising variability as a precursor of financial meltdowns albeit with a limitation that they may signal false alarms.     

Slowing down in spatially patterned ecosystems at the brink of collapse

Predicting the risk of critical transitions, such as the collapse of a population, is important in order to direct management efforts. In any system that is close to a critical transition, recovery upon small perturbations becomes slow, a phenomenon known as critical slowing down. It has been suggested that such slowing down may be detected indirectly through an increase in spatial and temporal correlation and variance. Here, we tested this idea in arid ecosystems, where vegetation may collapse to desert as a result of increasing water limitation. We used three models that describe desertification but differ in the spatial vegetation patterns they produce. In all models, recovery rate upon perturbation decreased before vegetation collapsed. However, in one of the models, slowing down failed to translate into rising variance and correlation. This is caused by the regular self-organized vegetation patterns produced by this model. This finding implies an important limitation of variance and correlation as indicators of critical transitions. However, changes in such self-organized patterns themselves are a reliable indicator of an upcoming transition. Our results illustrate that while critical slowing down may be a universal phenomenon at critical transitions, its detection through indirect indicators may have limitations in particular systems.     

Methods for detecting early warnings of critical transitions in time series illustrated using simulated ecological data

Many dynamical systems, including lakes, organisms, ocean circulation patterns, or financial markets, are now thought to have tipping points where critical transitions to a contrasting state can happen. Because critical transitions can occur unexpectedly and are difficult to manage, there is a need for methods that can be used to identify when a critical transition is approaching. Recent theory shows that we can identify the proximity of a system to a critical transition using a variety of so-called 'early warning signals', and successful empirical examples suggest a potential for practical applicability. However, while the range of proposed methods for predicting critical transitions is rapidly expanding, opinions on their practical use differ widely, and there is no comparative study that tests the limitations of the different methods to identify approaching critical transitions using time-series data. Here, we summarize a range of currently available early warning methods and apply them to two simulated time series that are typical of systems undergoing a critical transition. In addition to a methodological guide, our work offers a practical toolbox that may be used in a wide range of fields to help detect early warning signals of critical transitions in time series data.     

Network based early warning indicators of vegetation changes in a land–atmosphere model

Numerous model studies demonstrate that ecosystems might not shift smoothly with a gradual change in resource concentration. At specific points, vegetation can suddenly shift from one stable state to another. To predict such undesirable shifts, statistical indicators are proposed for early warning prediction. These so-called classical indicators can address whether vegetation state is moving towards the tipping point of an abrupt transition, however when the transition will occur is hard to predict. Recent studies suggest that complex network based indicators can improve early warning signals of abrupt transitions in complex dynamic systems. In this study, both classical and network based indicators are tested in a coupled land–atmosphere ecological model in which a scale-dependent hydrology-infiltration feedback and a large scale vegetation–precipitation feedback are represented. Multiple biomass equilibria are found in the model and abrupt transitions can occur when rainfall efficiency is decreased. Interaction network based indicators of these transitions are compared with classical indicators, such as the lag-1 autocorrelation and Moran's coefficient, with particular focus on the transition associated with desertification. Two criteria are used to evaluate the quality of these early warning indicators and several high quality network based indicators are identified.     

How Stochasticity Influences Leading Indicators of Critical Transitions

Many complex systems exhibit critical transitions. Of considerable interest are bifurcations, small smooth changes in underlying drivers that produce abrupt shifts in system state. Before reaching the bifurcation point, the system gradually loses stability (‘critical slowing down’). Signals of critical slowing down may be detected through measurement of summary statistics, but how extrinsic and intrinsic noises influence statistical patterns prior to a transition is unclear. Here, we consider a range of stochastic models that exhibit transcritical, saddle-node and pitchfork bifurcations. Noise was assumed to be either intrinsic or extrinsic. We derived expressions for the stationary variance, autocorrelation and power spectrum for all cases. Trends in summary statistics signaling the approach of each bifurcation depend on the form of noise. For example, models with intrinsic stochasticity may predict an increase in or a decline in variance as the bifurcation parameter changes, whereas models with extrinsic noise applied additively predict an increase in variance. The ability to classify trends of summary statistics for a broad class of models enhances our understanding of how critical slowing down manifests in complex systems approaching a transition.     

Slow Recovery from Local Disturbances as an Indicator for Loss of Ecosystem Resilience

A range of indicators have been proposed for identifying the elevated risk of critical transitions in ecosystems. Most indicators are based on the idea that critical slowing down can be inferred from changes in statistical properties of natural fluctuations and spatial patterns. However, identifying these signals in nature has remained challenging. An alternative approach is to infer changes in resilience from differences in standardized experimental perturbations. However, system-wide experimental perturbations are rarely feasible. Here we evaluate the potential to infer the risk of large-scale systemic transitions from local experimental or natural perturbations. We use models of spatially explicit landscapes to illustrate how recovery rates upon small-scale perturbations decrease as an ecosystem approaches a tipping point for a large-scale collapse. We show that the recovery trajectory depends on: (1) the resilience of the ecosystem at large scale, (2) the dispersal rate of organisms, and (3) the scale of the perturbation. In addition, we show that recovery of natural disturbances in a heterogeneous environment can potentially function as an indicator of resilience of a large-scale ecosystem. Our analyses reveal fundamental differences between large-scale weak and local-scale strong perturbations, leading to an overview of opportunities and limitations of the use of local disturbance-recovery experiments.     

Spatially heterogeneous pressure raises risk of catastrophic shifts

Ecosystems may exhibit catastrophic shifts, i.e. abrupt and irreversible responses of ecosystem functions and services to continuous changes in external conditions. The search for early warning signs of approaching shifts has so far mainly been conducted on theoretical models assuming spatially-homogeneous external pressures (e.g. climatic). Here, we investigate how a spatially explicit pressure may affect ecosystems’ risk of catastrophic shifts and the associated spatial early-warning signs. As a case study, we studied a dryland vegetation model assuming ‘associational resistance’, i.e. the mutual reduction of local grazing impact by neighboring plants sharing the investment in defensive traits. Consequently, grazing pressure depends on the local density of plants and is thus spatially-explicit. We focus on the distribution of vegetation patch sizes, which can be assessed using remote sensing and are candidate early warning signs of catastrophic shifts in drylands. We found that spatially explicit grazing affected both the resilience and the spatial patterns of the landscape. Grazing impact became self-enhancing in more fragmented landscapes, disrupted patch growth and put apparently ‘healthy’ drylands under high risks of catastrophic shifts. Our study highlights that a spatially explicit pressure may affect the nature of the spatial pattern observed and thereby change the interpretation of the early warning signs. This may generalize to other ecosystems exhibiting self-organized spatial patterns, where a spatially-explicit pressure may interfere with pattern formation.     

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.