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.     

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.     

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.     

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.     

An ecological resilience perspective on cancer: Insights from a toy model

In this paper we propose an ecological resilience point of view on cancer. This view is based on the analysis of a simple ODE model for the interactions between cancer and normal cells. The model presents two regimes for tumor growth. In the first, cancer arises due to three reasons: a partial corruption of the functions that avoid the growth of mutated cells, an aggressive phenotype of tumor cells and exposure to external carcinogenic factors. In this case, treatments may be effective if they drive the system to the basin of attraction of the cancer cure state. In the second regime, cancer arises because the repair system is intrinsically corrupted. In this case, the complete cure is not possible since the cancer cure state is no more stable, but tumor recurrence may be delayed if treatment is prolongued. We review three indicators of the resilience of a stable equilibrium, related with size and shape of its basin of attraction: latitude, precariousness and resistance. A novel method to calculate these indicators is proposed. This method is simpler and more efficient than those currently used, and may be easily applied to other population dynamics models. We apply this method to the model and investigate how these indicators behave with parameters changes. Finally, we present some simulations to illustrate how the resilience analysis can be applied to validated models in order to obtain indicators for personalized cancer treatments.     

基于临界慢化模型和长时间序列叶面积指数的植被及其恢复力遥感监测研究——以三峡库区为例

基于临界慢化模型,利用长时间序列叶面积指数(GLASS LAI)数据,进行时间序列分解后,计算了LAI及其时间自相关指数作为指标,对三峡库区植被及其恢复力进行监测,通过案例模型对临界慢化模型精度进行了验证,分析了三峡库区植被及其植被恢复力的时空分布特征,探索基于临界慢化模型的植被恢复力遥感定量估算方法的适用性。结果表明：(1)2000—2018年三峡库区LAI平均值为3.4,重庆段LAI较低,湖北段LAI较高;三峡库区LAI整体呈上升趋势,重庆段LAI呈现降低趋势,显著下降区域占重庆段面积的21.75%,湖北段LAI呈现升高趋势,显著上升区域占湖北段面积的21.22%;(2)2000—2018年三峡库区重庆市北碚区、大渡口区、渝北区植被恢复力较低,宜昌市兴山县、夷陵区、点军区植被恢复力较高;(3)模型精度方面,在两个地质灾害扰动事件中案例模型结果与临界慢化模型结果呈现较高的一致性。本文对三峡库区2000—2018年的植被恢复力进行了定量估算,同时通过案例模型对临界慢化模型在恢复力监测上的有效性进行了验证,为三峡库区制定相应生态环境管理决策提供理论基础,为保障西南地区生态安全提供决策依据...     

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.     

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.     

How Hives Collapse: Allee Effects,Ecological Resilience,and the Honey Bee

We construct a mathematical model to quantify the loss of resilience in collapsing honey bee colonies due to the presence of a strong Allee effect. In the model, recruitment and mortality of adult bees have substantial social components, with recruitment enhanced and mortality reduced by additional adult bee numbers. The result is an Allee effect, a net per-individual rate of hive increase that increases as a function of adult bee numbers. The Allee effect creates a critical minimum size in adult bee numbers, below which mortality is greater than recruitment, with ensuing loss of viability of the hive. Under ordinary and favorable environmental circumstances, the critical size is low, and hives remain large, sending off viably-sized swarms (naturally or through beekeeping management) when hive numbers approach an upper stable equilibrium size (carrying capacity). However, both the lower critical size and the upper stable size depend on many parameters related to demographic rates and their enhancement by bee sociality. Any environmental factors that increase mortality, decrease recruitment, or interfere with the social moderation of these rates has the effect of exacerbating the Allee effect by increasing the lower critical size and substantially decreasing the upper stable size. As well, the basin of attraction to the upper stable size, defined by the model potential function, becomes narrower and shallower, indicating the loss of resilience as the hive becomes subjected to increased risk of falling below the critical size. Environmental effects of greater severity can cause the two equilibria to merge and the basin of attraction to the upper stable size to disappear, resulting in collapse of the hive from any initial size. The model suggests that multiple proximate causes, among them pesticides, mites, pathogens, and climate change, working singly or in combinations, could trigger hive collapse.     

Early warning signals also precede non‐catastrophic transitions

Synthesis The quickly expanding literature on early warning signals for critical transitions in ecosystems suggests that critical slowing down is a key phenomenon to measure the distance to a tipping point in ecosystems. Such work is broadly misinterpreted as showing that slowing down is specific to tipping points. In this contribution, we show why this is not the case. Early warning signals based on critical slowing down indicate a broader class of situations where a system becomes increasingly sensitive to perturbations. Ecosystem responses to external changes can surprise us by their abruptness and irreversibility. Models have helped identifying indicators of impending catastrophic shifts, referred to as ‘generic early warning signals’. These indicators are linked to a phenomenon known as ‘critical slowing down’ which describes the fact that the recovery rate of a system after a perturbation decreases when the system approaches a bifurcation – such as the classical fold bifurcation associated to catastrophic shifts. However, contrary to what has sometimes been suggested in the literature, a decrease in recovery rate cannot be considered as specific to approaching catastrophic shifts. Here, we analyze the behavior of early warning signals based on critical slowing down in systems approaching a range of catastrophic and non‐catastrophic situations. Our results show that slowing down generally happens in situations where a system is becoming increasingly sensitive to external perturbations, independently of whether the impeding change is catastrophic or not. These results highlight that indicators specific to catastrophic shifts are still lacking. More importantly, they also imply that in systems where we have no reason to expect catastrophic transitions, slowing down may still be used in a more general sense as a warning signal for a potential decrease in stability.     

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.     

Precritical State Transition Dynamics in the Attractor Landscape of a Molecular Interaction Network Underlying Colorectal Tumorigenesis

From the perspective of systems science, tumorigenesis can be hypothesized as a critical transition (an abrupt shift from one state to another) between proliferative and apoptotic attractors on the state space of a molecular interaction network, for which an attractor is defined as a stable state to which all initial states ultimately converge, and the region of convergence is called the basin of attraction. Before the critical transition, a cellular state might transit between the basin of attraction for an apoptotic attractor and that for a proliferative attractor due to the noise induced by the inherent stochasticity in molecular interactions. Such a flickering state transition (state transition between the basins of attraction for alternative attractors from the impact of noise) would become more frequent as the cellular state approaches near the boundary of the basin of attraction, which can increase the variation in the estimate of the respective basin size. To investigate this for colorectal tumorigenesis, we have constructed a stochastic Boolean network model of the molecular interaction network that contains an important set of proteins known to be involved in cancer. In particular, we considered 100 representative sequences of 20 gene mutations that drive colorectal tumorigenesis. We investigated the appearance of cancerous cells by examining the basin size of apoptotic, quiescent, and proliferative attractors along with the sequential accumulation of gene mutations during colorectal tumorigenesis. We introduced a measure to detect the flickering state transition as the variation in the estimate of the basin sizes for three-phenotype attractors from the impact of noise. Interestingly, we found that this measure abruptly increases before a cell becomes cancerous during colorectal tumorigenesis in most of the gene mutation sequences under a certain level of stochastic noise. This suggests that a frequent flickering state transition can be a precritical phenomenon of colorectal tumorigenesis.     

The effect of time delays on Caribbean coral-algal interactions

The analysis of ecological models often focuses on their asymptotic behavior, but there is increasing recognition that it is important to understand the role of transient behavior. By introducing a time delay into a model of coral-algal interactions in Caribbean coral reefs that exhibits alternative stable states (a favorable coral rich state and a degraded coral-depleted state), we demonstrate the criticality of understanding the basins of attraction for stable equilibria in addition to the systems' asymptotic behavior. Specifically, we show that although the introduction of a time delay into the model does not change the asymptotic stability of the stable equilibria, there are significant changes to their basins of attraction. An understanding of these effects is necessary when determining appropriate reef management options. We then demonstrate that this is a general phenomenon by considering similar behavior underlying the changes in the basins of attraction in a simple Lotka-Volterra model of competition.     

Spatial correlation as leading indicator of catastrophic shifts

Generic early-warning signals such as increased autocorrelation and variance have been demonstrated in time-series of systems with alternative stable states approaching a critical transition. However, lag times for the detection of such leading indicators are typically long. Here, we show that increased spatial correlation may serve as a more powerful early-warning signal in systems consisting of many coupled units. We first show why from the universal phenomenon of critical slowing down, spatial correlation should be expected to increase in the vicinity of bifurcations. Subsequently, we explore the applicability of this idea in spatially explicit ecosystem models that can have alternative attractors. The analysis reveals that as a control parameter slowly pushes the system towards the threshold, spatial correlation between neighboring cells tends to increase well before the transition. We show that such increase in spatial correlation represents a better early-warning signal than indicators derived from time-series provided that there is sufficient spatial heterogeneity and connectivity in the system.     

Changing skewness: an early warning signal of regime shifts in ecosystems

Empirical evidence for large-scale abrupt changes in ecosystems such as lakes and vegetation of semi-arid regions is growing. Such changes, called regime shifts, can lead to degradation of ecological services. We study simple ecological models that show a catastrophic transition as a control parameter is varied and propose a novel early warning signal that exploits two ubiquitous features of ecological systems: nonlinearity and large external fluctuations. Either reduced resilience or increased external fluctuations can tip ecosystems to an alternative stable state. It is shown that changes in asymmetry in the distribution of time series data, quantified by changing skewness, is a model-independent and reliable early warning signal for both routes to regime shifts. Furthermore, using model simulations that mimic field measurements and a simple analysis of real data from abrupt climate change in the Sahara, we study the feasibility of skewness calculations using data available from routine monitoring.     

Conditional heteroscedasticity as a leading indicator of ecological regime shifts

Regime shifts are massive, often irreversible, rearrangements of nonlinear ecological processes that occur when systems pass critical transition points. Ecological regime shifts sometimes have severe consequences for human well-being, including eutrophication in lakes, desertification, and species extinctions. Theoretical and laboratory evidence suggests that statistical anomalies may be detectable leading indicators of regime shifts in ecological time series, making it possible to foresee and potentially avert incipient regime shifts. Conditional heteroscedasticity is persistent variance characteristic of time series with clustered volatility. Here, we analyze conditional heteroscedasticity as a potential leading indicator of regime shifts in ecological time series. We evaluate conditional heteroscedasticity by using ecological models with and without four types of critical transition. On approaching transition points, all time series contain significant conditional heteroscedasticity. This signal is detected hundreds of time steps in advance of the regime shift. Time series without regime shifts do not have significant conditional heteroscedasticity. Because probability values are easily associated with tests for conditional heteroscedasticity, detection of false positives in time series without regime shifts is minimized. This property reduces the need for a reference system to compare with the perturbed system.     

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...     

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.     

Robustness of early warning signals of regime shifts in time-delayed ecological models

Various ecological and other complex dynamical systems may exhibit abrupt regime shifts or critical transitions, wherein they reorganize from one stable state to another over relatively short time scales. Because of potential losses to ecosystem services, forecasting such unexpected shifts would be valuable. Using mathematical models of regime shifts, ecologists have proposed various early warning signals of imminent shifts. However, their generality and applicability to real ecosystems remain unclear because these mathematical models are considered too simplistic. Here, we investigate the robustness of recently proposed early warning signals of regime shifts in two well-studied ecological models, but with the inclusion of time-delayed processes. We find that the average variance may either increase or decrease prior to a regime shift and, thus, may not be a robust leading indicator in time-delayed ecological systems. In contrast, changing average skewness, increasing autocorrelation at short time lags, and reddening power spectra of time series of the ecological state variable all show trends consistent with those of models with no time delays. Our results provide insights into the robustness of early warning signals of regime shifts in a broader class of ecological systems.     

Seeking alternative stable states in a deep lake

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