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

Regime shifts driven by dynamic correlations in gene expression noise

Gene expression is a noisy process that leads to regime shifts between alternative steady states among individual living cells, inducing phenotypic variability. The effects of white noise on the regime shift in bistable systems have been well characterized, however little is known about such effects of colored noise (noise with nonzero correlation time). Here, we show that noise correlation time, by considering a genetic circuit of autoactivation, can have a significant effect on the regime shift between distinct phenotypic states in gene expression. We demonstrate this theoretically, using stochastic potential, stationary probability density function, and first-passage time based on the Fokker-Planck description, where the Ornstein-Uhlenbeck process is used to model colored noise. We find that an increase in noise correlation time in the degradation rate can induce a regime shift from a low to a high protein concentration state and enhance the bistable regime, while an increase in noise correlation time in the basal rate retains the bimodal distribution. We then show how cross-correlated colored noises in basal and degradation rates can induce regime shifts from a low to a high protein concentration state, but reduce the bistable regime. We also validate these results through direct numerical simulations of the stochastic differential equation. In gene expression understanding the causes of regime shift to a harmful phenotype could improve early therapeutic intervention in complex human diseases.

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.

Ranking of critical species to preserve the functionality of mutualistic networks using the k-core decomposition

Background

Network analysis has become a relevant approach to analyze cascading species extinctions resulting from perturbations on mutualistic interactions as a result of environmental change. In this context, it is essential to be able to point out key species, whose stability would prevent cascading extinctions, and the consequent loss of ecosystem function. In this study, we aim to explain how the k-core decomposition sheds light on the understanding the robustness of bipartite mutualistic networks.

Methods

We defined three k-magnitudes based on the k-core decomposition: k-radius, k-degree, and k-risk. The first one, k-radius, quantifies the distance from a node to the innermost shell of the partner guild, while k-degree provides a measure of centrality in the k-shell based decomposition. k-risk is a way to measure the vulnerability of a network to the loss of a particular species. Using these magnitudes we analyzed 89 mutualistic networks involving plant pollinators or seed dispersers. Two static extinction procedures were implemented in which k-degree and k-risk were compared against other commonly used ranking indexes, as for example MusRank, explained in detail in Material and Methods.

Results

When extinctions take place in both guilds, k-risk is the best ranking index if the goal is to identify the key species to preserve the giant component. When species are removed only in the primary class and cascading extinctions are measured in the secondary class, the most effective ranking index to identify the key species to preserve the giant component is k-degree. However, MusRank index was more effective when the goal is to identify the key species to preserve the greatest species richness in the second class.

Discussion

The k-core decomposition offers a new topological view of the structure of mutualistic networks. The new k-radius, k-degree and k-risk magnitudes take advantage of its properties and provide new insight into the structure of mutualistic networks. The k-risk and k-degree ranking indexes are especially effective approaches to identify key species to preserve when conservation practitioners focus on the preservation of ecosystem functionality over species richness.

Probability of noise- and rate-induced tipping

We propose an approximation for the probability of tipping when the speed of parameter change and additive white noise interact to cause tipping. Our approximation is valid for small to moderate drift speeds and helps to estimate the probability of false positives and false negatives in early-warning indicators in the case of rate- and noise-induced tipping. We illustrate our approximation on a prototypical model for rate-induced tipping with additive noise using Monte Carlo simulations. The formula can be extended to close encounters of rate-induced tipping and is otherwise applicable to other forms of tipping. We also provide an asymptotic formula for the critical ramp speed of the parameter in the absence of noise for a general class of systems undergoing rate-induced tipping.

Immanent conditions determine imminent collapses: nutrient regimes define the resilience of macroalgal communities

Predicting where state-changing thresholds lie can be inherently complex in ecosystems characterized by nonlinear dynamics. Unpacking the mechanisms underlying these transitions can help considerably reduce this unpredictability. We used empirical observations, field and laboratory experiments, and mathematical models to examine how differences in nutrient regimes mediate the capacity of macrophyte communities to sustain sea urchin grazing. In relatively nutrient-rich conditions, macrophyte systems were more resilient to grazing, shifting to barrens beyond 1 800 g m-2 (urchin biomass), more than twice the threshold of nutrient-poor conditions. The mechanisms driving these differences are linked to how nutrients mediate urchin foraging and algal growth: controlled experiments showed that low-nutrient regimes trigger compensatory feeding and reduce plant growth, mechanisms supported by our consumer-resource model. These mechanisms act together to halve macrophyte community resilience. Our study demonstrates that by mediating the underlying drivers, inherent conditions can strongly influence the buffer capacity of nonlinear systems.

Elevated nonlinearity as an indicator of shifts in the dynamics of populations under stress

Populations occasionally experience abrupt changes, such as local extinctions, strong declines in abundance or transitions from stable dynamics to strongly irregular fluctuations. Although most of these changes have important ecological and at times economic implications, they remain notoriously difficult to detect in advance. Here, we study changes in the stability of populations under stress across a variety of transitions. Using a Ricker-type model, we simulate shifts from stable point equilibrium dynamics to cyclic and irregular boom-bust oscillations as well as abrupt shifts between alternative attractors. Our aim is to infer the loss of population stability before such shifts based on changes in nonlinearity of population dynamics. We measure nonlinearity by comparing forecast performance between linear and nonlinear models fitted on reconstructed attractors directly from observed time series. We compare nonlinearity to other suggested leading indicators of instability (variance and autocorrelation). We find that nonlinearity and variance increase in a similar way prior to the shifts. By contrast, autocorrelation is strongly affected by oscillations. Finally, we test these theoretical patterns in datasets of fisheries populations. Our results suggest that elevated nonlinearity could be used as an additional indicator to infer changes in the dynamics of populations under stress.

Towards understanding the integrative approach of the water, energy and food nexus

The water, energy and food nexus (WEF nexus) is currently quite popular in environmental management. The concept found a fertile ground in science and policymaking, but there is no consistent view on the meaning of integration within the nexus. Here, a wealth of publications is reviewed in an endeavour to: (1) reveal the lines of justification for the need of the WEF nexus debate and (2) identify the range of tools for analysing the interdependent resource issues of the nexus using an integrated framework of science and policy. There are three drivers behind the emergence of the nexus thinking. These are a) increasing resource interlinks due to growing scarcities, b) recent resource supply crises, and c) failures of sector-driven management strategies. Evaluation of the WEF nexus integrative debate can be carried out using four key criteria, namely ability to change current policy debates, issue and thinking novelty, practicability and measurability, and clearness and implementation roadmap. It is clear that, although the nexus has been quite successful in changing policy debates, issue prioritization is missing and seems to be left to specific case studies and policymakers' choices. There is a high need for 'incorporation' and 'cross-linking' of issues between the three resources. In this regard, nexus governance is the missing link in the nexus debate.

Reversal of a cyanobacterial bloom in response to early warnings

Directional change in environmental drivers sometimes triggers regime shifts in ecosystems. Theory and experiments suggest that regime shifts can be detected in advance, and perhaps averted, by monitoring resilience indicators such as variance and autocorrelation of key ecosystem variables. However, it is uncertain whether management action prompted by a change in resilience indicators can prevent an impending regime shift. We caused a cyanobacterial bloom by gradually enriching an experimental lake while monitoring an unenriched reference lake and a continuously enriched reference lake. When resilience indicators exceeded preset boundaries, nutrient enrichment was stopped in the experimental lake. Concentrations of algal pigments, dissolved oxygen saturation, and pH rapidly declined following cessation of nutrient enrichment and became similar to the unenriched lake, whereas a large bloom occurred in the continuously enriched lake. This outcome suggests that resilience indicators may be useful in management to prevent unwanted regime shifts, at least in some situations. Nonetheless, a safer approach to ecosystem management would build and maintain the resilience of desirable ecosystem conditions, for example, by preventing excessive nutrient input to lakes and reservoirs.

Early warning signals of regime shifts in coupled human-environment systems

In complex systems, a critical transition is a shift in a system's dynamical regime from its current state to a strongly contrasting state as external conditions move beyond a tipping point. These transitions are often preceded by characteristic early warning signals such as increased system variability. However, early warning signals in complex, coupled human-environment systems (HESs) remain little studied. Here, we compare critical transitions and their early warning signals in a coupled HES model to an equivalent environment model uncoupled from the human system. We parameterize the HES model, using social and ecological data from old-growth forests in Oregon. We find that the coupled HES exhibits a richer variety of dynamics and regime shifts than the uncoupled environment system. Moreover, the early warning signals in the coupled HES can be ambiguous, heralding either an era of ecosystem conservationism or collapse of both forest ecosystems and conservationism. The presence of human feedback in the coupled HES can also mitigate the early warning signal, making it more difficult to detect the oncoming regime shift. We furthermore show how the coupled HES can be "doomed to criticality": Strategic human interactions cause the system to remain perpetually in the vicinity of a collapse threshold, as humans become complacent when the resource seems protected but respond rapidly when it is under immediate threat. We conclude that the opportunities, benefits, and challenges of modeling regime shifts and early warning signals in coupled HESs merit further research.

Evaluating early-warning indicators of critical transitions in natural aquatic ecosystems

Ecosystems can show sudden and persistent changes in state despite only incremental changes in drivers. Such critical transitions are difficult to predict, because the state of the system often shows little change before the transition. Early-warning indicators (EWIs) are hypothesized to signal the loss of system resilience and have been shown to precede critical transitions in theoretical models, paleo-climate time series, and in laboratory as well as whole lake experiments. The generalizability of EWIs for detecting critical transitions in empirical time series of natural aquatic ecosystems remains largely untested, however. Here we assessed four commonly used EWIs on long-term datasets of five freshwater ecosystems that have experienced sudden, persistent transitions and for which the relevant ecological mechanisms and drivers are well understood. These case studies were categorized by three mechanisms that can generate critical transitions between alternative states: competition, trophic cascade, and intraguild predation. Although EWIs could be detected in most of the case studies, agreement among the four indicators was low. In some cases, EWIs were detected considerably ahead of the transition. Nonetheless, our results show that at present, EWIs do not provide reliable and consistent signals of impending critical transitions despite using some of the best routinely monitored freshwater ecosystems. Our analysis strongly suggests that a priori knowledge of the underlying mechanisms driving ecosystem transitions is necessary to identify relevant state variables for successfully monitoring EWIs.

Characterizing Species Interactions to Understand Press Perturbations: What Is the Community Matrix?

The community matrix is among ecology's most important mathematical abstractions, formally encapsulating the interconnected network of effects that species have on one another's populations. Despite its importance, the term “community matrix” has been applied to multiple types of matrices that have differing interpretations. This has hindered the application of theory for understanding community structure and perturbation responses. Here, we clarify the correspondence and distinctions among the Interaction matrix, the Alpha matrix, and the Jacobian matrix, terms that are frequently used interchangeably as well as synonymously with the term “community matrix.” We illustrate how these matrices correspond to different ways of characterizing interaction strengths, how they permit insights regarding different types of press perturbations, and how these are related by a simple scaling relationship. Connections to additional interaction strength characterizations encapsulated by the Beta matrix, the Gamma matrix, and the Removal matrix are also discussed. Our synthesis highlights the empirical challenges that remain in using these tools to understand actual communities.

Early warning signals for critical transitions in a thermoacoustic system

Dynamical systems can undergo critical transitions where the system suddenly shifts from one stable state to another at a critical threshold called the tipping point. The decrease in recovery rate to equilibrium (critical slowing down) as the system approaches the tipping point can be used to identify the proximity to a critical transition. Several measures have been adopted to provide early indications of critical transitions that happen in a variety of complex systems. In this study, we use early warning indicators to predict subcritical Hopf bifurcation occurring in a thermoacoustic system by analyzing the observables from experiments and from a theoretical model. We find that the early warning measures perform as robust indicators in the presence and absence of external noise. Thus, we illustrate the applicability of these indicators in an engineering system depicting critical transitions.

The impact of chemical pollution on the resilience of soils under multiple stresses: A conceptual framework for future research

Soils are faced with man-made chemical stress factors, such as the input of organic or metal-containing pesticides, in combination with non-chemical stressors like soil compaction and natural disturbance like drought. Although multiple stress factors are typically co-occurring in soil ecosystems, research in soil sciences on this aspect is limited and focuses mostly on single structural or functional endpoints. A mechanistic understanding of the reaction of soils to multiple stressors is currently lacking. Based on a review of resilience theory, we introduce a new concept for research on the ability of polluted soil (xenobiotics or other chemical pollutants as one stressor) to resist further natural or anthropogenic stress and to retain its functions and structure. There is strong indication that pollution as a primary stressor will change the system reaction of soil, i.e., its resilience, stability and resistance. It can be expected that pollution affects the physiological adaption of organisms and the functional redundancy of the soil to further stress. We hypothesize that the recovery of organisms and chemical-physical properties after impact of a follow-up stressor is faster in polluted soil than in non-polluted soil, i.e., polluted soil has a higher dynamical stability (dynamical stability=1/recovery time), whereas resilience of the contaminated soil is lower compared to that of not or less contaminated soil. Thus, a polluted soil might be more prone to change into another system regime after occurrence of further stress. We highlight this issue by compiling the literature exemplarily for the effects of Cu contamination and compaction on soil functions and structure. We propose to intensify research on effects of combined stresses involving a multidisciplinary team of experts and provide suggestions for corresponding experiments. Our concept offers thus a framework for system level analysis of soils paving the way to enhance ecological theory.

The fundamental role of ecological feedback mechanisms for the adaptive management of seagrass ecosystems - a review

Seagrass meadows are vital ecosystems in coastal zones worldwide, but are also under global threat. One of the major hurdles restricting the success of seagrass conservation and restoration is our limited understanding of ecological feedback mechanisms. In these ecosystems, multiple, self-reinforcing feedbacks can undermine conservation efforts by masking environmental impacts until the decline is precipitous, or alternatively they can inhibit seagrass recovery in spite of restoration efforts. However, no clear framework yet exists for identifying or dealing with feedbacks to improve the management of seagrass ecosystems. Here we review the causes and consequences of multiple feedbacks between seagrass and biotic and/or abiotic processes. We demonstrate how feedbacks have the potential to impose or reinforce regimes of either seagrass dominance or unvegetated substrate, and how the strength and importance of these feedbacks vary across environmental gradients. Although a myriad of feedbacks have now been identified, the co-occurrence and likely interaction among feedbacks has largely been overlooked to date due to difficulties in analysis and detection. Here we take a fundamental step forward by modelling the interactions among two distinct above- and belowground feedbacks to demonstrate that interacting feedbacks are likely to be important for ecosystem resilience. On this basis, we propose a five-step adaptive management plan to address feedback dynamics for effective conservation and restoration strategies. The management plan provides guidance to aid in the identification and prioritisation of likely feedbacks in different seagrass ecosystems.

Detecting the Collapse of Cooperation in Evolving Networks

The sustainability of biological, social, economic and ecological communities is often determined by the outcome of social conflicts between cooperative and selfish individuals (cheaters). Cheaters avoid the cost of contributing to the community and can occasionally spread in the population leading to the complete collapse of cooperation. Although such collapse often unfolds unexpectedly, it is unclear whether one can detect the risk of cheater’s invasions and loss of cooperation in an evolving community. Here, we combine dynamical networks and evolutionary game theory to study the abrupt loss of cooperation with tools for studying critical transitions. We estimate the risk of cooperation collapse following the introduction of a single cheater under gradually changing conditions. We observe an increase in the average time it takes for cheaters to be eliminated from the community as the risk of collapse increases. We argue that such slow system response resembles slowing down in recovery rates prior to a critical transition. In addition, we show how changes in community structure reflect the risk of cooperation collapse. We find that these changes strongly depend on the mechanism that governs how cheaters evolve in the community. Our results highlight novel directions for detecting abrupt transitions in evolving networks.

Percolation-based precursors of transitions in extended systems

Abrupt transitions are ubiquitous in the dynamics of complex systems. Finding precursors, i.e. early indicators of their arrival, is fundamental in many areas of science ranging from electrical engineering to climate. However, obtaining warnings of an approaching transition well in advance remains an elusive task. Here we show that a functional network, constructed from spatial correlations of the system’s time series, experiences a percolation transition way before the actual system reaches a bifurcation point due to the collective phenomena leading to the global change. Concepts from percolation theory are then used to introduce early warning precursors that anticipate the system’s tipping point. We illustrate the generality and versatility of our percolation-based framework with model systems experiencing different types of bifurcations and with Sea Surface Temperature time series associated to El Niño phenomenon.

Including trait-based early warning signals helps predict population collapse

Foreseeing population collapse is an on-going target in ecology, and this has led to the development of early warning signals based on expected changes in leading indicators before a bifurcation. Such signals have been sought for in abundance time-series data on a population of interest, with varying degrees of success. Here we move beyond these established methods by including parallel time-series data of abundance and fitness-related trait dynamics. Using data from a microcosm experiment, we show that including information on the dynamics of phenotypic traits such as body size into composite early warning indices can produce more accurate inferences of whether a population is approaching a critical transition than using abundance time-series alone. By including fitness-related trait information alongside traditional abundance-based early warning signals in a single metric of risk, our generalizable approach provides a powerful new way to assess what populations may be on the verge of collapse.

Predicting population collapse by monitoring key early warning signals in time-series data may highlight when interventions are needed. Here, the authors show that including information on phenotypic traits like body size can more accurately predict critical transitions than abundance data alone.

Anticipating regime shifts in gene expression: The case of an autoactivating positive feedback loop

Considerable evidence suggests that anticipating sudden shifts from one state to another in bistable dynamical systems is a challenging task; examples include ecosystems, financial markets, and complex diseases. In this paper, we investigate the effects of additive, multiplicative, and cross-correlated stochastic perturbations on determining the regime shifts in a bistable gene regulatory system, which gives rise to two distinct states of low and high concentrations of protein. We obtain the stationary probability density and mean first-passage time of the system. We show that increasing the additive (multiplicative) noise intensity induces a regime shift from a low (high) to a high (low) protein concentration state. However, an increase in the cross-correlation intensity always induces regime shifts from a high to a low protein concentration state. For both bifurcation-induced (often called the tipping point) and noise-induced (called stochastic switching) regime shifts, we further explore the robustness of recently developed critical-down-based early warning signal (EWS) indicators (e.g., rising variance and lag-1 autocorrelation) on our simulated time-series data. We identify that using EWS indicators, prediction of an impending bifurcation-induced regime shift is relatively easier than that of a noise-induced regime shift in the considered system. Moreover, the success of EWS indicators also strongly depends upon the nature of the noise.

Towards a critical transition theory under different temporal scales and noise strengths

The mechanism of critical phenomena or critical transitions has been recently studied from various aspects, in particular considering slow parameter change and small noise. In this article, we systematically classify critical transitions into three types based on temporal scales and noise strengths of dynamical systems. Specifically, the classification is made by comparing three important time scales τ(λ), τ(tran), and τ(ergo), where τ(λ) is the time scale of parameter change (e.g., the change of environment), τ(tran) is the time scale when a particle or state transits from a metastable state into another, and τ(ergo) is the time scale when the system becomes ergodic. According to the time scales, we classify the critical transition behaviors as three types, i.e., state transition, basin transition, and distribution transition. Moreover, for each type of transition, there are two cases, i.e., single-trajectory transition and multitrajectory ensemble transition, which correspond to the transition of individual behavior and population behavior, respectively. We also define the critical point for each type of critical transition, derive several properties, and further propose the indicators for predicting critical transitions with numerical simulations. In addition, we show that the noise-to-signal ratio is effective to make the classification of critical transitions for real systems.

Catastrophic Regime Shift in Water Reservoirs and São Paulo Water Supply Crisis

The relation between rainfall and water accumulated in reservoirs comprises nonlinear feedbacks. Here we show that they may generate alternative equilibrium regimes, one of high water-volume, the other of low water-volume. Reservoirs can be seen as socio-environmental systems at risk of regime shifts, characteristic of tipping point transitions. We analyze data from stored water, rainfall, and water inflow and outflow in the main reservoir serving the metropolitan area of São Paulo, Brazil, by means of indicators of critical regime shifts, and find a strong signal of a transition. We furthermore build a mathematical model that gives a mechanistic view of the dynamics and demonstrates that alternative stable states are an expected property of water reservoirs. We also build a stochastic version of this model that fits well to the data. These results highlight the broader aspect that reservoir management must account for their intrinsic bistability, and should benefit from dynamical systems theory. Our case study illustrates the catastrophic consequences of failing to do so.

Dynamical disease: Challenges for nonlinear dynamics and medicine

Dynamical disease refers to illnesses that are associated with striking changes in the dynamics of some bodily function. There is a large literature in mathematics and physics which proposes mathematical models for the physiological systems and carries out analyses of the properties of these models using nonlinear dynamics concepts involving analyses of the stability and bifurcations of attractors. This paper discusses how these concepts can be applied to medicine.

Ecosystems: the rocky road to regime-shift indicators

Ecosystems can undergo dramatic shifts from one stable state to another. While indicators of such shifts are well known, experimental tests are few and far between. A new study on rocky shore ecosystems now offers a test of warning indicators.

Relation between stability and resilience determines the performance of early warning signals under different environmental drivers

Shifting patterns of temporal fluctuations have been found to signal critical transitions in a variety of systems, from ecological communities to human physiology. However, failure of these early warning signals in some systems calls for a better understanding of their limitations. In particular, little is known about the generality of early warning signals in different deteriorating environments. In this study, we characterized how multiple environmental drivers influence the dynamics of laboratory yeast populations, which was previously shown to display alternative stable states [Dai et al., Science, 2012]. We observed that both the coefficient of variation and autocorrelation increased before population collapse in two slowly deteriorating environments, one with a rising death rate and the other one with decreasing nutrient availability. We compared the performance of early warning signals across multiple environments as "indicators for loss of resilience." We find that the varying performance is determined by how a system responds to changes in a specific driver, which can be captured by a relation between stability (recovery rate) and resilience (size of the basin of attraction). Furthermore, we demonstrate that the positive correlation between stability and resilience, as the essential assumption of indicators based on critical slowing down, can break down in this system when multiple environmental drivers are changed simultaneously. Our results suggest that the stability-resilience relation needs to be better understood for the application of early warning signals in different scenarios.

Food-web stability signals critical transitions in temperate shallow lakes

A principal aim of ecologists is to identify critical levels of environmental change beyond which ecosystems undergo radical shifts in their functioning. Both food-web theory and alternative stable states theory provide fundamental clues to mechanisms conferring stability to natural systems. Yet, it is unclear how the concept of food-web stability is associated with the resilience of ecosystems susceptible to regime change. Here, we use a combination of food web and ecosystem modelling to show that impending catastrophic shifts in shallow lakes are preceded by a destabilizing reorganization of interaction strengths in the aquatic food web. Analysis of the intricate web of trophic interactions reveals that only few key interactions, involving zooplankton, diatoms and detritus, dictate the deterioration of food-web stability. Our study exposes a tight link between food-web dynamics and the dynamics of the whole ecosystem, implying that trophic organization may serve as an empirical indicator of ecosystem resilience.

How mechanisms underlying food-web stability may influence ecosystem regime shifts is not well understood. Combining food-web and ecosystem modelling, Kuiper et al. show that destabilizing reorganization of a small number of key trophic interactions precede catastrophic changes in shallow lake ecosystems.

Predicting the onset of period-doubling bifurcations in noisy cardiac systems

Biological, physical, and social systems often display qualitative changes in dynamics. Developing early warning signals to predict the onset of these transitions is an important goal. The current work is motivated by transitions of cardiac rhythms, where the appearance of alternating features in the timing of cardiac events is often a precursor to the initiation of serious cardiac arrhythmias. We treat embryonic chick cardiac cells with a potassium channel blocker, which leads to the initiation of alternating rhythms. We associate this transition with a mathematical instability, called a period-doubling bifurcation, in a model of the cardiac cells. Period-doubling bifurcations have been linked to the onset of abnormal alternating cardiac rhythms, which have been implicated in cardiac arrhythmias such as T-wave alternans and various tachycardias. Theory predicts that in the neighborhood of the transition, the system's dynamics slow down, leading to noise amplification and the manifestation of oscillations in the autocorrelation function. Examining the aggregates' interbeat intervals, we observe the oscillations in the autocorrelation function and noise amplification preceding the bifurcation. We analyze plots--termed return maps--that relate the current interbeat interval with the following interbeat interval. Based on these plots, we develop a quantitative measure using the slope of the return map to assess how close the system is to the bifurcation. Furthermore, the slope of the return map and the lag-1 autocorrelation coefficient are equal. Our results suggest that the slope and the lag-1 autocorrelation coefficient represent quantitative measures to predict the onset of abnormal alternating cardiac rhythms.

Organising a safe space for navigating social-ecological transformations to sustainability

The need for developing socially just living conditions for the world’s growing population whilst keeping human societies within a ‘safe operating space’ has become a modern imperative. This requires transformative changes in the dominant social norms, behaviours, governance and management regimes that guide human responses in areas such as urban ecology, public health, resource security (e.g., food, water, energy access), economic development and biodiversity conservation. However, such systemic transformations necessitate experimentation in public arenas of exchange and a deepening of processes that can widen multi-stakeholder learning. We argue that there is an emergent potential in bridging the sustainability transitions and resilience approaches to create new scientific capacity that can support large-scale social-ecological transformations (SETs) to sustainability globally, not just in the West. In this article, we elucidate a set of guiding principles for the design of a ‘safe space’ to encourage stronger interactions between these research areas and others that are relevant to the challenges faced. We envisage new opportunities for transdisciplinary collaboration that will develop an adaptive and evolving community of practice. In particular, we emphasise the great opportunity for engaging with the role of emerging economies in facilitating safe space experimentation.

A holistic view of marine regime shifts

Understanding marine regime shifts is important not only for ecology but also for developing marine management that assures the provision of ecosystem services to humanity. While regime shift theory is well developed, there is still no common understanding on drivers, mechanisms and characteristic of abrupt changes in real marine ecosystems. Based on contributions to the present theme issue, we highlight some general issues that need to be overcome for developing a more comprehensive understanding of marine ecosystem regime shifts. We find a great divide between benthic reef and pelagic ocean systems in how regime shift theory is linked to observed abrupt changes. Furthermore, we suggest that the long-lasting discussion on the prevalence of top-down trophic or bottom-up physical drivers in inducing regime shifts may be overcome by taking into consideration the synergistic interactions of multiple stressors, and the special characteristics of different ecosystem types. We present a framework for the holistic investigation of marine regime shifts that considers multiple exogenous drivers that interact with endogenous mechanisms to cause abrupt, catastrophic change. This framework takes into account the time-delayed synergies of these stressors, which erode the resilience of the ecosystem and eventually enable the crossing of ecological thresholds. Finally, considering that increased pressures in the marine environment are predicted by the current climate change assessments, in order to avoid major losses of ecosystem services, we suggest that marine management approaches should incorporate knowledge on environmental thresholds and develop tools that consider regime shift dynamics and characteristics. This grand challenge can only be achieved through a holistic view of marine ecosystem dynamics as evidenced by this theme issue.

The importance of within-system spatial variation in drivers of marine ecosystem regime shifts

Comparative analyses of the dynamics of exploited marine ecosystems have led to differing hypotheses regarding the primary causes of observed regime shifts, while many ecosystems have apparently not undergone regime shifts. These varied responses may be partly explained by the decade-old recognition that within-system spatial heterogeneity in key climate and anthropogenic drivers may be important, as recent theoretical examinations have concluded that spatial heterogeneity in environmental characteristics may diminish the tendency for regime shifts. Here, we synthesize recent, empirical within-system spatio-temporal analyses of some temperate and subarctic large marine ecosystems in which regime shifts have (and have not) occurred. Examples from the Baltic Sea, Black Sea, Bengula Current, North Sea, Barents Sea and Eastern Scotian Shelf reveal the largely neglected importance of considering spatial variability in key biotic and abiotic influences and species movements in the context of evaluating and predicting regime shifts. We highlight both the importance of understanding the scale-dependent spatial dynamics of climate influences and key predator–prey interactions to unravel the dynamics of regime shifts, and the utility of spatial downscaling of proposed mechanisms (as evident in the North Sea and Barents Sea) as a means of evaluating hypotheses originally derived from among-system comparisons.