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

Applicability of spatial early warning signals to complex network dynamics

Early warning signals (EWSs) for complex dynamical systems aim to anticipate tipping points before they occur. While signals computed from time-series data, such as temporal variance, are useful for this task, they are costly to obtain in practice because they need many samples over time to calculate. Spatial EWSs use just a single sample per spatial location and aggregate the samples over space rather than time to try to mitigate this limitation. However, although many complex systems in nature and society form diverse networks, the performance of spatial EWSs is mostly unknown for general networks because the vast majority of studies of spatial EWSs have been on regular lattice networks. Therefore, we have carried out a comprehensive investigation of six major spatial EWSs on various networks. We find that the winning EWS depends on tipping scenarios, although the coefficient of variation and spatial skewness tend to outperform alternative EWSs. We also find that spatial EWSs behave in a drastically different manner between the square lattice and complex networks and tend to be more reliable for the latter than the former. The present results encourage further studies of spatial EWSs on complex networks.

Resilience response of China's terrestrial ecosystem gross primary productivity under environmental stress

Climate change perturbations contribute to the alteration of ecosystem functions and the reduction of their resilience. Understanding this reduction in resilience is fundamental for formulating strategies for sustainable ecosystem management. Consequently, a systematic evaluation of factors influencing ecosystem resilience is imperative. This involves elucidating the resilience trends and stress conditions within Chinese ecosystems spatially, thereby enabling a holistic assessment of their health status. This study assesses resilience across China by analyzing the one-month lagged time autocorrelation of terrestrial Gross Primary Production (GPP) across China. It differentiates between various stress states within the study region and employs interpretable machine learning methods to examine the relationship between terrestrial ecosystem resilience and environmental changes under different stress conditions. The findings reveal that approximately 20 % of Chinese regions are undergoing a decrease in ecosystem resilience, with over half experiencing stressed conditions. This is particularly pronounced in the northeastern and central regions of China, where a more significant and widespread decrease in resilience is observed. In the stressed areas of China, the effect of each environmental factor on the decrease in resilience is greater, where attention needs to be paid to areas with higher maximum temperature, precipitation, vapor pressure deficit, and radiation. The study highlights the variability in ecosystem resilience under different environmental conditions and their varied responses to environmental changes. This provides a scientific basis for protecting ecological balance and promoting the sustainable development of ecosystems.

Early detection of disease outbreaks and non-outbreaks using incidence data: A framework using feature-based time series classification and machine learning

Forecasting the occurrence and absence of novel disease outbreaks is essential for disease management, yet existing methods are often context-specific, require a long preparation time, and non-outbreak prediction remains understudied. To address this gap, we propose a novel framework using a feature-based time series classification (TSC) method to forecast outbreaks and non-outbreaks. We tested our methods on synthetic data from a Susceptible-Infected-Recovered (SIR) model for slowly changing, noisy disease dynamics. Outbreak sequences give a transcritical bifurcation within a specified future time window, whereas non-outbreak (null bifurcation) sequences do not. We identified incipient differences, reflected in 22 statistical features and 5 early warning signal indicators, in time series of infectives leading to future outbreaks and non-outbreaks. Classifier performance, given by the area under the receiver-operating curve (AUC), ranged from 0 . 99 for large expanding windows of training data to 0 . 7 for small rolling windows. The framework is further evaluated on four empirical datasets: COVID-19 incidence data from Singapore, 18 other countries, and Edmonton, Canada, as well as SARS data from Hong Kong, with two classifiers exhibiting consistently high accuracy. Our results highlight detectable statistical features distinguishing outbreak and non-outbreak sequences well before potential occurrence, in both synthetic and real-world datasets presented in this study.

Time-series-analysis-based detection of critical transitions in real-world non-autonomous systems

Real-world non-autonomous systems are open, out-of-equilibrium systems that evolve in and are driven by temporally varying environments. Such systems can show multiple timescale and transient dynamics together with transitions to very different and, at times, even disastrous dynamical regimes. Since such critical transitions disrupt the systems' intended or desired functionality, it is crucial to understand the underlying mechanisms, to identify precursors of such transitions, and to reliably detect them in time series of suitable system observables to enable forecasts. This review critically assesses the various steps of investigation involved in time-series-analysis-based detection of critical transitions in real-world non-autonomous systems: from the data recording to evaluating the reliability of offline and online detections. It will highlight pros and cons to stimulate further developments, which would be necessary to advance understanding and forecasting nonlinear behavior such as critical transitions in complex systems.

Reduced precipitation can induce ecosystem regime shifts in lakes by increasing internal nutrient recycling

Eutrophication is a main threat to continental aquatic ecosystems. Prevention and amelioration actions have been taken under the assumption of a stable climate, which needs reconsideration. Here, we show that reduced precipitation can bring a lake ecosystem to a more productive regime even with a decline in nutrient external load. By analyzing time series of several decades in the largest lake of the Iberian Peninsula, we found autocorrelated changes in the variance of state variables (i.e., chlorophyll and oxygen) indicative of a transient situation towards a new ecosystem regime. Indeed, exceptional planktonic diatom blooms have occurred during the last few years, and the sediment record shows a shift in phytoplankton composition and an increase in nutrient retention. Reduced precipitation almost doubled the water residence time in the lake, enhancing the relevance of internal processes. This study demonstrates that ecological quality targets for aquatic ecosystems must be tailored to the changing climatic conditions for appropriate stewardship.

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

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

Anticipating regime shifts by mixing early warning signals from different nodes

Real systems showing regime shifts, such as ecosystems, are often composed of many dynamical elements interacting on a network. Various early warning signals have been proposed for anticipating regime shifts from observed data. However, it is unclear how one should combine early warning signals from different nodes for better performance. Based on theory of stochastic differential equations, we propose a method to optimize the node set from which to construct an early warning signal. The proposed method takes into account that uncertainty as well as the magnitude of the signal affects its predictive performance, that a large magnitude or small uncertainty of the signal in one situation does not imply the signal's high performance, and that combining early warning signals from different nodes is often but not always beneficial. The method performs well particularly when different nodes are subjected to different amounts of dynamical noise and stress.

Early warning signals have limited applicability to empirical lake data

Research aimed at identifying indicators of persistent abrupt shifts in ecological communities, a.k.a regime shifts, has led to the development of a suite of early warning signals (EWSs). As these often perform inaccurately when applied to real-world observational data, it remains unclear whether critical transitions are the dominant mechanism of regime shifts and, if so, which EWS methods can predict them. Here, using multi-trophic planktonic data on multiple lakes from around the world, we classify both lake dynamics and the reliability of classic and second generation EWSs methods to predict whole-ecosystem change. We find few instances of critical transitions, with different trophic levels often expressing different forms of abrupt change. The ability to predict this change is highly processing dependant, with most indicators not performing better than chance, multivariate EWSs being weakly superior to univariate, and a recent machine learning model performing poorly. Our results suggest that predictive ecology should start to move away from the concept of critical transitions, developing methods suitable for predicting resilience loss not limited to the strict bounds of bifurcation theory.

Connecting distinct realms along multiple dimensions: A meta-ecosystem resilience perspective

Resilience research is central to confront the sustainability challenges to ecosystems and human societies in a rapidly changing world. Given that social-ecological problems span the entire Earth system, there is a critical need for resilience models that account for the connectivity across intricately linked ecosystems (i.e., freshwater, marine, terrestrial, atmosphere). We present a resilience perspective of meta-ecosystems that are connected through the flow of biota, matter and energy within and across aquatic and terrestrial realms, and the atmosphere. We demonstrate ecological resilience sensu Holling using aquatic-terrestrial linkages and riparian ecosystems more generally. A discussion of applications in riparian ecology and meta-ecosystem research (e.g., resilience quantification, panarchy, meta-ecosystem boundary delineations, spatial regime migration, including early warning indications) concludes the paper. Understanding meta-ecosystem resilience may have potential to support decision making for natural resource management (scenario planning, risk and vulnerability assessments).

Evolutionary emergence of alternative stable states in shallow lakes

Ecosystems under stress may respond abruptly and irreversibly through tipping points. Although mechanisms leading to alternative stable states are much studied, little is known about how such ecosystems could have emerged in the first place. We investigate whether evolution by natural selection along resource gradients leads to bistability, using shallow lakes as an example. There, tipping points occur between two alternative states dominated by either submersed or floating macrophytes depending on nutrient loading. We model the evolution of macrophyte depth in the lake, identify the conditions under which the ancestor population diversifies and investigate whether alternative stable states dominated by different macrophyte phenotypes occur. We find that eco-evolutionary dynamics may lead to alternative stable states, but under restrictive conditions. Such dynamics require sufficient asymmetries in the acquisition of both light and nutrient. Our analysis suggests that competitive asymmetries along opposing resource gradients may allow bistability to emerge by natural selection.

Early warnings for multi-stage transitions in dynamics on networks

Successfully anticipating sudden major changes in complex systems is a practical concern. Such complex systems often form a heterogeneous network, which may show multi-stage transitions in which some nodes experience a regime shift earlier than others as an environment gradually changes. Here we investigate early warning signals for networked systems undergoing a multi-stage transition. We found that knowledge of both the ongoing multi-stage transition and network structure enables us to calculate effective early warning signals for multi-stage transitions. Furthermore, we found that small subsets of nodes could anticipate transitions as well as or even better than using all the nodes. Even if we fix the network and dynamical system, no single best subset of nodes provides good early warning signals, and a good choice of sentinel nodes depends on the tipping direction and the current stage of the dynamics within a multi-stage transition, which we systematically characterize.

Early warning of red tides using bacterial and eukaryotic communities in nearshore waters

Anthropogenic discharge activities have increased nutrient pollution in coastal areas, leading to algal blooms and microbial community changes. Particularly, microbial communities could easily be affected with variation in nutrient pollution, and thus offered a promising strategy to predict early red tides warning via microbial community-levels variation and their keystone taxa hysteretic responses to nutrient pollution. Herein high-throughput sequencing technology from 52 samples were used to explore the variation of microbial communities and find the significant tipping points with aggravating nutrient conditions in Xiaoping Island coastal area. Results indicated that bacterial and microeukaryote communities were generally spatial and seasonal heterogeneity and were influenced by the different nutrient conditions. Procrustes test results showed that the comprehensive index of organics polluting (OPI), total nitrogen (TN), inorganic nitrogen (DIN), and total phosphorus (TP) were significantly correlated with the composition of bacteria and microeukaryotes. A SEGMENTED analysis revealed that the threshold of TN, DIN, and NH₄-N for bacterial community were 0.23 ± 0.091 mg/L, 0.21 ± 0.084 mg/L, 0.09 ± 0.057 mg/L, respectively. Tipping points for TN, DIN, and NH₄-N agreed with the concentration during Ceratium tripos and Skeletonema costatum blooms. Co-occurrence network results found that Planktomarina, Acinetobacter, and Verrucomicrobiaceae were keystone and OPI-discriminatory taxa. The abundant changes of Planktomarina at station A1 were significantly correlated with the development of C. tripos blooms (r = 0.55, p < 0.05), and also significantly correlated with TN, DIN, and NO₃-N (r≥|0.55|, p < 0.05). The abundant changes of Acinetobacter and Verrucomicrobiaceae at station C1 were significantly correlated with the development of C. tripos blooms (r ≥ 0.77, p < 0.05), and also significantly correlated with PO₄-P (r ≥ 0.64, p < 0.05). The dynamic abundance of keystone taxa showed that the trend of rapid changes could be monitored 1.5 months before the occurrence of red tide. Therefore, this study provides an assessment method for early warning of red tide occurrence and factors that trigger red tide.

Quantifying resilience and the risk of regime shifts under strong correlated noise

Early warning indicators often suffer from the shortness and coarse-graining of real-world time series. Furthermore, the typically strong and correlated noise contributions in real applications are severe drawbacks for statistical measures. Even under favourable simulation conditions the measures are of limited capacity due to their qualitative nature and sometimes ambiguous trend-to-noise ratio. In order to solve these shortcomings, we analyze the stability of the system via the slope of the deterministic term of a Langevin equation, which is hypothesized to underlie the system dynamics close to the fixed point. The open-source available method is applied to a previously studied seasonal ecological model under noise levels and correlation scenarios commonly observed in real world data. We compare the results to autocorrelation, standard deviation, skewness, and kurtosis as leading indicator candidates by a Bayesian model comparison with a linear and a constant model. We show that the slope of the deterministic term is a promising alternative due to its quantitative nature and high robustness against noise levels and types. The commonly computed indicators apart from the autocorrelation with deseasonalization fail to provide reliable insights into the stability of the system in contrast to a previously performed study in which the standard deviation was found to perform best. In addition, we discuss the significant influence of the seasonal nature of the data to the robust computation of the various indicators, before we determine approximately the minimal amount of data per time window that leads to significant trends for the drift slope estimations.

Understanding the depth limit of the seagrass Cymodocea nodosa as a critical transition: Field and modeling evidence

Changes in light and sediment conditions can sometimes trigger abrupt regime shifts in seagrass meadows resulting in dramatic and unexpected die-offs of seagrass. Light attenuates rapidly with depth, and in seagrass systems with non-linear behaviours, can serve as a sharp boundary beyond which the meadow transitions to bare sand. Determining system behaviour is therefore essential to ensuring resilience is maintained and to prevent stubborn critical ecosystem transitions caused by declines in water quality. Here we combined field and modelling studies to explore the transition from meadow to bare sand in the seagrass Cymodocea nodosa at the limit of its depth distribution in a shallow, light-limited bay. We first describe the relationship between light availability and seagrass density along a depth gradient in an extensive unfragmented meadow (Alfacs bay, NE Spain). We then develop a simple mechanistic model to characterise system behaviour. In the field, we identified sharp decline in shoot density beyond a threshold of ∼1.9 m depth, shifting from a vegetated state to bare sand. The dynamic population model we developed assumes light-dependent growth and an inverse density-dependent mortality due to facilitation between shoots (mortality rate decreases as shoot density increases). The model closely tracked our empirical observations, and both the model and the field data showed signs of bistability. This strongly suggests that the depth limit of C. nodosa is a critical transition driven by photosynthetic light requirements. While the mechanisms still need to be confirmed with experimental evidence, recognizing the non-linear behaviour of C. nodosa meadows is vital not only in improving our understanding of light effects on seagrass dynamics, but also in managing shallow-water meadows. Given the shallow threshold (<2m), light-limited systems may experience significant and recalcitrant meadow retractions with even small changes in sediment and light conditions. Understanding the processes underlying meadow resilience can inform the maintenance and restoration of meadows worldwide.

Applying early warning indicators to predict critical transitions in a lake undergoing multiple changes

Lakes are dynamic ecosystems that can transition among stable states. Since ecosystem-scale transitions can be detrimental and difficult to reverse, being able to predict impending critical transitions in state variables has become a major area of research. However, not all transitions are detrimental, and there is considerable interest in better evaluating the success of management interventions to support adaptive management strategies. Here, we retrospectively evaluated the agreement between time series statistics (i.e., standard deviation, autocorrelation, skewness, and kurtosis-also known as early warning indicators) and breakpoints in state variables in a lake (Lake Simcoe, Ontario, Canada) that has improved from a state of eutrophication. Long-term (1980 to 2019) monitoring data collected fortnightly throughout the ice-free season were used to evaluate historical changes in 15 state variables (e.g., dissolved organic carbon, phosphorus, chlorophyll a) and multivariate-derived time series at three monitoring stations (shallow, middepth, deep) in Lake Simcoe. Time series results from the two deep-water stations indicate that over this period Lake Simcoe transitioned from an algal-dominated state toward a state with increased water clarity (i.e., Secchi disk depth) and silica and lower nutrient and chlorophyll a concentrations, which coincided with both substantial management intervention and the establishment of invasive species (e.g., Dreissenid mussels). Consistent with improvement, Secchi depth at the deep-water stations demonstrated expected trends in statistical indicators prior to identified breakpoints, whereas total phosphorus and chlorophyll a revealed more nuanced patterns. Overall, state variables were largely found to yield inconsistent trends in statistical indicators, so many breakpoints were likely not reflective of traditional bifurcation critical transitions. Nevertheless, statistical indicators of state variable time series may be a valuable tool for the adaptive management and long-term monitoring of lake ecosystems, but we call for more research within the domain of early warning indicators to establish a better understanding of state variable behavior prior to lake changes.

Regime shifts, trends, and variability of lake productivity at a global scale

Lakes can change dramatically following a slow change in conditions. They can abruptly shift from being oligotrophic to eutrophic or vice versa, in what is called a regime shift. Despite the important consequences for ecosystems and human activities of abrupt shifts, we do not know how frequent they are or how they are distributed globally. To answer these questions, we analyze lake productivity dynamics of 1,015 lakes worldwide. Our results show few experienced regime shifts, yet the occurrence of observed regime shifts is increasing over time. Our analysis' global scope allows us to better understand the occurrence of regime shifts and the socioeconomic drivers associated with them. This knowledge will help manage lakes' response to global change.

Lakes are often described as sentinels of global change. Phenomena like lake eutrophication, algal blooms, or reorganization in community composition belong to the most studied ecosystem regime shifts. However, although regime shifts have been well documented in several lakes, a global assessment of the prevalence of regime shifts is still missing, and, more in general, of the factors altering stability in lake status, is missing. Here, we provide a first global assessment of regime shifts and stability in the productivity of 1,015 lakes worldwide using trophic state index (TSI) time series derived from satellite imagery. We find that 12.8% of the lakes studied show regime shifts whose signatures are compatible with tipping points, while the number of detected regime shifts from low to high TSI has increased over time. Although our results suggest an overall stable picture for global lake dynamics, the limited instability signatures do not mean that lakes are insensitive to global change. Modeling the interaction between lake climatic, geophysical, and socioeconomic features and their stability properties, we find that the probability of a lake experiencing a tipping point increases with human population density in its catchment, while it decreases as the gross domestic product of that population increases. Our results show how quantifying lake productivity dynamics at a global scale highlights socioeconomic inequalities in conserving natural environments.

Scaling up our understanding of tipping points

Anthropogenic activities are increasingly affecting ecosystems across the globe. Meanwhile, empirical and theoretical evidence suggest that natural systems can exhibit abrupt collapses in response to incremental increases in the stressors, sometimes with dramatic ecological and economic consequences. These catastrophic shifts are faster and larger than expected from the changes in the stressors and happen once a tipping point is crossed. The primary mechanisms that drive ecosystem responses to perturbations lie in their architecture of relationships, i.e. how species interact with each other and with the physical environment and the spatial structure of the environment. Nonetheless, existing theoretical work on catastrophic shifts has so far largely focused on relatively simple systems that have either few species and/or no spatial structure. This work has laid a critical foundation for understanding how abrupt responses to incremental stressors are possible, but it remains difficult to predict (let alone manage) where or when they are most likely to occur in more complex real-world settings. Here, we discuss how scaling up our investigations of catastrophic shifts from simple to more complex-species rich and spatially structured-systems could contribute to expanding our understanding of how nature works and improve our ability to anticipate the effects of global change on ecological systems. This article is part of the theme issue 'Ecological complexity and the biosphere: the next 30 years'.

Evaluating the performance of temporal and spatial early warning statistics of algal blooms

Regime shifts have large consequences for ecosystems and the services they provide. However, understanding the potential for, causes of, proximity to, and thresholds for regime shifts in nearly all settings is difficult. Generic statistical indicators of resilience have been proposed and studied in a wide range of ecosystems as a method to detect when regime shifts are becoming more likely without direct knowledge of underlying system dynamics or thresholds. These early warning statistics (EWS) have been studied separately but there have been few examples that directly compare temporal and spatial EWS in ecosystem-scale empirical data. To test these methods, we collected high-frequency time series and high-resolution spatial data during a whole-lake fertilization experiment while also monitoring an adjacent reference lake. We calculated two common EWS, standard deviation and autocorrelation, in both time series and spatial data to evaluate their performance prior to the resulting algal bloom. We also applied the quickest detection method to generate binary alarms of resilience change from temporal EWS. One temporal EWS, rolling window standard deviation, provided advanced warning in most variables prior to the bloom, showing trends and between-lake patterns consistent with theory. In contrast, temporal autocorrelation and both measures of spatial EWS (spatial SD, Moran's  I) provided little or no warning. By compiling time series data from this and past experiments with and without nutrient additions, we were able to evaluate temporal EWS performance for both constant and changing resilience conditions. True positive alarm rates were 2.5-8.3 times higher for rolling window standard deviation when a lake was being pushed towards a bloom than the rate of false positives when it was not. For rolling window autocorrelation, alarm rates were much lower and no variable had a higher true positive than false positive alarm rate. Our findings suggest temporal EWS provide advanced warning of algal blooms and that this approach could help managers prepare for and/or minimize negative bloom impacts.

Structure-based identification of sensor species for anticipating critical transitions

Ecological systems can undergo sudden, catastrophic changes known as critical transitions. Anticipating these critical transitions remains challenging in systems with many species because the associated early warning signals can be weakly present or even absent in some species, depending on the system dynamics. Therefore, our limited knowledge of ecological dynamics may suggest that it is hard to identify those species in the system that display early warning signals. Here, we show that, in mutualistic ecological systems, it is possible to identify species that early anticipate critical transitions by knowing only the system structure-that is, the network topology of plant-animal interactions. Specifically, we leverage the mathematical theory of structural observability of dynamical systems to identify a minimum set of "sensor species," whose measurement guarantees that we can infer changes in the abundance of all other species. Importantly, such a minimum set of sensor species can be identified by using the system structure only. We analyzed the performance of such minimum sets of sensor species for detecting early warnings using a large dataset of empirical plant-pollinator and seed-dispersal networks. We found that species that are more likely to be sensors tend to anticipate earlier critical transitions than other species. Our results underscore how knowing the structure of multispecies systems can improve our ability to anticipate critical transitions.

Abrupt shifts in 21st-century plankton communities

Marine microbial communities sustain ocean food webs and mediate global elemental cycles. These communities will change with climate; these changes can be gradual or foreseeable but likely have much more substantial consequences when sudden and unpredictable. In a complex virtual marine microbial ecosystem, we find that climate change–driven shifts over the 21st century are often abrupt, large in amplitude and extent, and unpredictable using standard early warning signals. Phytoplankton with unique resource needs, especially fast-growing species such as diatoms, are more prone to abrupt shifts. Abrupt shifts in biomass, productivity, and community structure are concentrated in Atlantic and Pacific subtropics. Abrupt changes in environmental variables such as temperature and nutrients rarely precede these ecosystem shifts, indicating that rapid community restructuring can occur in response to gradual environmental changes, particularly in nutrient supply rate ratios.

Collective rule-breaking

Rules form an important part of our everyday lives. Here we explore the role of social influence in rule-breaking. In particular, we identify some of the cognitive mechanisms underlying rule-breaking and propose approaches for how they can be scaled up to the level of groups or crowds to better understand the emergence of collective rule-breaking. Social contagion plays an important role in such processes and different dynamics such as linear or rapid nonlinear spreading can have important consequences for interventions in rule-breaking. A closer integration of cognitive psychology, microsociology and mathematical modelling will be key to a deeper understanding of collective rule-breaking to turn this field of research into a predictive science.

Detecting alternative attractors in ecosystem dynamics

Dynamical systems theory suggests that ecosystems may exhibit alternative dynamical attractors. Such alternative attractors, as for example equilibria and cycles, have been found in the dynamics of experimental systems. Yet, for natural systems, where multiple biotic and abiotic factors simultaneously affect population dynamics, it is more challenging to distinguish alternative dynamical behaviors. Although recent research exemplifies that some natural systems can exhibit alternative states, a robust methodology for testing whether these constitute distinct dynamical attractors is currently lacking. Here, using attractor reconstruction techniques we develop such a test. Applications of the methodology to simulated, experimental and natural time series data, reveal that alternative dynamical behaviors are hard to distinguish if population dynamics are governed by purely stochastic processes. However, if population dynamics are brought about also by mechanisms internal to the system, alternative attractors can readily be detected. Since many natural populations display evidence of such internally driven dynamics, our approach offers a method for empirically testing whether ecosystems exhibit alternative dynamical attractors.

When and Where We Can Expect to See Early Warning Signals in Multispecies Systems Approaching Tipping Points: Insights from Theory

AbstractEarly warning signals (EWSs) have the potential to predict tipping points where catastrophic changes occur in ecological systems. However, EWSs are plagued by false negatives, leading to undetected catastrophes. One reason may be because EWSs do not occur equally for all species in a system, so whether and how strongly EWSs are detected depends on which species is being observed. Here, we illustrate how the strength of EWSs is determined by each species' relationship to properties of the noise, the system's response to that noise, and the occurrence of critical slowing down (the dynamical phenomenon that gives rise to EWSs). Using these relationships, we present general rules for maximizing EWS detection in ecological communities. We find that for two-species competitive and mutualistic systems, one should generally monitor the species experiencing smaller intraspecific effects to maximize EWS performance, while in consumer-resource systems, one should monitor the species imposing the smaller interspecific effects. These guidelines appear to hold for at least some larger communities as well. We close by extending the theoretical basis for our rules to systems with any number of species and more complex forms of noise. Our findings provide important guidance on how to monitor systems for EWSs to maximize detection of tipping points.

The role of freshwater eutrophication in greenhouse gas emissions: A review

Greenhouse gases (GHGs) have long received public attention because they affect the Earth's climate by producing the greenhouse effect. Freshwaters are an important source of GHGs, and the enhancement in their eutrophic status affects GHG emissions. Along with the increasing eutrophication of water bodies, the relevant quantitative and qualitative studies of the effects of freshwater eutrophication on GHG emissions have made substantial progress, particularly in the past 5 years. However, to our knowledge, this is the first critical review to focus on the role of freshwater eutrophication in GHG emissions. In this review, the emissions of common GHGs from freshwater are quantitatively described. Importantly, direct (i.e., dissolved oxygen, organic carbon, and nutrients) and indirect factors (i.e., dominant primary producer and algal blooms) affecting GHG emissions from eutrophic freshwater are systematically analyzed. In particular, the existence and significance of feedback loops between freshwater eutrophication and GHG emissions are emphasized considering the difficulties managing freshwater ecosystems and the Earth's climate. Finally, several future research directions as well as mitigation measures are described to provide useful insight into the dynamics and control of GHG emissions.

Non-additive effects of foundation species determine the response of aquatic ecosystems to nutrient perturbation

Eutrophication is a persistent threat to aquatic ecosystems worldwide. Foundation species, namely those that play a central role in the structuring of communities and functioning of ecosystems, are likely important for the resilience of aquatic ecosystems in the face of disturbance. However, little is known about how interactions among such species influence ecosystem responses to nutrient perturbation. Here, using an array (N = 20) of outdoor experimental pond ecosystems (15,000 L), we manipulated the presence of two foundation species, the macrophyte Myriophyllum spicatum and the mussel Dreissena polymorpha, and quantified ecosystem responses to multiple nutrient disturbances, spread over two years. In the first year, we added five nutrient pulses, ramping up from 10 to 50 μg P/L over a 10-week period from mid-July to mid-October, and in the second year, we added a single large pulse of 50 μg P/L in mid-October. We used automated sondes to measure multiple ecosystems properties at high frequency (15-minute intervals), including phytoplankton and dissolved organic matter fluorescence, and to model whole-ecosystem metabolism. Overall, both foundation species strongly affected the ecosystem responses to nutrient perturbation, and, as expected, initially suppressed the increase in phytoplankton abundance following nutrient additions. However, when both species were present, phytoplankton biomass increased substantially relative to other treatment combinations: non-additivity was evident for multiple ecosystem metrics following the nutrient perturbations in both years but was diminished in the intervening months between our perturbations. Overall, these results demonstrate how interactions between foundation species can cause surprisingly strong deviations from the expected responses of aquatic ecosystems to perturbations such as nutrient additions.

Methodological framework for identifying sustainability intervention priority areas on coastal landscapes and its application in China

In regional sustainability evaluation and policy analysis, the paradigm of safe operating spaces (SOS) has been widely applied. Yet, SOS is not readily useful for informing policy interventions toward sustainability transition. This study reports on a methodological framework that operationalizes SOS at the regional scale for designing spatially targeted sustainability interventions. In particular, this framework accounts for teleology by integrating policy orientations of the place-variant "major function" of development, and provides early-warnings by integrating long-term social-environmental trends. The framework we proposed has been applied by the Chinese government in a coastal province (Liaoning) for a landscape sustainability project, which is introduced here step-by-step. The four main steps include: (1) Quantifying SOS status across multiple "what to sustain" dimensions, e.g., land scarcity, water scarcity, pollutant discharge, and ecosystem health for the inland, and coastal exploitation intensity, marine environmental quality, and marine ecosystem biodiversity for the sea. (2) Quantifying SOS status in terms of the place-variant "what to develop" dimensions, e.g., urbanization-oriented, agriculture-stock-oriented, versus conservation-oriented development. (3) Integrating the two as a composite indicator of three ordinal levels to classify the current SOS status. (4) Developing a multi-level sustainability early-warning system by cross-analysis of the SOS status and social-environmental interaction trends (e.g., changes in, e.g., resource utilization efficiency, pollutant discharge, and eco-environmental quality). The potential use of the framework is demonstrated through the case of Liaoning Province, China, which helps policy-makers to identify priority areas for sustainability interventions. Methodological robustness and future directions of applying this multi-level sustainability early-warning system are further discussed.

Models for Eco-Evolutionary Extinction Vortices under Balancing Selection

AbstractThe smaller a population is, the faster it loses genetic diversity as a result of genetic drift. Loss of genetic diversity can reduce population growth rate, making populations even smaller and more vulnerable to loss of genetic diversity. Ultimately, the population can be driven to extinction by this "eco-evolutionary extinction vortex." While there are already quantitative models for extinction vortices resulting from inbreeding depression and mutation accumulation, to date extinction vortices resulting from loss of genetic diversity at loci under various forms of balancing selection have been mainly described verbally. To understand better when such extinction vortices arise and to develop methods for detecting them, we propose quantitative eco-evolutionary models, both stochastic individual-based simulations and deterministic approximations, linking loss of genetic diversity and population decline. Using mathematical analysis and simulations, we identify parameter combinations that exhibit strong interactions between population size and genetic diversity and match our definition of an eco-evolutionary vortex (i.e., per capita population decline rates and per-locus fixation rates increase with decreasing population size and number of polymorphic loci). We further highlight cues that may be exhibited by such populations but find that classical early-warning signals are of limited use in detecting populations undergoing an eco-evolutionary extinction vortex.

Forecasting resilience profiles of the run-up to regime shifts in nearly-one-dimensional systems

The forecasting of sudden, irreversible shifts in natural systems is a challenge of great importance, whose realization could allow pre-emptive action to be taken to avoid or mitigate catastrophic transitions, or to help systems adapt to them. In recent years, there have been many advances in the development of such early warning signals. However, much of the current toolbox is based around the tracking of statistical trends and therefore does not aim to estimate the future time scale of transitions or resilience loss. Metric-based indicators are also difficult to implement when systems have inherent oscillations which can dominate the indicator statistics. To resolve these gaps in the toolbox, we use additional system properties to fit parsimonious models to dynamics in order to predict transitions. Here, we consider nearly-one-dimensional systems-higher dimensional systems whose dynamics can be accurately captured by one-dimensional discrete time maps. We show how the nearly one-dimensional dynamics can be used to produce model-based indicators for critical transitions which produce forecasts of the resilience and the time of transitions in the system. A particularly promising feature of this approach is that it allows us to construct early warning signals even for critical transitions of chaotic systems. We demonstrate this approach on two model systems: of phosphorous recycling in a shallow lake, and of an overcompensatory fish population.

Exploring How Cyanobacterial Traits Affect Nutrient Loading Thresholds in Shallow Lakes: A Modelling Approach

Globally, many shallow lakes have shifted from a clear macrophyte-dominated state to a turbid phytoplankton-dominated state due to eutrophication. Such shifts are often accompanied by toxic cyanobacterial blooms, with specialized traits including buoyancy regulation and nitrogen fixation. Previous work has focused on how these traits contribute to cyanobacterial competitiveness. Yet, little is known on how these traits affect the value of nutrient loading thresholds of shallow lakes. These thresholds are defined as the nutrient loading at which lakes shift water quality state. Here, we used a modelling approach to estimate the effects of traits on nutrient loading thresholds. We incorporated cyanobacterial traits in the process-based ecosystem model PCLake+, known for its ability to determine nutrient loading thresholds. Four scenarios were simulated, including cyanobacteria without traits, with buoyancy regulation, with nitrogen fixation, and with both traits. Nutrient loading thresholds were obtained under N-limited, P-limited, and colimited conditions. Results show that cyanobacterial traits can impede lake restoration actions aimed at removing cyanobacterial blooms via nutrient loading reduction. However, these traits hardly affect the nutrient loading thresholds for clear lakes experiencing eutrophication. Our results provide references for nutrient loading thresholds and draw attention to cyanobacterial traits during the remediation of eutrophic water bodies.

Keystone taxa of water microbiome respond to environmental quality and predict water contamination

The human activity introduces strong environmental stresses, and results in great spatiotemporal heterogeneity for the environment. Although the effects of environmental factors on the microbial diversity and succession have been widely studied, knowledge about how keystone taxa respond to environmental stresses remains poorly understood. We examined bacterial and archaeal communities from 45 wetland ponds covering a wide range of waters in Hangzhou. We found that shifts in bacterial and archaeal communities were strongly correlated with water pollution as indicated by the comprehensive water quality identification (CWQI). The SEGMENTED analysis suggested that there were non-linear responses of microbial communities and keystone taxa to the water pollution gradient. Moreover, these significant tipping points (e.g., CWQI > 4.0) would afford a warning line for urban wetland management. Notably, keystone taxa of bacterial communities could be used to successfully (~88.9% accuracy) predict water contamination levels. This study provides new insights into the potential for keystone bacterial taxa to predict water contamination.

Rethinking discretization to advance limnology amid the ongoing information explosion

Limnologists often adhere to a discretized view of waterbodies-they classify them, divide them into zones, promote discrete management targets, and use research tools, experimental designs, and statistical analyses focused on discretization. By offering useful shortcuts, this approach to limnology has profoundly benefited the way we understand, manage, and communicate about waterbodies. But the research questions and the research tools in limnology are changing rapidly in the era of big data, with consequences for the relevance of our current discretization schemes. Here, I examine how and why we discretize and argue that selectively rethinking the extent to which we must discretize gives us an exceptional chance to advance limnology in new ways. To help us decide when to discretize, I offer a framework (discretization evaluation framework) that can be used to compare the usefulness of various discretization approaches to an alternative which relies less on discretization. This framework, together with a keen awareness of discretization's advantages and disadvantages, may help limnologists benefit from the ongoing information explosion.

A review of social-ecological system resilience: Mechanism, assessment and management

Social-ecological system (SES) resilience involves the large information and complex relationships of nature, society and economy. To promote multi-disciplinary integration to jointly balance current well-being and long-term sustainability, it is necessary to sort resilience studies on different perspectives into a comprehensive framework to establish interdisciplinary consensus. Based on literature analysis and review, this paper presents an analytical framework for resilience in regional management, and gives a review of SES resilience studies in terms of mechanism, assessment, and management. We outline the current state of resilience research, identify the remaining challenges, and make key recommendations for future research. Our recommendations include promoting interdisciplinary consensus, emphasising dynamic adaptation processes, synthesizing multiple systems and scales, building comprehensive databases, and using mixed methods approach. The paper offers a framework for researchers, practitioners and policy makers to have a more comprehensive understanding of resilience as a whole, and thus helps navigate more fully the challenge of adapting complex resource and environmental problems.

Resilience of aquatic systems: Review and management implications

Our understanding of how ecosystems function has changed from an equilibria-based view to one that recognizes the dynamic, fluctuating, nonlinear nature of aquatic systems. This current understanding requires that we manage systems for resilience. In this review, we examine how resilience has been defined, measured and applied in aquatic systems, and more broadly, in the socioecological systems in which they are embedded. Our review reveals the importance of managing stressors adversely impacting aquatic system resilience, as well as understanding the environmental and climatic cycles and changes impacting aquatic resources. Aquatic resilience may be enhanced by maintaining and enhancing habitat connectivity as well as functional redundancy and physical and biological diversity. Resilience in aquatic socioecological system may be enhanced by understanding and fostering linkages between the social and ecological subsystems, promoting equity among stakeholders, and understanding how the system is impacted by factors within and outside the area of immediate interest. Management for resilience requires implementation of adaptive and preferably collaborative management. Implementation of adaptive management for resilience will require an effective monitoring framework to detect key changes in the coupled socioecological system. Research is needed to (1) develop sensitive indicators and monitoring designs, (2) disentangle complex multi-scalar interactions and feedbacks, and (3) generalize lessons learned across aquatic ecosystems and apply them in new contexts.

Evidence for crisis-induced intermittency during geomagnetic superchron transitions

The geomagnetic field's dipole undergoes polarity reversals in irregular time intervals. Particularly long periods without reversals (of the order of 10^{7} yr), called superchrons, have occurred at least three times in the Phanerozoic (since 541 million years ago). We provide observational evidence for high non-Gaussianity in the vicinity of a transition to and from a geomagnetic superchron, consisting of a sharp increase in high-order moments (skewness and kurtosis) of the dipole's distribution. Such an increase in the moments is a universal feature of crisis-induced intermittency in low-dimensional dynamical systems undergoing global bifurcations. This implies a temporal variation of the underlying parameters of the physical system. Through a low-dimensional system that models the geomagnetic reversals, we show that the increase in the high-order moments during transitions to geomagnetic superchrons is caused by the progressive destruction of global periodic orbits exhibiting both polarities as the system approaches a merging bifurcation. We argue that the non-Gaussianity in this system is caused by the redistribution of the attractor around local cycles as global ones are destroyed.

Anticipating critical transitions in epithelial-hybrid-mesenchymal cell-fate determination

In the vicinity of a tipping point, critical transitions occur when small changes in an input condition cause sudden, large, and often irreversible changes in the state of a system. Many natural systems ranging from ecosystems to molecular biosystems are known to exhibit critical transitions in their response to stochastic perturbations. In diseases, an early prediction of upcoming critical transitions from a healthy to a disease state by using early-warning signals is of prime interest due to potential application in forecasting disease onset. Here, we analyze cell-fate transitions between different phenotypes (epithelial, hybrid-epithelial/mesenchymal [E/M], and mesenchymal states) that are implicated in cancer metastasis and chemoresistance. These transitions are mediated by a mutually inhibitory feedback loop-microRNA-200/ZEB-driven by the levels of transcription factor SNAIL. We find that the proximity to tipping points enabling these transitions among different phenotypes can be captured by critical slowing down-based early-warning signals, calculated from the trajectory of ZEB messenger RNA level. Further, the basin stability analysis reveals the unexpectedly large basin of attraction for a hybrid-E/M phenotype. Finally, we identified mechanisms that can potentially elude the transition to a hybrid-E/M phenotype. Overall, our results unravel the early-warning signals that can be used to anticipate upcoming epithelial-hybrid-mesenchymal transitions. With the emerging evidence about the hybrid-E/M phenotype being a key driver of metastasis, drug resistance, and tumor relapse, our results suggest ways to potentially evade these transitions, reducing the fitness of cancer cells and restricting tumor aggressiveness.

Research questions to facilitate the future development of European long-term ecosystem research infrastructures: A horizon scanning exercise

Distributed environmental research infrastructures are important to support assessments of the effects of global change on landscapes, ecosystems and society. These infrastructures need to provide continuity to address long-term change, yet be flexible enough to respond to rapid societal and technological developments that modify research priorities. We used a horizon scanning exercise to identify and prioritize emerging research questions for the future development of ecosystem and socio-ecological research infrastructures in Europe. Twenty research questions covered topics related to (i) ecosystem structures and processes, (ii) the impacts of anthropogenic drivers on ecosystems, (iii) ecosystem services and socio-ecological systems and (iv), methods and research infrastructures. Several key priorities for the development of research infrastructures emerged. Addressing complex environmental issues requires the adoption of a whole-system approach, achieved through integration of biotic, abiotic and socio-economic measurements. Interoperability among different research infrastructures needs to be improved by developing standard measurements, harmonizing methods, and establishing capacities and tools for data integration, processing, storage and analysis. Future research infrastructures should support a range of methodological approaches including observation, experiments and modelling. They should also have flexibility to respond to new requirements, for example by adjusting the spatio-temporal design of measurements. When new methods are introduced, compatibility with important long-term data series must be ensured. Finally, indicators, tools, and transdisciplinary approaches to identify, quantify and value ecosystem services across spatial scales and domains need to be advanced.

Time scale of resilience loss: Implications for managing critical transitions in water quality

Regime shifts involving critical transitions are a type of rapid ecological change that are difficult to predict, but may be preceded by decreases in resilience. Time series statistics like lag-1 autocorrelation may be useful for anticipating resilience declines; however, more study is needed to determine whether the dynamics of autocorrelation depend on the resolution of the time series being analyzed, i.e., whether they are time-scale dependent. Here, we examined timeseries simulated from a lake eutrophication model and gathered from field measurements. The field study involved collecting high frequency chlorophyll fluorescence data from an unmanipulated reference lake and a second lake undergoing experimental fertilization to induce a critical transition in the form of an algal bloom. As part of the experiment, the fertilization was halted in response to detected early warnings of the algal bloom identified by increased autocorrelation. We tested these datasets for time-scale dependence in the dynamics of lag-1 autocorrelation and found that in both the simulation and field experiment, the dynamics of autocorrelation were similar across time scales. In the simulated time series, autocorrelation increased exponentially approaching algal bloom development, and in the field experiment, the difference in autocorrelation between the manipulated and reference lakes increased sharply. These results suggest that, as an early warning indicator, autocorrelation may be robust to the time scale of the analysis. Given that a time scale can be shortened by increasing sampling frequency, or lengthened by aggregating data during analysis, these results have important implications for management as they demonstrate the potential for detecting early warning signals over a wide range of monitoring frequencies and without requiring analysts to make situation-specific decisions regarding aggregation. Such an outcome provides promise that data collection procedures, especially by automated sensors, may be used to monitor and manage ecosystem resilience without the need for strict attention to time scale.

The Impact of Lake Water Quality on the Performance of Mature Artificial Recharge Ponds

Artificial groundwater recharge is commonly used for drinking water supply. The resulting water quality is highly dependent on the raw water quality. In many cases, pretreatment is required. Pretreatment improves the drinking water quality, although how and to what extent it affects the subsequent pond water quality and infiltration process, is still unknown. We evaluated two treatment systems by applying different pretreatment methods for raw water from a eutrophic and temperate lake. An artificial recharge pond was divided into two parts, where one received raw water, only filtered through a microscreen with 500 µm pores (control treatment), while the other part received pretreated lake water using chemical flocculation with polyaluminum chloride (PACl) combined with sand filtration, i.e., continuous contact filtration (contact filter treatment). Water quality factors such as cyanobacterial biomass, microcystin, as well as organic matter and nutrients were measured in both treatment processes. Microcystin condition was screened by an immunoassay and a few selected samples were examined by ultra-high-performance liquid chromatography tandem mass spectrometry (UPLC–MS/MS) which is a chemistry technique that combines the physical separation capabilities of liquid chromatography with the mass analysis capabilities of mass spectrometry. Results showed that cyanobacterial biomass and microcystin after the contact filter treatment were significantly different from the control treatment and also significantly different in the pond water. In addition, with contact filter treatment, total phosphorus (TP) and organic matter removal were significantly improved in the end water, TP was reduced by 96% (<20 µg/L) and the total organic carbon (TOC) was reduced by 66% instead of 55% (TOC content around 2.1 mg/L instead of 3.0 mg/L). This full-scale onsite experiment demonstrated effective pretreatment would benefit a more stable water quality system, with less variance and lower microcystin risk. From a broader drinking water management perspective, the presented method is promising for reducing cyanotoxin risk, as well as TP and TOC, which are all predicted to increase with global warming and extreme weather.

Network dynamics of positive and negative affect in bipolar disorder

BACKGROUND

The network approach to psychopathology has become increasingly popular. Little research has examined the dynamic network structure of mental disorders, and, to date, no study has investigated the network dynamics of positive affect, negative affect, and physical activity in bipolar disorder. This represents the first study to estimate the dynamic network structure of affect and physical activity in individuals with and without bipolar I disorder.

METHODS

An intensive longitudinal design was used to assess positive affect, negative affect, and actigraphy-based estimates of physical activity. The overall sample consisted of 32 adults with bipolar I disorder and 36 healthy control participants. Eligible participants underwent an 8-week assessment period, in which once-per-day ratings of affect and actigraphy estimates were obtained. Participants were re-assessed on baseline measures afterwards. Dynamic network analysis was used to examine the network structure of affect and physical activity over time. Multilevel models were used to examine the relationship between autocorrelation and changes in depression symptoms among participants with bipolar disorder.

LIMITATIONS

The network analyses assume stationarity. Future research should consider time-varying multilevel network models to better account for time trends.

RESULTS

The results of the temporal networks indicated that the directed edges between positive and negative affect were mostly positive among individuals with bipolar I disorder. Among healthy control participants, the directed edges between positive and negative affect were mostly negative in direction. Physical activity, as assessed by daily actigraphy indices, was more densely connected in the healthy control network than the bipolar disorder network. Furthermore, the results indicated that critical slowing down predicted worsening of mood symptoms in the bipolar I disorder group.

CONCLUSIONS

This study suggests that certain dynamic patterns of affect may be an underlying process that contributes to the maintenance of bipolar disorder. These results have both theoretical and practical implications.

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

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

Early Warnings for State Transitions

New concepts have emerged in theoretical ecology with the intent to quantify complexities in ecological change that are unaccounted for in state-and-transition models and to provide applied ecologists with statistical early warning metrics able to predict and prevent state transitions. With its rich history of furthering ecological theory and its robust and broad-scale monitoring frameworks, the rangeland discipline is poised to empirically assess these newly proposed ideas while also serving as early adopters of novel statistical metrics that provide advanced warning of a pending shift to an alternative ecological regime. Were view multivariate early warning and regime shift detection metrics, identify situations where various metrics will be most useful for rangeland science, and then highlight known shortcomings. Our review of a suite of multivariate-based regime shift/early warning indicators provides a broad range of metrics applicable to a wide variety of data types or contexts, from situations where a great deal is known about the key system drivers and a regime shift is hypothesized a priori, to situations where the key drivers and the possibility of a regime shift are both unknown. These metrics can be used to answer ecological state-and-transition questions, inform policymakers, and provide quantitative decision-making tools for managers.

The role of the individual in the coming era of process-based therapy

For decades the development of evidence-based therapy has been based on experimental tests of protocols designed to impact psychiatric syndromes. As this paradigm weakens, a more process-based therapy approach is rising in its place, focused on how to best target and change core biopsychosocial processes in specific situations for given goals with given clients. This is an inherently more idiographic question than has normally been at issue in evidence-based therapy over the last few decades. In this article we explore methods of assessment and analysis that can integrate idiographic and nomothetic approaches in a process-based era.

Profiling physicochemical and planktonic features from discretely/continuously sampled surface water

There is an increasing need for assessing aquatic ecosystems that are globally endangered. Since aquatic ecosystems are complex, integrated consideration of multiple factors utilizing omics technologies can help us better understand aquatic ecosystems. An integrated strategy linking three analytical (machine learning, factor mapping, and forecast-error-variance decomposition) approaches for extracting the features of surface water from datasets comprising ions, metabolites, and microorganisms is proposed herein. The three developed approaches can be employed for diverse datasets of sample sizes and experimentally analyzed factors. The three approaches are applied to explore the features of bay water surrounding Odaiba, Tokyo, Japan, as a case study. Firstly, the machine learning approach separated 681 surface water samples within Japan into three clusters, categorizing Odaiba water into seawater with relatively low inorganic ions, including Mg, Ba, and B. Secondly, the factor mapping approach illustrated Odaiba water samples from the summer as rich in multiple amino acids and some other metabolites and poor in inorganic ions relative to other seasons based on their seasonal dynamics. Finally, forecast-error-variance decomposition using vector autoregressive models indicated that a type of microalgae (Raphidophyceae) grows in close correlation with alanine, succinic acid, and valine on filters and with isobutyric acid and 4-hydroxybenzoic acid in filtrate, Ba, and average wind speed. Our integrated strategy can be used to examine many biological, chemical, and environmental physical factors to analyze surface water.

Detecting critical transitions in the case of moderate or strong noise by binomial moments

Detecting critical transitions in the case of moderate or strong noise (collectively referred to as big noise) is challenging, since such noise can make a critical transition point far from the bifurcation point, leading to the failure of traditional small-noise methods. To handle this tough issue, we first transform a generic noisy system into a linear set of binomial moment equations (BMEs). Then, we can solve a closed set of BMEs obtained by truncation and use the resulting binomial moments to reconstruct a joint probability distribution of the state variables of the original system. Third, we derive a leading indicator from the closed set of BMEs. Importantly, the reconstructed distribution determines the way of critical transition (i.e., critical transition is distribution transition rather than state transition in the strong-noise case) as the system comes close to the critical transition point, whereas the derived indicator anticipates when the distribution transition occurs. Our theory has broad applications, and artificial and data examples exhibit its power.