ICES Journal of Marine Science: Journal du Conseil

Griffith, Gary, P

doi: 10.1093/icesjms/fsaa087pmid: N/A

Abstract The themed article set (TS) “Science in support of a nonlinear non-equilibrium world” reflects the challenge presented by the increasing potential for complex non-linear behaviour in marine ecosystems, many of which are undergoing dramatic changes due to anthropogenic perturbations. Marine ecosystems are complex adaptive systems, yet management strategies are often guided by a linear, stable perspective that excludes non-linearities and the possibility for evolution and adaptation. Rapidly increasing amounts of observational data, the interdisciplinary development of powerful mathematical approaches from complexity theory, and the evolving complex adaptive systems approach that includes human behaviour promise to substantially advance the development of management strategies. For these reasons, the ICES Journal of Marine Science solicited contributions to a TS that would take up these issues. In this introduction, I focus on three important areas—causality, prediction, and emergence—where a “non-linear” perspective can advance our understanding and better support sustainable management of ecosystems. I briefly present the nine contributions that are included in the themed set and suggest some ways forward. We hope that these articles serve to convince managers and marine scientists of the benefits of incorporating approaches and results from research on non-linear dynamics. Background to this themed article set “Science in a nonlinear, non-equilibrium world” reflects a growing awareness of the dramatic changes in marine systems due to anthropogenic perturbations. The viewpoint that marine systems are stable dynamic systems is increasingly challenged with observational evidence showing that marine dynamics (e.g. fluctuations in species abundance) emerge from an interplay between deterministic non-linear dynamics and stochastic forces (Cenci et al., 2019). The analytical framework currently used to guide many management decisions assumes that the causal relationships arise from “well-behaved” linear dynamics or non-linear dynamics approximated at equilibrium (Fogarty et al., 2016). In contrast, it is well known that such assumptions are not universally applicable for marine systems, when state-dependent behaviour characteristic of complex non-linear systems is ubiquitous (Sugihara and May, 1990). “Non-linearity”, often termed “non-linear dynamical behaviour” or simply “dynamic complexity”, has emerged from a range of paradigms broadly considered as complexity science to explain one or more aspects of non-linear dynamics. These include chaos theory (Alligood et al., 1996; Gleick, 2011), catastrophe theory (Zeeman, 1976; Thom, 1989), and bifurcation theory (Kuznetsov, 2004). Each of these paradigms are linked by a common theme that non-linear dynamics increases the potential for sudden and often unexpected changes in the system such as regime shifts (Scheffer et al., 2015; Fogarty et al., 2016; Moore, 2018, and https://www.complexityexplorer.org/). While non-linear dynamics are essentially deterministic, observed fluctuations, such as those in time series of species abundance, result from the complex interplay between deterministic non-linear dynamics and stochastic forces (Cenci et al., 2019). The consequences of exogenous environmental forcing added to endogenous dynamics have profound consequences for the variability of species abundance over different time scales. Rather than attempting to investigate system dynamics through complex equation-based models (e.g., end-to-end ecosystem models, Olsen et al., 2018) that explicitly account for each interaction, the “non-linear dynamic” perspective draws on the concept from statistical physics that any variable in a state-dependent dynamic system encodes the history of its interactions with other system variables. That is, the data itself provide the evidentiary portal to identifying the causal relationships in complex real-world system dynamics. To many in marine management, this concept is challenging despite recent work demonstrating the utility of this approach to a wide range of topics in marine ecology (Ye et al., 2015; Deyle et al., 2016; Munch et al., 2018) and associated environmental problems (Goitom et al., 2018; Rypdal and Sugihara, 2019). However, a rapidly increasing amount of observational data, combined with progress in computational power and the continuous development of applied techniques can substantially advance our understanding of non-linear ecosystems and socio-ecological systems in three key areas: identifying causal relationships, improving prediction, and emergence. Causality Manifestations of non-linear dynamics include mirage correlations where causally inferred variables can appear to be in-phase, asynchronous, or anti-correlated over different time periods. In such cases, classical correlation methods are often ambiguous to interpret and can lead to incorrect conclusions regarding the causal drivers (Deyle et al., 2013). While the truism that correlation does not imply causation holds, the key concept that underlies dynamic complexity follows Reichenbach’s common cause principle that, if variables are state dependent, then they are causal to each other or driven by a common causal driver (Reichenbach, 1956). Equilibrium-based equation (parametric) models widely applied to understand, predict, and manage populations and marine ecosystems fall short if the underlying system exhibits non-linear dynamics (Ye and Sugihara 2016). Two model types: single-species fishery stock assessment and multispecies end-to-end ecosystem models (e.g., Ecopath, https://ecopath.org, and Atlantis, Olsen et al., 2018) underlie many efforts to inform fisheries and marine management. However, it is not clear how well underlying real-world dynamic complexity is implemented or retained through the model output (Fulton et al., 2011; Griffith and Fulton, 2014). For instance, the classic Ricker model that underlies many stock assessment models (Ricker, 1975) can generate non-linear deterministic dynamics or linear dynamics depending on the population growth rate (Munch et al., 2018). The assumption of a fixed constant is not valid if the quantities such as growth rate or recruitment, when tested against observational data, are themselves state dependent (Ye et al., 2015). Model output may underestimate variability and overestimate stability, an issue of some concern when the output is used for prediction (Storch et al., 2017). However, these issues are well recognized and there are continuing efforts to critically evaluate how well these models represent real world non-linear dynamics (Olsen et al., 2018). Rather than a parametric approach of constructing complex equation-based models, the dynamic systems perspective is evolving around a range of applications within a nonparametric framework of mathematically rigorous procedures using real-world observational data without a pre-supposed set of equations (Perretti et al., 2013; Huffaker et al., 2017). These methods assume any single variable or combination of variables encodes the history of its causal interactions with other system variables. Phase-space or state-space reconstruction remains the core of these approaches based on an abstract mathematical construction defined by the co-ordinate axes of time evolving variables (Huffaker et al., 2017). The state of the system changes according to the trajectory of the co-ordinate axes forming a geometric object known as an attractor (or an attractor manifold) that describes empirically how variables relate to each other in time. Consequently, any point on the attractor encapsulates the dynamic behaviour of the system. State-space reconstruction takes advantage of the powerful approach of embedding theorems (Takens, 1981) that show how lagged variables of a single time series can be used as proxy variables to reconstruct an attractor for the underlying dynamic system (Deyle and Sugihara, 2011; Chang et al., 2017). The most commonly applied approach of state-space reconstruction is empirical dynamic modelling (e.g., Deyle et al., 2016, 2018; McGowan et al., 2017), which uses deterministic low-dimensional non-linear attractor manifolds reconstructed from time series data. If the system behaviour is governed by deterministic rules, then attractor manifolds representing the underlying non-linear dynamics can be constructed from the time lags of a single variable or a combination of variables (see http://tinyurl.com/EDM-intro and Chang et al., 2017). Some researchers question the validity of non-linear time series analysis, such as phase-space reconstruction techniques when estimated from the often short and noisy real-world data common in marine systems (McSharry, 2011). It remains challenging to measure noise and nonequilibrium behaviour in natural systems (Cobey and Baskerville, 2016). If the dynamics of the system occur on time scales shorter than the sampling interval, then the empirical data may not be enough to reconstruct the full dynamics of the system, limiting the result to a skeleton of a real-world attractor that is not informative (Schreiber, 1999). Real world applications of empirical dynamic modelling to 185 fish populations show that time series data needs to be ten times the mean age of stock maturation (Munch et al., 2018). Recent developments have shown that the issues of short and noisy data can be overcome by the straightforward approach of creating multiple viewpoints of the attractor created from combinations of the variables and then combined into a single model (Ye and Sugihara, 2016). While such approaches can overcome the issue of “mirage correlations”, “false positives” are possible where a causal relationship is detected between two variables that do not exist (Sugihara et al., 2012; Ye et al., 2015). Nevertheless, there is considerable value in first assuming that the data is generated by deterministic and non-linear real-world dynamics before presuming linear dynamics that potentially misrepresent (e.g. mirage correlations) real-world dynamics. Non-linear time series analysis can help uncover the potential underlying dynamics and causal drivers, whether they are linear, non-linear, deterministic, or stochastic (Huffaker et al., 2017). Nonparametric models can also be of great help as a complementary method to improving complex parametric mechanistic models. For instance, state-space reconstruction can help identify the important causal species interactions (Deyle et al., 2016) that can then be a focus in the development of food web and end-to-end ecosystem models. Prediction Predictability, defined as how well future values can be forecast based on past and present values, remains the cornerstone of fisheries stock–recruitment models. Despite the substantial increase in our understanding of stock–recruitment variability, predicting recruitment remains problematic. Traditional management approaches have been questioned on their ability to show evidence of separability and independence between stock size and recruitment (Hilborn et al., 2017). Recent research shows the benefit of applying a non-linear perspective complementary to traditional stock–recruitment modelling to improve forecast recruitment (Ye et al., 2015; Deyle et al., 2018; Munch et al., 2018). This raises the realistic possibility of setting precautionary margins on prediction rather than uncertainty. Further development of nonparametric approaches can help develop multispecies management strategies incorporating different life histories (Munch et al., 2018). There is considerable benefit to management of using nonparametric models to test the predictive estimates of parametric complex end-to-end ecosystem models that may underestimate variability and overestimate ecosystem stability in the presence of underlying non-linear dynamics (Storch et al., 2017). Scenario exploration that considers the presence of non-linear dynamics alongside traditional linear approaches is crucial to understanding model uncertainty and provides a more realistic picture of the range of possible dynamical responses from proposed management strategies. Emergence The complex adaptive systems perspective extends the traditional view of complexity science (bifurcation theory, catastrophe theory, and chaos theory) to investigate how self-organization emerges from the interplay between processes (e.g. nutrient cycling, species interactions, trait biogeography) at different scales (Hagstrom and Levin, 2017). That is, how macroscale structures emerge from complex microscale interactions. The complex adaptive systems framework emphasizes non-linear dynamics and stochasticity that goes beyond species and ecosystem management to considering the interactions of humans with the marine environment, integrating dynamic complexity across multiple scales of organization. The features of complex adaptive systems such as emergence, alternative stable states, novel types, evolution of cooperative behaviour, and critical phenomena foil many management strategies (Hagstrom and Levin, 2017). An inability of management to understand the complex interactions at disparate social and ecological timescales (e.g. different adaptive cycles, Gunderson, 2000) often results in protective procedures being implemented after ecosystem collapse resulting in the system being trapped in an undesirable stable state for long periods of time. Recent research using the complex adaptive systems approach has shown how considering resilience as an emergent property of the self-organizing of key ecosystem processes can help management understand how rapidly changing complex adaptive systems such as the Arctic marine system are absorbing environmental perturbations and potentially adapting with the emergence of novel species interactions (Griffith et al., 2019). Promising new techniques based on cavity and random reference models show that complex ecological species interactions exhibit generic behaviour that can be predicted from a few emergent properties such as equilibrium diversity, functioning, and stability (Barbier et al., 2018). As with other approaches such as phase-space methods, complex adaptive systems tools and approaches can also help further quantify model uncertainty, particularly around the probability of critical transitions. With the growing awareness of the potential for complex dynamical behaviour in exploited marine ecosystems and its management implications, the ICES Journal of Marine Science solicited contributions to the themed article set (TS), “Science in support of a nonlinear non-equilibrium world”. This TS includes contributions to a diverse range of topics on novel advances to investigate causal relationships between species and environmental conditions; multi-model approaches that consider non-linearity, linearity, and thresholds; new ecological indicators of resilience and sustainability; and investigating stability in marine phytoplankton and a model approach that considers the uncertainty and predictability in a participatory management framework. The contributions to this TS provide evidence of novel tools, methods, and thinking to facilitate a greater consideration of complex non-linear dynamics for effective marine management. About the articles in this TS Dispelling the mystery of non-linear dynamics Munch et al. (2020) address some of the most frequently asked questions about non-linear causal dynamics and non-linear forecasting from ecologists without a formal background in dynamic systems theory. They focus on the empirical dynamic modelling framework and show four important applications to marine ecosystem research and management: first, applying phase-space reconstruction approaches to evaluate if the dynamics of a system are non-linear; successful reconstruction of a shadow attractor provides preliminary empirical evidence that the data may be generated by deterministic dynamics; second, testing for causality between variables; this is a great strength of non-linear state-space methods for fisheries management; third, as a direct estimate of the net effect of species interactions at each point in time; this is one application of phase-space reconstruction that potentially is a powerful practical tool for marine management to track changing species interactions (e.g. predator–prey) with environmental perturbation; and fourth, applying in a complementary way to help construct and validate parametric models such as mechanistic population models (Thorson et al., 2014). Importantly, they also address the important issue of when empirical dynamic modelling does not work, for instance when trying to make predictions for species whose lifespans exceed the length of the available time series. Non-linearity in stock–recruitment Nakayama et al. (2020) investigate the potential of applying the convergent cross mapping method within the empirical dynamic modelling framework to the relationship between the biomass of Pacific Bluefin (Thunnus orientalis) and the amount of recruitment. Recent work has questioned the ultimate predictability or randomness of recruitment in Pacific Bluefin, despite considerable stock assessment data (Schindler and Hilborn, 2015; Deyle et al., 2018). A common explanation is the inability to incorporate environmental information and other ecosystem processes such as food availability into recruitment forecasts using traditional stock–recruitment relationships (SSRs) such as Ricker and Beverton–Holt, where the dynamics are always stable and in equilibrium (Deyle et al., 2018). Nakayama et al. (2020) show that SSRs formulated from empirical dynamic modelling incorporating causal relationships between sea surface temperature and density-dependent recruitment are improvements on traditional methods. This study joins a growing number of studies showing the usefulness of non-linear state-space methods for applied fisheries management (e.g. Deyle et al., 2018; Munch et al., 2018). Sguotti et al. (2020) show the value of a multi-model approach and accounting for environmental factors when applied to the SSR of Atlantic cod (Gadus morhua). They compared the traditional Ricker model with two nonparametric approaches: first, multivariate simplex projection (Deyle et al., 2018) and, second, the stochastic cusp model based on the catastrophe theory (Sguotti et al., 2019). They found that predicting fish recruitment depends on the dynamical properties of each individual stock. Cod stocks with gradual and mostly linear dynamics are best predicted by the traditional Ricker model. The stochastic cusp model was well suited to stocks that experience sudden abrupt changes in recruitment and stock size with the additional utility to applied management of providing a form of vulnerability assessment for regime shift dynamics. In comparison, multivariate simplex projection captured the most complex dynamics and most appropriate for stocks displaying more fluctuating behaviour. Satterthwaite et al. (2020) examined the potential environmental drivers that can plausibly affect Chinook salmon (Oncorhynchus tshawytscha) throughout their lifecycle. They then investigated if current forecast models can capture non-linear threshold (tipping points) relationships in those environmental drivers. They show that this type of exploratory approach is particularly useful for identifying potential non-linear relationships between preseason abundance forecasts and environmental drivers. A promising near-term application of their results is the identification of thresholds in environmental states, beyond which management should consider a precautionary approach. They recommend extending their approach to ensemble modelling to improve greater statistical robustness (Large et al., 2013). They suggest that for real-world use, the threshold indices should cover a range of time lags (i.e. at least a year in advance) to inform management and help with further investigation before a final preseason decision is made. Non-linearity and ecological thresholds Fu et al. (2020) used a comparative multi-ecosystem, multi-model simulation experiment to explore the responses (linearity, non-linearity, threshold) of 14 ecological indicators to fishing mortality and changes in primary productivity (a proxy for environmental change). The over application of threshold responses (i.e. tipping points) suggesting non-linearity and potential sudden ecological transitions of ecological concern has caused confusion among managers (Litzow and Hunsicker, 2016). This study shows that most of the indicators have a linear response to fishing mortality and that the thresholds for a non-linear mortality response to changes in primary productivity are infrequent. There are three important messages from this work for applied management: first, the value of a multi-model simulation approach where the different assumptions and parametric structure of each can compensate for the practical problems of typically short and autocorrelated observational data and improve the predictability of the underlying system dynamics; second, the importance of considering the response of the ecological indicators to different fishing strategies; and third, the importance of considering the various responses (linearity, non-linearity, and threshold) and then eliminating the linear threshold response (predicable and reversible) to focus the management response on the more concerning non-linear threshold. Boschetti et al. (2020) introduce new measures of ecosystem change, resilience, and sustainability that can be applied to describe the behaviour of complex ecological systems exhibiting non-linear dynamics. Using information theory (Cover and Thomas, 2012), they introduce a measure of ecosystem change—“Total Divergence”—that accounts for the changes in overall biomass and system diversity. Using “Total Divergence”, four measures of sustainability, baseline system shift, resilience to local perturbation, and resilience to global perturbation are defined. Resilience in this case is defined as the context-dependent ability of the system to return to a pre-defined baseline (e.g. biomass) in response to perturbation. The results from three different simulations (Ecopath with Ecosim, Coral Reef model, and predator–prey model) show the utility of the indicators to simplify system analysis and monitoring of state-dependent ecosystem behaviour. Pedersen et al. (2020) propose a new approach to characterize regime shifts in complex and spatially structured communities. Applying nonparametric spatio-temporal regression models, they provide new insights into the dynamics of the Newfoundland groundfish collapse and partial recovery between 1981 and 2017. Stability in a non-linear world Kovac et al. (2020) investigate stability in marine phytoplankton biomass. They report the results of an analysis of a simple nutrient-phytoplankton marine ecosystem model representing the surface mixed layer influenced by self-shading of phytoplankton and nutrient limitation. In this context, stability results from mechanisms by which fluctuations in biomass can be dampened and the speed at which a steady state can be achieved. The causes of stability are shown to be self-shading and nutrient limitation. They also show that the system remains stable with continuously varying mixed layer depth and for an abrupt change in mixed layer depth. The return times of the phytoplankton biomass following perturbation of a parameter (e.g. nutrients) are proposed as a metric for the resilience of the pelagic ecosystem. Exploring multiple possibilities and uncertainty Planque and Mullon (2020) show how chance and necessity modelling can provide a participatory management framework to reconcile the misunderstanding and miscommunication about the uncertainties and predictability associated with non-linearity. By applying this approach to the Barents Sea Ecosystem, they show how a combination of chance and necessity provides a simple, transparent and easily communicable way to explore potential complex system dynamics with only partial knowledge of the ecosystem. The model structure and constraints are based on the knowledge of the system such as fisheries and environmental time series. What is not known of the system is considered in terms of chance and necessity (Monod, 1971). Important constraints (e.g. trophic flow from prey to predator is always positive) are defined and quantified with the ensemble of constraints forming the necessity part of the approach. Chance is defined as the set of possibilities of the processes and interactions operating in an ecosystem (e.g., a population can grow or decline). In the context of participatory management, Planque and Mullon (2020) suggest that the ICES-integrated ecosystem assessment groups provide an ideal platform to move chance and necessity modelling from concept to a proof of concept. Conclusions and future challenges The studies included in this TS demonstrate the growing recognition of the utility of tools, methods, and results developed around the concept of complex non-linear dynamics. The articles highlight that the non-linear perspective should be viewed as complementary to traditional approaches. It should not be viewed as the magic elixir or a new paradigm but an additional viewpoint to understand real-world dynamics that naturally consist of linear, non-linear, stable, unstable, deterministic, and stochastic behaviour. Several of the studies in this article TS also highlight the benefit of analysing observational data using non-linear time series analysis prior to selecting an appropriate method, rather than assuming linear dynamics. While non-linear approaches are being increasingly applied in fisheries management, progress in broader marine and socio-ecological management has been slower. Adoption in marine and socio-ecological management has been hampered by the belief that dynamic complexity is mathematically complex and difficult to understand. As Planque and Mullon (2020) show with chance and necessity modelling, dynamically complex natural systems can be considered in the context of participatory management between scientists and non-scientists to resolve the uncertainties around dynamic complexity for applied management. Many of the methods reported in the articles in this TS are data-driven causal inference methods aimed at discovering and quantifying the causal interdependencies of the underlying marine system. They are not black-box machine-learning approaches (e.g., Deep Learning, Goodfellow et al., 2016) that are increasingly viewed as having potential predictive and classification power. Although interpreting deep learning models is an active research area, it is often difficult to interpret the underlying causal relationships (Runge et al., 2019). Moving forward, an integration of machine-learning methods with causal inference methods will be an important step to understanding and predicting the interplay between non-linear deterministic and stochastic dynamics. Moving beyond the approaches applied in the articles in this TS, there are causal inference methods from other applied fields such as statistical mechanics, social sciences, and earth sciences based around the concept of networks that have the potential to be useful for marine management. Runge et al. (2019) provide a very useful summary of current methods from non-linear state-space methods to Bayesian score-based approaches with key strengths and future research directions. For instance, causal network algorithms classified by their search methods for adding or removing links between variables have the potential to overcome the issue of high dimensionality and for detecting unobserved direct or indirect potential causal relationships. Such approaches are potentially very useful for investigating spatio-temporal problems such as the effect of anthropogenic contaminants and other cumulative stressor on the energetics of complex marine food webs. While challenging, integrating network analysis with complex adaptive systems offers the potential of causal hypothesis testing and identifying causal pathways at different scales in response to environmental perturbation. For socio-ecological issues, such an integrative and interdisciplinary approach may be able to infer the key causal interactions (human and environmental) to identify the cooperative behaviour required to preserve critical ecosystem services. The growing amount of observational data raises the realistic possibility of going beyond existing causal methods, such as phase-space reconstruction to other approaches such as phenomenological modelling that allow the data “to speak” of the underlying dynamic complexity data where one or more observed time variables can be distilled into a set of equations that provide a parsimonious representation of non-linear dynamic complexity (Brunton et al., 2016). While there are many challenges to overcome in using the tools developed from the non-linearity perspective for solving real-world marine management problems, we must move towards applied management strategies that include non-linearity, the potential for evolution and adaptation of organisms and ecosystems, and the time-scale separation between the ecological dynamics and the social dynamics. The rapidly increasing amount of observational data, combined with the evolving nonlinear perspective sensibly applied and complementary to traditional linear approaches, promises to substantially advance the development of effective management strategies of marine systems. Acknowledgements I thank Howard Browman for comments on earlier drafts. I also thank Harald Steen, Haakon Hop, Mikko Vihtakari, Benjamin Planque, Freya Griffith, and Martin Rypdal for useful discussions. Funding GPG’s contribution to this special issue was supported by the Norwegian Research Council, project 287114: Integrated risk assessment framework for evaluating the combined impacts of multiple pressures on Arctic ecosystems. References Alligood K. T. , Sauer T. D., Springer J. A. Y. 1996 . CHAOS: An Introduction to Dynamical Systems . Springer , New York . Google Scholar Crossref Search ADS Google Scholar Google Preview WorldCat COPAC Boschetti F. , Prunera K., Vanderklift M. A., Thomson D. P., Babcock R. C., Doropoulos C., Cresswell A., et al. 2020. Information-theoretic measures of ecosystem change, sustainability, and resilience . ICES Journal of Marine Science , doi: 10.1093/icesjms/fsz105. OpenURL Placeholder Text WorldCat Barbier M. , Arnoldi J.-F., Bunin G., Loreau M. 2018 . Generic assembly patterns in complex ecosystems . Proceedings of the National Academy of Sciences of the United States of America , 115 : 2156 – 2161 . Google Scholar Crossref Search ADS PubMed WorldCat Brunton S. L. , Proctor J. L., Kutz J. N. 2016 . Discovering governing equations from data by sparse identification of nonlinear dynamical systems . Proceedings of the National Academy of Sciences of the United States of America , 113 : 3932 – 3937 . Google Scholar Crossref Search ADS PubMed WorldCat Cenci S. , Sugihara G., Saavedra S. 2019 . Regularised S-map for inference and forecasting with noisy ecological time series . Methods in Ecology and Evolution , 10 : 650 – 660 . Google Scholar Crossref Search ADS WorldCat Chang C. W. , Ushio M., Hsieh C. h. 2017 . Empirical dynamic modeling for beginners . Ecological Research , 32 : 785 – 796 . Google Scholar Crossref Search ADS WorldCat Cobey S. , Baskerville E. B. 2016 . Limits to causal inference with state-space reconstruction for infectious disease . PLoS One , 11 : e0169050 . Google Scholar Crossref Search ADS PubMed WorldCat Cover T. M. , Thomas J. A. 2012 . Elements of Information Theory . John Wiley & Sons , New Jersey. Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Deyle E. R. , Fogarty M., Hsieh C-h., Kaufman L., MacCall A. D., Munch S. B., Perretti C. T., et al. 2013 . Predicting climate effects on Pacific sardine . Proceedings of the National Academy of Sciences of the United States of America , 110 : 6430 – 6435 . Google Scholar Crossref Search ADS PubMed WorldCat Deyle E. R. , May R. M., Munch S. B., Sugihara G. 2016 . Tracking and forecasting ecosystem interactions in real time . Proceedings of the Royal Society B: Biological Sciences , 283 : 20152258 . Google Scholar Crossref Search ADS WorldCat Deyle E. R. , Schueller A. M., Ye H., Pao G. M., Sugihara G. 2018 . Ecosystem-based forecasts of recruitment in two menhaden species. Fish and Fisheries, doi: 10.1111/faf.12287. Deyle E. R. , Sugihara G. 2011 . Generalized theorems for nonlinear state space reconstruction . PLoS One , 6 : e18295 . Google Scholar Crossref Search ADS PubMed WorldCat Fogarty M. J. , Gamble R., Perretti C. T. 2016 . Dynamic complexity in exploited marine ecosystems . Frontiers in Ecology and Evolution , 4 : 1 – 20 . Google Scholar Crossref Search ADS WorldCat Fu C. , Xu Y., Grüss A., Bundy A., Shannon L., Heymans J. J., Halouani G., et al. 2020 . Responses of ecological indicators to fishing pressure under environmental change: exploring non-linearity and thresholds . ICES Journal of Marine Science . doi:10.1093/icesjms/fsz182. OpenURL Placeholder Text WorldCat Fulton E. A. , Link J. S., Kaplan I. C., Savina-Rolland M., Johnson P., Ainsworth C., Horne P., et al. 2011 . Lessons in modelling and management of marine ecosystems: the Atlantis experience . Fish and Fisheries , 12 : 171 – 188 . Google Scholar Crossref Search ADS WorldCat Gleick J. 2011 . Chaos: Making a New Science. Open Road Media, New York. Goitom E. , Kilsdonk L., Brans K., Jansen M., Lemmens P., De Meester L. 2018 . Rapid evolution leads to differential population dynamics and top-down control in resurrected Daphnia populations . Evolutionary Applications , 11 : 96 – 16 . Google Scholar Crossref Search ADS PubMed WorldCat Goodfellow I. , Bengio Y., Courville A. 2016 . Deep Learning . MIT Press , Cambridge, MA, USA . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Griffith G. P. , Fulton E. A. 2014 . New approaches to simulating the complex interaction effects of multiple human impacts on the marine environment . ICES Journal of Marine Science , 71 : 764 – 774 . Google Scholar Crossref Search ADS WorldCat Griffith G. P. , Hop H., Vihtakari M., Wold A., Kalhagen K., Gabrielsen G. W. 2019 . Ecological resilience of Arctic marine food webs to climate change . Nature Climate Change , 9 : 868 – 872 . Google Scholar Crossref Search ADS WorldCat Gunderson L. H. 2000 . Ecological resilience—in theory and application . Annual Review of Ecology and Systematics , 31 : 425 – 439 . Google Scholar Crossref Search ADS WorldCat Hagstrom G. I. , Levin S. A. 2017 . Marine ecosystems as complex adaptive systems: emergent patterns, critical transitions, and public goods . Ecosystems , 20 : 458 – 476 . Google Scholar Crossref Search ADS WorldCat Hilborn R. , Amoroso R. O., Bogazzi E., Jensen O. P., Parma A. M., Szuwalski C., Walters C. J. 2017 . When does fishing forage species affect their predators? Fisheries Research , 191 : 211 – 221 . Google Scholar Crossref Search ADS WorldCat Huffaker R. , Bittelli M., Rosa R. 2017 . Nonlinear Time Series Analysis . Oxford University Press , Oxford. Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Kovac Z. , Platt T., Sathyendranath S. 2020 . Stability and resilience in a nutrient-phytoplankton marine ecosystem model . ICES Journal of Marine Science, doi: 10.1093/icesjms/fsaa067 . OpenURL Placeholder Text WorldCat Kuznetsov Y. A. 2004 . Elements of Applied Bifurcation Theory , 2nd edn. Springer , New York. Google Scholar Crossref Search ADS Google Scholar Google Preview WorldCat COPAC Large S. I. , Fay G., Friedland K. D., Link J. S. 2013 . Defining trends and thresholds in responses of ecological indicators to fishing and environmental pressures . ICES Journal of Marine Science , 70 : 755 – 767 . Google Scholar Crossref Search ADS WorldCat Litzow M. A. , Hunsicker M. E. 2016 . Early warning signal, nonlinearity, and signs of hysteresis in real ecosystems . Ecosphere , 7 : e01614 . Google Scholar Crossref Search ADS WorldCat McGowan J. A. , Deyle E. R., Ye H., Carter M. L., Perretti C. T., Seger K. D., de Verneil A., et al. 2017 . Predicting coastal algal blooms in southern California . Ecology , 98 : 1419 – 1433 . Google Scholar Crossref Search ADS PubMed WorldCat McSharry P. 2011 . The danger of wishing for chaos. In Nonlinear Dynamical Systems Analysis for the Behavioral Sciences Using Real Data , pp. 539 – 558 . Ed. by Guastello S., Gregson R.. CRC Press , Boca Raton, FL . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Monod J. 1971 . Chance and Necessity: An Essay on the Natural Philosophy of Modern Biology . Collins-Fontana Books , New York. Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Moore J. C. 2018 . Predicting tipping points in complex environmental systems . Proceedings of the National Academy of Sciences of the United States of America , 115 : 635 – 636 . Google Scholar Crossref Search ADS PubMed WorldCat Munch S. B. , Brias A., Sugihara G., Rogers T. L. 2020 . Frequently asked questions about nonlinear dynamics and empirical dynamic modelling . ICES Journal of Marine Science , doi: 10.1093/icesjms/fsz209. OpenURL Placeholder Text WorldCat Munch S. B. , Giron-Nava A., Sugihara G. 2018 . Nonlinear dynamics and noise in fisheries recruitment: a global meta-analysis . Fish and Fisheries , 19 : 964 – 973 . Google Scholar Crossref Search ADS WorldCat Nakayama S.-I. , Fukuda H., Nakatsuka S. 2020 . Model-free time series analysis detected the contributions of middle-age spawner biomass and the environment on Pacific bluefin tuna recruitment . ICES Journal of Marine Science , doi: 10.1093/icesjms/fsz129. OpenURL Placeholder Text WorldCat Olsen E. , Kaplan I. C., Ainsworth C., Fay G., Gaichas S., Gamble R., Girardin R., et al. 2018 . Ocean futures under ocean acidification, marine protection, and changing fishing pressures explored using a worldwide suite of ecosystem models . Frontiers in Marine Science , 5 : 64 . Google Scholar Crossref Search ADS WorldCat Pedersen E. J. , Koen-Alonso M., Tunney T. D. 2020 . Detecting regime shifts in communities using estimated rates of change . ICES Journal of Marine Science, doi:10.1093/icesjms/fsaa056 . OpenURL Placeholder Text WorldCat Perretti C. T. , Munch S. B., Sugihara G. 2013 . Model-free forecasting outperforms the correct mechanistic model for simulated and experimental data . Proceedings of the National Academy of Sciences of the United States of America , 110 : 5253 – 5257 . Google Scholar Crossref Search ADS PubMed WorldCat Planque B. , Mullon C. 2020 . Modelling chance and necessity in natural systems . ICES Journal of Marine Science , doi: 10.1093/icesjms/fsz173. OpenURL Placeholder Text WorldCat Reichenbach H. 1956 . The Direction of Time . University of California Press , Berkeley/Los Angeles, CA, USA . Google Scholar Crossref Search ADS Google Scholar Google Preview WorldCat COPAC Ricker W. E. 1975 . Computation and interpretation of biological statistics of fish populations. Bulletin of the Fisheries Research Board of Canada, Bulletin 191. Runge J. , Bathiany S., Bollt E., Camps-Valls G., Coumou D., Deyle E., Glymour G., et al. 2019 . Inferring causation from time series in Earth system sciences . Nature Communications ,. doi:10.10.1038/s41467-019-10105-3. OpenURL Placeholder Text WorldCat Rypdal M. , Sugihara G. 2019 . Inter-outbreak stability reflects the size of the susceptible pool and forecasts magnitudes of seasonal epidemics . Nature Communications , doi: 10.1038/s41467-019-10099-y. OpenURL Placeholder Text WorldCat Satterthwaite, W. H., Andrews, K. S., Burke, B. J., Gosselin, J. L., Greene, C. M., Harvey, C. J. et al. 2020. Ecological thresholds in forecast performance for key United States West Coast Chinook salmon stocks. ICES Journal of Marine Science, doi: 10.1093/icesjms/fsz189. Scheffer M. , Carpenter S. R., Dakos V., van Nes E. H. 2015 . Generic indicators of ecological resilience: inferring the chance of a critical transition . Annual Review of Ecology, Evolution, and Systematics , 46 : 145 – 167 . Google Scholar Crossref Search ADS WorldCat Schindler D. E. , Hilborn R. 2015 . Prediction, precaution, and policy under global climate change . Science , 347 : 953 – 954 . Google Scholar Crossref Search ADS PubMed WorldCat Sguotti C. , Otto S. A., Cormon X., Werner K. M., Deyle E., Sugihara G., Möllmann C. 2020 . Non-linearity in stock–recruitment of Atlantic cod: insights from a multi-model approach . ICES Journal of Marine Science , doi:10.1093/icesjms/fsz113. OpenURL Placeholder Text WorldCat Sguotti C. , Otto S. A., Frelat R., Langbehn T. J., Ryberg M. P., Lindegren M., Durant J. M., et al. 2019 . Catastrophic dynamics limit Atlantic cod recovery . Proceedings of the Royal Society B: Biological Sciences , 286 . https://royalsocietypublishing.org/doi/pdf/10.1098/rspb.2018.2877 (last accessed 5 May 2020). OpenURL Placeholder Text WorldCat Schreiber T. 1999 . Interdisciplinary application of nonlinear time series methods . Physics Reports , 308 : 1 – 64 . Google Scholar Crossref Search ADS WorldCat Storch L. S. , Glaser S. M., Ye H., Rosenberg A. A. 2017 . Stock assessment and end-to-end ecosystem models alter dynamics of fisheries data . PLoS One , 12 : e0171644 . Google Scholar Crossref Search ADS PubMed WorldCat Sugihara G. , May R. M. 1990 . Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series . Nature , 344 : 734 – 741 . Google Scholar Crossref Search ADS PubMed WorldCat Sugihara G. , May R., Ye H., Hsieh C. H., Deyle E., Fogarty M., Munch S. 2012 . Detecting causality in complex ecosystems . Science , 338 : 496 – 500 . Google Scholar Crossref Search ADS PubMed WorldCat Takens F. 1981 . Detecting strange attractors in turbulence. In Dynamical Systems and Turbulence, Warwick 1980. Lecture Notes in Mathematics , 898 , pp. 366 – 381 . Ed. by Rand D. A., Young L.S.. Springer , New York, NY/Berlin, Heidelberg . Google Scholar Crossref Search ADS Google Scholar Google Preview WorldCat COPAC Thom R. 1989 . Structural Stability and Morphogenesis: An Outline of a General Theory of Models . Addison-Welsey , Reading, MA . Google Scholar Google Preview OpenURL Placeholder Text WorldCat COPAC Thorson J. T. , Ono K., Munch S. B. 2014 . A Bayesian approach to identifying and compensating for model misspecification in population models . Ecology , 95 : 329 – 341 . Google Scholar Crossref Search ADS PubMed WorldCat Ye H. , Beamish R. J., Glaser S. M., Grant S. C., Hsieh C. H., Richards L. J., Schnute J. T., Sugihara G. 2015 . Equation-free mechanistic ecosystem forecasting using empirical dynamic modeling . Proceedings of the National Academy of Sciences of the United States of America , 112 : E1569 – E1576 . Google Scholar Crossref Search ADS PubMed WorldCat Ye H. , Sugihara G. 2016 . Information leverage in interconnected ecosystems: overcoming the curse of dimensionality . Science , 353 : 922 – 925 . Google Scholar Crossref Search ADS PubMed WorldCat Zeeman E. 1976 . Catastrophe theory . Scientific American , 234 : 65 – 83 . Google Scholar Crossref Search ADS WorldCat © International Council for the Exploration of the Sea 2020. All rights reserved. For permissions, please email: [email protected] This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)