Anticipating the occurrence and type of critical transitions

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.

Optimal sampling of spatial patterns improves deep learning-based early warning signals of critical transitions

Complex spatio-temporal systems like lakes, forests and climate systems exhibit alternative stable states. In such systems, as the threshold value of the driver is crossed, the system may experience a sudden (discontinuous) transition or smooth (continuous) transition to an undesired steady state. Theories predict that changes in the structure of the underlying spatial patterns precede such transitions. While there has been a large body of research on identifying early warning signals of critical transitions, the problem of forecasting the type of transitions (sudden versus smooth) remains an open challenge. We address this gap by developing an advanced machine learning (ML) toolkit that serves as an early warning indicator of spatio-temporal critical transitions, Spatial Early Warning Signal Network (S-EWSNet). ML models typically resemble a black box and do not allow envisioning what the model learns in discerning the labels. Here, instead of naively relying upon the deep learning model, we let the deep neural network learn the latent features characteristic of transitions via an optimal sampling strategy (OSS) of spatial patterns. The S-EWSNet is trained on data from a stochastic cellular automata model deploying the OSS, providing an early warning indicator of transitions while detecting its type in simulated and empirical samples.

Resilience and high compositional variability reflect the complex response of river waters to global drivers: The Eastern Siberian River Chemistry database

The chemical composition of river waters represents an important matter of investigation to understand environment modifications in response to climate changes and global warming. Prolonged dry periods, heavy flood events, degradation of the lands and ice thawing, modify the chemical composition of river waters influencing the drivers governing the complex dynamics of river catchments where everything comes together. In this framework, Compositional Data Analysis (CoDA) offers methods in which the complex structure of the river water composition and the interrelationships among the various components are put into the proper context for their statistical analysis. In this research, we propose a new CoDA approach combining the robust Mahalanobis distance (D) calculus of ilr-transformed chemical variables and the perturbation difference, both with respect to a pristine compositional benchmark. The aim was to trace the change in the chemical composition of the Eastern Siberian River Chemistry Database where degradation of the permafrost for global warming produces important effects on natural waters. The findings indicate complex multiplicative laws and feedback mechanisms governing solutes in Eastern Siberian rivers, with high values of D found where permafrost is more discontinuous. Perturbations clearly discriminate chemical components more resilient to stresses induced by global changes (Ca2+, Mg2+ and HCO3-) from those whose variability is not maintained under control (Cl-, Na+, SO42-). These outcomes open up a new scenario in searching for spatiotemporal resilience metrics to reveal rivers response to environmental changes.

Detecting shifts in nonlinear dynamics using Empirical Dynamic Modeling with Nested-Library Analysis

Abrupt changes in system states and dynamical behaviors are often observed in natural systems; such phenomena, named regime shifts, are explained as transitions between alternative steady states (more generally, attractors). Various methods have been proposed to detect regime shifts from time series data, but a generic detection method with theoretical linkage to underlying dynamics is lacking. Here, we provide a novel method named Nested-Library Analysis (NLA) to retrospectively detect regime shifts using empirical dynamic modeling (EDM) rooted in theory of attractor reconstruction. Specifically, NLA determines the time of regime shift as the cutting point at which sequential reduction of the library set (i.e., the time series data used to reconstruct the attractor for forecasting) optimizes the forecast skill of EDM. We illustrate this method on a chaotic model of which changing parameters present a critical transition. Our analysis shows that NLA detects the change point in the model system and outperforms existing approaches based on statistical characteristics. In addition, NLA empirically detected a real-world regime shift event revealing an abrupt change of Pacific Decadal Oscillation index around the mid-1970s. Importantly, our method can be easily generalized to various systems because NLA is equation-free and requires only a single time series.

Antifragility as a complex system's response to perturbations, volatility, and time

Antifragility characterizes the benefit of a dynamical system derived from the variability in environmental perturbations. Antifragility carries a precise definition that quantifies a system's output response to input variability. Systems may respond poorly to perturbations (fragile) or benefit from perturbations (antifragile). In this manuscript, we review a range of applications of antifragility theory in technical systems (e.g., traffic control, robotics) and natural systems (e.g., cancer therapy, antibiotics). While there is a broad overlap in methods used to quantify and apply antifragility across disciplines, there is a need for precisely defining the scales at which antifragility operates. Thus, we provide a brief general introduction to the properties of antifragility in applied systems and review relevant literature for both natural and technical systems' antifragility. We frame this review within three scales common to technical systems: intrinsic (input-output nonlinearity), inherited (extrinsic environmental signals), and interventional (feedback control), with associated counterparts in biological systems: ecological (homogeneous systems), evolutionary (heterogeneous systems), and interventional (control). We use the common noun in designing systems that exhibit antifragile behavior across scales and guide the reader along the spectrum of fragility-adaptiveness-resilience-robustness-antifragility, the principles behind it, and its practical implications.

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.

Reactivity of complex communities can be more important than stability

Understanding stability-whether a community will eventually return to its original state after a perturbation-is a major focus in the study of various complex systems, particularly complex ecosystems. Here, we challenge this focus, showing that short-term dynamics can be a better predictor of outcomes for complex ecosystems. Using random matrix theory, we study how complex ecosystems behave immediately after small perturbations. Our analyses show that many communities are expected to be 'reactive', whereby some perturbations will be amplified initially and generate a response that is directly opposite to that predicted by typical stability measures. In particular, we find reactivity is prevalent for complex communities of mixed interactions and for structured communities, which are both expected to be common in nature. Finally, we show that reactivity can be a better predictor of extinction risk than stability, particularly when communities face frequent perturbations, as is increasingly common. Our results suggest that, alongside stability, reactivity is a fundamental measure for assessing ecosystem health.