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1. Non-equilibrium early-warning signals for critical transitions in ecological systems

   https://doi.org/10.1073/pnas.2218663120  

   Xu, Li; Patterson, Denis; Levin, Simon Asher; Wang, Jin (January 2023, Proceedings of the National Academy of Sciences)

   Complex systems can exhibit sudden transitions or regime shifts from one stable state to another, typically referred to as critical transitions. It becomes a great challenge to identify a robust warning sufficiently early that action can be taken to avert a regime shift. We employ landscape-flux theory from nonequilibrium statistical mechanics as a general framework to quantify the global stability of ecological systems and provide warning signals for critical transitions. We quantify the average flux as the nonequilibrium driving force and the dynamical origin of the nonequilibrium transition while the entropy production rate as the nonequilibrium thermodynamic cost and thermodynamic origin of the nonequilibrium transition. Average flux, entropy production, nonequilibrium free energy, and time irreversibility quantified by the difference in cross-correlation functions forward and backward in time can serve as early warning signals for critical transitions much earlier than other conventional predictors. We utilize a classical shallow lake model as an exemplar for our early warning prediction. Our proposed method is general and can be readily applied to assess the resilience of many other ecological systems. The early warning signals proposed here can potentially predict critical transitions earlier than established methods and perhaps even sufficiently early to avert catastrophic shifts. 

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2. Critical transitions in the hydrological system: early-warning signals and network analysis

   https://doi.org/10.5194/hess-26-1845-2022  

   Yang, Xueli; Wang, Zhi-Hua; Wang, Chenghao (January 2022, Hydrology and Earth System Sciences)

   Abstract. One critical challenge of studying Earth's hydroclimate system, in the face of global environmental changes, is to predict whether the system approaches a critical threshold. Here, we identified the critical transitions of hydrological processes, including precipitation and potential evapotranspiration, by analyzing their early-warning signals and system-based network structures. The statistical early-warning signals are manifest in increasing trends of autocorrelation and variance in the hydrologic system ranging from regional to global scales, prior to climate shifts in the 1970s and 1990s, in agreement with observations. We further extended the conventional statistics-based measures of early-warning signals to system-based network analysis in urban areas across the contiguous United States. The topology of an urban precipitation network features hub-periphery (clustering) and modular organization, with strong intra-regional connectivity and inter-regional gateways (teleconnection). We found that several network parameters (mean correlation coefficient, density, and clustering coefficient) gradually increased prior to the critical transition in the 1990s, signifying the enhanced synchronization among urban precipitation patterns. These topological parameters can not only serve as novel system-based early-warning signals for critical transitions in hydrological processes but also shed new light on structure–dynamic interactions in the complex hydrological system. 

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3. Early warning signals of changing resilience in the biogeochemistry and biology of a northern hardwood forest

   https://doi.org/10.1088/1748-9326/acf3fe  

   Contosta, Alexandra R.; Battles, John J.; Campbell, John L.; Driscoll, Charles T.; Garlick, Sarah R.; Holmes, Richard T.; Likens, Gene E.; Rodenhouse, Nicholas L.; Rogers, Shannon H.; Templer, Pamela H.; et al (September 2023, Environmental Research Letters)

   Abstract Resilience is the ability of ecosystems to maintain function while experiencing perturbation. Globally, forests are experiencing disturbances of unprecedented quantity, type, and magnitude that may diminish resilience. Early warning signals are statistical properties of data whose increase over time may provide insights into decreasing resilience, but there have been few applications to forests. We quantified four early warning signals (standard deviation, lag-1 autocorrelation, skewness, and kurtosis) across detrended time series of multiple ecosystem state variables at the Hubbard Brook Experimental Forest, New Hampshire, USA and analyzed how these signals have changed over time. Variables were collected over periods from 25 to 55 years from both experimentally manipulated and reference areas and were aggregated to annual timesteps for analysis. Long-term (>50 year) increases in early warning signals of stream calcium, a key biogeochemical variable at the site, illustrated declining resilience after decades of acid deposition, but only in watersheds that had previously been harvested. Trends in early warning signals of stream nitrate, a critical nutrient and water pollutant, likewise exhibited symptoms of declining resilience but in all watersheds. Temporal trends in early warning signals of some of groups of trees, insects, and birds also indicated changing resilience, but this pattern differed among, and even within, groups. Overall, ∼60% of early warning signals analyzed indicated decreasing resilience. Most of these signals occurred in skewness and kurtosis, suggesting ‘flickering’ behavior that aligns with emerging evidence of the forest transitioning into an oligotrophic condition. The other ∼40% of early warning signals indicated increasing or unchanging resilience. Interpretation of early warning signals in the context of system specific knowledge is therefore essential. They can be useful indicators for some key ecosystem variables; however, uncertainties in other variables highlight the need for further development of these tools in well-studied, long-term research sites. 

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4. Anticipating regime shifts by mixing early warning signals from different nodes

   https://doi.org/10.1038/s41467-024-45476-9  

   Masuda, Naoki; Aihara, Kazuyuki; MacLaren, Neil G (December 2024, Nature Communications)

   Abstract 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. 

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5. Variations in stability revealed by temporal asymmetries in contraction of phase space flow

   https://doi.org/10.1038/s41598-021-84865-8  

   Williams, Zachary C.; McNamara, Dylan E. (March 2021, Scientific Reports)

   Abstract Empirical diagnosis of stability has received considerable attention, often focused on variance metrics for early warning signals of abrupt system change or delicate techniques measuring Lyapunov spectra. The theoretical foundation for the popular early warning signal approach has been limited to relatively simple system changes such as bifurcating fixed points where variability is extrinsic to the steady state. We offer a novel measurement of stability that applies in wide ranging systems that contain variability in both internal steady state dynamics and in response to external perturbations. Utilizing connections between stability, dissipation, and phase space flow, we show that stability correlates with temporal asymmetry in a measure of phase space flow contraction. Our method is general as it reveals stability variation independent of assumptions about the nature of system variability or attractor shape. After showing efficacy in a variety of model systems, we apply our technique for measuring stability to monthly returns of the S&P 500 index in the time periods surrounding the global stock market crash of October 1987. Market stability is shown to be higher in the several years preceding and subsequent to the 1987 market crash. We anticipate our technique will have wide applicability in climate, ecological, financial, and social systems where stability is a pressing concern. 

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