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

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.

Anticipating the occurrence and type of critical transitions

Critical transition can occur in many real-world systems. The ability to forecast the occurrence of transition is of major interest in a range of contexts. Various early warning signals (EWSs) have been developed to anticipate the coming critical transition or distinguish types of transition. However, no effective method allows to establish practical threshold indicating the condition when the critical transition is most likely to occur. Here, we introduce a powerful EWS, named dynamical eigenvalue (DEV), that is rooted in bifurcation theory of dynamical systems to estimate the dominant eigenvalue of the system. Theoretically, the absolute value of DEV approaches 1 when the system approaches bifurcation, while its position in the complex plane indicates the type of transition. We demonstrate the efficacy of the DEV approach in model systems with known bifurcation types and also test the DEV approach on various critical transitions in real-world systems.

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.

A novel nonlinear analysis of blood flow dynamics applied to the human lung

The spatial/temporal dynamics of blood flow in the human lung can be measured noninvasively with magnetic resonance imaging (MRI) using arterial spin labeling (ASL). We report a novel data analysis method using nonlinear prediction to identify dynamic interactions between blood flow units (image voxels), potentially providing a probe of underlying vascular control mechanisms. The approach first estimates the linear relationship (predictability) of one voxel time series with another using correlation analysis, and after removing the linear component estimates the nonlinear relationship with a numerical mutual information approach. Dimensionless global metrics for linear prediction (F_L) and nonlinear prediction (F_NL) represent the average amplitude of fluctuations in one voxel estimated by another voxel, as a percentage of the global average voxel flow. A proof-of-principle test of this approach analyzed experimental data from a study of high-altitude pulmonary edema (HAPE), providing two groups exhibiting known differences in vascular reactivity. Subjects were mountaineers divided into HAPE-susceptible (S, n=4) and HAPE-resistant (R, n=5) groups based on prior history at high altitude. Dynamic ASL measurements in the lung in normoxia (N, FIO₂=0.21) and hypoxia (H, FIO₂=0.13±0.01) were compared. The nonlinear prediction metric F_NL decreased with hypoxia (7.4±1.3(N) vs. 6.3±0.7(H), P=0.03) and was significantly different between groups (7.4±1.2 (R) vs. 6.2±14.1 (S), P=0.03). This proof-of-principle test demonstrates that this nonlinear analysis approach applied to ASL data is sensitive to physiological effects even in small subject cohorts, and potentially can be used in a wide range of studies in health and disease in the lung and other organs.

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

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.

Detecting and distinguishing tipping points using spectral early warning signals

Theory and observation tell us that many complex systems exhibit tipping points—thresholds involving an abrupt and irreversible transition to a contrasting dynamical regime. Such events are commonly referred to as critical transitions. Current research seeks to develop early warning signals (EWS) of critical transitions that could help prevent undesirable events such as ecosystem collapse. However, conventional EWS do not indicate the type of transition, since they are based on the generic phenomena of critical slowing down. For instance, they may fail to distinguish the onset of oscillations (e.g. Hopf bifurcation) from a transition to a distant attractor (e.g. Fold bifurcation). Moreover, conventional EWS are less reliable in systems with density-dependent noise. Other EWS based on the power spectrum (spectral EWS) have been proposed, but they rely upon spectral reddening, which does not occur prior to critical transitions with an oscillatory component. Here, we use Ornstein–Uhlenbeck theory to derive analytic approximations for EWS prior to each type of local bifurcation, thereby creating new spectral EWS that provide greater sensitivity to transition proximity; higher robustness to density-dependent noise and bifurcation type; and clues to the type of approaching transition. We demonstrate the advantage of applying these spectral EWS in concert with conventional EWS using a population model, and show that they provide a characteristic signal prior to two different Hopf bifurcations in data from a predator–prey chemostat experiment. The ability to better infer and differentiate the nature of upcoming transitions in complex systems will help humanity manage critical transitions in the Anthropocene Era.

Insights into coral reef benthic dynamics from nonlinear spatial forecasting

Nonlinear time-series forecasting, or empirical dynamic modelling, has been used extensively in the past two decades as a tool for distinguishing between random temporal behaviour and nonlinear deterministic dynamics. Previous authors have extended nonlinear time-series forecasting to continuous spatial data. Here, we adjust spatial forecasting to handle discrete data and apply the technique to explore the ubiquity of nonlinear determinism in irregular spatial configurations of coral and algal taxa from Palmyra Atoll, a relatively pristine reef in the central Pacific Ocean. We find that the spatial distributions of coral and algal taxa show signs of nonlinear determinism in some locations and that these signals can change through time. We introduce the hypothesis that nonlinear spatial determinism may be a signal of systems in intermediate developmental (i.e. successional) stages, with spatial randomness characterizing early (i.e. recruitment dominated) and late-successional (i.e. ‘climax’ or attractor) phases. Common state-based metrics that sum community response to environmental forcing lack resolution to detect dynamics of (potential) recovery phases; incorporating signal of spatial patterning among sessile taxa holds unique promise to elucidate dynamical characters of complex ecological systems, thereby enhancing study and response efforts.

Nonlinear population dynamics are ubiquitous in animals

Nonlinear dynamics, where a change in the input is not proportional to a change in the output, are often found throughout nature, for example in biochemical kinetics. Because of the complex suite of interacting abiotic and biotic variables present in ecosystems, animal population dynamics are often thought to be driven in a nonlinear, state-dependent fashion. However, so far these have only been identified in model organisms and some natural systems. Here we show that nonlinear population dynamics are ubiquitous in nature. We use nonlinear forecasting to analyse 747 datasets of 228 species to find that insect population trends were highly nonlinear (74%), followed by mammals (58%), bony fish (49%) and birds (35%). This indicates that linear, equilibrium-based model assumptions may fail at predicting population dynamics across a wide range of animal taxa. We show that faster-reproducing animals are more likely to have nonlinear and high-dimensional dynamics, supporting past ecological theory. Lastly, only a third of time series were predictable beyond two years; therefore, the ability to predict animal population trends using these methods may be limited. Our results suggest that the complex dynamics necessary to cause regime shifts and other transitions may be inherent in a wide variety of animals.

Causality analysis and prediction of 2-methylisoborneol production in a reservoir using empirical dynamic modeling

2-Methylisobornel (MIB) is one of the most widespread and problematic biogenic compounds causing taste-and-odor problems in freshwater. To investigate the causes of MIB production and develop models to predict the MIB concentration, we have applied empirical dynamic modeling (EDM), a nonlinear approach based on Chaos theory, to the long-term water quality dataset of Kamafusa Reservoir in Japan. The study revealed the dynamic nature of MIB production in the reservoir, and determined causal variables for MIB production, including water temperature, pH, transparency, light intensity, and Green Phormidium. Moreover, EDM established that the system is three-dimensional, and the approach found elevated nonlinearity (from 1.5 to 3) across the whole study period (1996-2015). By taking only one or two candidate predictors with varying time lags, multivariate models for predicting MIB production (best model: r = 0.83, p < 0.001, root mean squared error = 3.1 ng/L) were successfully established. The modeling approach used in this study is a powerful tool for causality identification and odor prediction, thus making important contributions to reservoir management.

Predicting tipping points of dynamical systems during a period-doubling route to chaos

Classical indicators of tipping points have limitations when they are applied to an ecological and a biological model. For example, they cannot correctly predict tipping points during a period-doubling route to chaos. To counter this limitation, we here try to modify four well-known indicators of tipping points, namely the autocorrelation function, the variance, the kurtosis, and the skewness. In particular, our proposed modification has two steps. First, the dynamic of the considered system is estimated using its time-series. Second, the original time-series is divided into some sub-time-series. In other words, we separate the time-series into different period-components. Then, the four different tipping point indicators are applied to the extracted sub-time-series. We test our approach on an ecological model that describes the logistic growth of populations and on an attention-deficit-disorder model. Both models show different tipping points in a period-doubling route to chaos, and our approach yields excellent results in predicting these tipping points.

Survival and weak chaos

Survival analysis in biology and reliability theory in engineering concern the dynamical functioning of bio/electro/mechanical units. Here we incorporate effects of chaotic dynamics into the classical theory. Dynamical systems theory now distinguishes strong and weak chaos. Strong chaos generates Type II survivorship curves entirely as a result of the internal operation of the system, without any age-independent, external, random forces of mortality. Weak chaos exhibits (a) intermittency and (b) Type III survivorship, defined as a decreasing per capita mortality rate: engineering explicitly defines this pattern of decreasing hazard as 'infant mortality'. Weak chaos generates two phenomena from the normal functioning of the same system. First, infant mortality-sensu engineering-without any external explanatory factors, such as manufacturing defects, which is followed by increased average longevity of survivors. Second, sudden failure of units during their normal period of operation, before the onset of age-dependent mortality arising from senescence. The relevance of these phenomena encompasses, for example: no-fault-found failure of electronic devices; high rates of human early spontaneous miscarriage/abortion; runaway pacemakers; sudden cardiac death in young adults; bipolar disorder; and epilepsy.