Detecting nonlinearity in natural population time series with classical and proxy indicators

Ecosystem dynamics are inherently nonlinear due to complex inter- and intraspecific interactions, which drive the complex population dynamics observed in nature. The nonlinearity in empirical population time series manifests in diverse patterns and can be detected using a variety of methods. In this study, multiple nonlinearity detection methods were systematically applied to empirical population time series from two major databases—the Living Planet Database (LPD) and the Global Population Dynamics Database (GPDD). The methodological framework encompassed three primary approaches: (1) classical statistical tests and forecasting methods that directly analyze time series nonlinearities; (2) machine learning methods that classify time series based on time series morphologies; and (3) the calculation of Early Warning Signals (EWS) as proxies for nonlinearity. First, the classical methods effectively identified nonlinear trends, periodicity, and serial dependence, despite the low concordance (Fleiss’ Kappa: 0.156 and 0.225) among the six test methods. Subsequent application of four machine learning models to a subset of LPD series demonstrated their dual capability to not only detect nonlinearity but also classify series into distinct morphological categories (e.g., monotonic, U-shaped, multimodal). Lastly, extending the nonlinear analysis to higher-order phenomena, EWS indicators were computed for GPDD series, revealing tipping points in 95 series and underscoring the common occurrence of regime shifts in ecosystems. Our findings collectively affirm that complex nonlinear phenomena are pervasive in population time series. This study highlights the complementary value of classical statistical tests, machine learning classification, and EWS indicators in characterizing ecological nonlinearity. These results underscore the utility of dynamical systems theory and multivariate methodological approaches for detecting and interpreting nonlinear patterns and critical transitions in natural population dynamics. • Multiple detection methods confirm widespread nonlinearity in population data. • Machine learning successfully classifies time series by their morphology. • Early warning signals detected critical transitions and tipping points. • Classical tests, machine learning, and EWS indicators together reveal nonlinearity.