Anticipating the occurrence and type of critical transitions

Abstract

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.

EWSmethods: an R package to forecast tipping points at the community level using early warning signals, resilience measures, and machine learning models

“…All measures other than Fisher information increase prior to the tipping point with the multivariate Jacobian index displaying trends earliest. Similarly, the univariate Jacobian index also reaches 1.0 at the tipping point, matching its conceptual behaviour in Grziwotz et al (2023). Fisher information does not, however, qualitatively decrease prior to the tipping point though this may stem from an inappropriate window size or size‐of‐state.…”

“…The functions uniJI and multiJI exploit S‐map reconstruction to estimate the system's Jacobian under the empirical dynamic modelling (EDM) framework (Ushio et al 2018, Medeiros et al 2022, Grziwotz et al 2023). EDM builds upon Takens theorem (Takens 1981), which suggests that if all interacting elements of a system can be simultaneously measured (all species, environmental variables, interaction and response strengths etc), then the evolution of the state of the system can be known.…”

“…The primary difference between the univariate and multivariate form of local Lyapunov stability/Jacobian index provided by EWSmethods involves the target of embedding. The univariate index time embeds the focal time series against itself (Grziwotz et al 2023) whereas the multivariate embeds across all species/time series (Ushio et al 2018, Medeiros et al 2022). The multivariate index is therefore sensitive to the choice of time series included in the reconstruction (Ushio et al 2018), and the number of time series used must equal E. This limits the minimum length of uninterrupted time series required by multiJI at E (equal to number of measured species) + 1.…”

“…The multivariate index is therefore sensitive to the choice of time series included in the reconstruction (Ushio et al 2018), and the number of time series used must equal E. This limits the minimum length of uninterrupted time series required by multiJI at E (equal to number of measured species) + 1. The univariate index is consequently suggested to be superior if the system is not well measured (Grziwotz et al 2023).…”

“…Thirdly, if multiple measurements have been made of a single system – such as when monitoring multiple species in the same community – multivariate EWSs can be estimated using either rolling or expanding window approaches. And finally, EWS distinct resilience metrics can also be estimated for univariate and multivariate data following the work of Ushio et al (2018) and Grziwotz et al (2023). EWSmethods therefore represents a compilation of new and existing tools to support this expanding field in an easy to use and interpret form.…”