Eigenvalues of the covariance matrix as early warning signals for critical transitions in ecological systems

Shiyang Chen,John M Drake,Bogdan I Epureanu,Eamon B O’Dea

doi:10.1038/s41598-019-38961-5

Shiyang Chen, John M Drake + Show 2 more

Open Access

https://doi.org/10.1038/s41598-019-38961-5

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Abstract

Many ecological systems are subject critical transitions, which are abrupt changes to contrasting states triggered by small changes in some key component of the system. Temporal early warning signals such as the variance of a time series, and spatial early warning signals such as the spatial correlation in a snapshot of the system’s state, have been proposed to forecast critical transitions. However, temporal early warning signals do not take the spatial pattern into account, and past spatial indicators only examine one snapshot at a time. In this study, we propose the use of eigenvalues of the covariance matrix of multiple time series as early warning signals. We first show theoretically why these indicators may increase as the system moves closer to the critical transition. Then, we apply the method to simulated data from several spatial ecological models to demonstrate the method’s applicability. This method has the advantage that it takes into account only the fluctuations of the system about its equilibrium, thus eliminating the effects of any change in equilibrium values. The eigenvector associated with the largest eigenvalue of the covariance matrix is helpful for identifying the regions that are most vulnerable to the critical transition.

Highlights

Ecological systems can be found in highly contrasting states
Let z(t) = x(t)−μ denote a vector of deviations from the equilibrium, where μ denotes the equilibrium of the state variables in the absence of noise
Our study suggests that an increase in the largest eigenvalue of the covariance matrix of a multivariate time series and the percentage it accounts for of the total variation can serve as early warning signals for critical transitions in multivariate systems

Summary

Introduction

Ecological systems can be found in highly contrasting states. For example, shallow lakes may be either clear or turbid from an abundance of cyanobacteria[1], coral reefs may be highly diverse or dominated by macroalgae[2], areas of land can be either wooded, grassy and open[3], or altogether barren[4], and time series of climatic and biological indicators of oceans can contain abrupt jumps[5]. In models that include stochastic effects which interupt deterministic trajectories, other indicators have been proposed that estimate this return rate indirectly, such as the variance and autocorrelation of fluctuations in a time series of the observations of the state of a system Such indicators of critical slowing down are not mathematical constructions; indicators calculated from observations of natural systems have been found to prefigure critical transitions as predicted by models[15,16,17,18]. Spatial early warning signals, such as the spatial variance, spatial skewness and spatial correlation have been proposed as indicators for critical transitions of spatially extended systems By referring to these measurements as “spatial” we mean that they are calculated from observations at different locations at a given time, a set of observations which we refer to as a snapshot. For instance, to identify the temporal pattern that is associated with the critical transition using only a single snapshot

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