Early warning of critical transitions: distinguishing tipping points from Turing destabilizations

TL;DR

This work tackles the challenge of distinguishing tipping points from pattern-forming bifurcations in spatial systems by inferring a dispersion relation from pre-transition spatio-temporal data. The authors fit a linear reaction-diffusion model to fluctuations to obtain dispersion relations $\lambda(k)$ and identify the dominant unstable mode $k_*$; the sign of $k_*$ determines whether a transition is spatially homogeneous ($k_*=0$) or heterogeneous ($k_*>0$). Using synthetic data from an extended Klausmeier model, they show that the method can correctly classify tipping versus Turing bifurcations under varying noise and data conditions, and outline data requirements favoring longer observation times and adequate spatial resolution. Overall, the approach provides a spatial early warning framework that informs both when and what type of critical transition may occur, with broad relevance for climate and ecological subsystems.

Abstract

Current early warning signs for tipping points often fail to distinguish between catastrophic shifts and less dramatic state changes, such as spatial pattern formation. This paper introduces a novel method that addresses this limitation by providing more information about the type of bifurcation being approached starting from a spatially homogeneous system state. This method relies on estimates of the dispersion relation from noisy spatio-temporal data, which reveals whether the system is approaching a spatially homogeneous (tipping) or spatially heterogeneous (Turing patterning) bifurcation. Using a modified Klausmeier model, we validate this method on synthetic data, exploring its performance under varying conditions including noise properties and distance to bifurcation. We also determine the data requirements for optimal performance. Our results indicate the promise of a new spatial early warning system built on this method to improve predictions of future transitions in many climate subsystems and ecosystems, which is critical for effective conservation and management in a rapidly changing world.

Figures (13)

• Figure 1: Illustration of two bifurcation diagrams with distinct destabilizing bifurcations, which are identified using the method introduced in this paper based on the estimation of dispersion relations. The lines indicate the spatially homogeneous steady states, with solid lines indicating stable and dashed lines unstable states. The left shows a situation in which the orange state destabilizes via a saddle-node 'Tipping' bifurcation; the right shows a situation in which the it destabilizes via a spatial pattern forming Turing bifurcation. The insets show dispersion relations, relating spatial Fourier eigenmodes, characterized by their wavenumber $k$, to eigenvalues $\lambda(k)$, at various levels with varying closeness to bifurcation. It can be seen that at the bifurcation - and well before it - the dispersion relations are qualitatively different: saddle-node bifurcations are signaled by a peak in the dispersion relation at $k = 0$, whereas Turing bifurcations have peaks for $k \neq 0$. The method introduced in this paper makes use of this fact by providing estimates of these dispersion relations, and thus of the most unstable spatial eigenmodes $k_*$, to determine whether a spatially homogeneous (e.g., left - saddle-node bifurcation) or spatially heterogeneous (e.g., right - Turing bifurcation) destabilization is imminent. Figures are made with the modified Klausmeier model in Eq \ref{['eq:Extended_klausmeier_2']} with parameter values $m = 0.5$, $h = 0.1$, $\delta = 0.5$ (left) or $\delta = 0.01$ (right) and varying $p$ along horizontal axis; system state is represented by the variable $v$.
• Figure 2: Schematic for the proposed method. Spatio-temporal data on fluctuations before a transition is fitted to a linear partial differential equation, whose dispersion relation is computed, and then used to track the most unstable spatial eigenmode $k_*$ and associated eigenvalue $\lambda_* = \lambda(k_*)$.
• Figure 3: Estimated dispersion relations for different noise levels and correlation lengths, with $\delta = 0.5$ (Turing case). The true dispersion relation is shown in black, the ensemble average of estimated dispersion relations in blue, with its 5th and 95th percentiles in red and green dotted lines. The blue dots show the estimated ($k_*$, $\lambda_*$) values from each of the $100$ data sets. The pink areas indicate the scaled marginal densities of these pairs. An orange covariance ellipse, centered at the mean, represents the one-standard-deviation, $\sigma(k_*,\lambda_*)$, spread along the principal directions. The true dominant pair ($k_*^{\text{true}}$, $\lambda_*^{\text{true}}$) is shown with a star. Blue shaded insets report on statistics of error of the method in determining this dominant pair (see main text). The data has been generated using the parameter settings defined in Experiment 1 from table \ref{['tab:experiments_final']}.
• Figure 4: Estimated dispersion relations for different noise levels and correlation lengths, with $\delta=0.5$ (saddle-node case). Data is generated using Experiment 1 settings in Table \ref{['tab:experiments_final']}. See the caption of Figure \ref{['Fig:Noise_level_correlation_Turing']} for the details of the depicted lines, areas, circles and insets
• Figure 5: Estimated dispersion relations for different observation times and temporal sampling rates for $\delta = 0.01$ (Turing case). Data is generated using Experiment 2 settings in Table \ref{['tab:experiments_final']}. See the caption of Figure \ref{['Fig:Noise_level_correlation_Turing']} for the details of the depicted lines, areas, circles and insets.