Early warning signs for tipping points in systems with non-Gaussian $$\alpha$$-stable noise

Abstract

Abstract Forecasting rapid, non-linear change or so-called tipping points is a major concern in ecology and environmental science. Statistical early warning signs, based on the theory of stochastic dynamical systems, are now regularly applied to observational data streams. However, the reliability of these early warning signs relies on a number of key mathematical assumptions, most notably the presence of Gaussian noise, while many ecological systems exhibit non-Gaussianity. We here show that for systems driven by non-Gaussian, $$\alpha$$ -stable noise, the classical early warning signs of rising variance and autocorrelation are not supported by mathematical theory, and their use poses the danger of spurious, false-positive results. To address this, we provide a generalized approach by introducing the scaling factor $$\gamma _X$$ as an alternative early warning sign. We show that in the case of the linear Ornstein-Uhlenbeck process, there exists a direct inverse relationship between $$\gamma _{X}$$ and the bifurcation parameter, telling us that $$\gamma _{X}$$ will increase as we approach the bifurcation. Our numerical simulations confirm theoretical results and show that our findings generalize well to non-linear, non-equilibrium systems often employed in ecological systems. We thus provide a generalized, robust, and broadly applicable statistical early warning sign for systems driven by Gaussian and non-Gaussian $$\alpha$$ -stable noise.