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Rate-Induced Tipping: Beyond Classical Bifurcations in Ecology

Banff International Research Station for Mathematical Innovation and Discovery

Many systems from the natural world have to adapt to continuously changing external conditions. Some systems have dangerous levels of external conditions, defined by catastrophic bifurcations, above which they undergo a critical transition (B-tipping) to a different state; e.g. forest-desert transitions. Other systems can be very sensitive to how fast the external conditions change and have dangerous rates - they undergo an unexpected critical transition (R-tipping) if the external conditions change slowly but faster than some critical rate; e.g. critical rates of climatic changes. R-tipping is a genuine non-autonomous instability which captures ``failure to adapt to changing environments" [1,2]. However, it cannot be described by classical bifurcations and requires an alternative mathematical framework. In the first part of the talk, we demonstrate the nonlinear phenomenon of R-tipping in a simple ecosystem model where environmental changes are represented by time-varying parameters [Scheffer et al. Ecosystems 11 2008]. We define R-tipping as a critical transition from the herbivore-dominating equilibrium to the plant-only equilibrium, triggered by a smooth parameter shift [1]. We then show how to complement classical bifurcation diagrams with information on nonautonomous R-tipping that cannot be captured by the classical bifurcation analysis. We produce tipping diagrams in the plane of the magnitude and `rateâ of a parameter shift to uncover nontrivial R-tipping phenomena. In the second part of the talk, we develop a general framework for R-tipping based on thresholds, edge states and a suitable compactification of the nonautonomous system. This allows us to define R-tipping in terms of connecting heteroclinic orbits in the compactified system, which greatly simplifies the analysis. We explain the key concept of threshold instability and give rigorous testable criteria for R-tipping in arbitrary dimensions. References: [1] PE O'Keeffe and S Wieczorek,'Tipping phenomena and points of no return in ecosystems: beyond classical bifurcations', arXiv preprint arXiv:1902.01796 [2] A Vanselow, S Wieczorek, U Feudel, 'When very slow is too fast: Collapse of a predator-prey system' Journal of Theoretical Biology (2019)

Mathematics; Partial differential equations; Biology and other natural sciences; Mathematical biology

video/mp4

Author affiliation: University College Cork

2020-09-09

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/75928

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/