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 `rate’ 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).