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This paper introduces an analytically tractable predator-prey model with a strong Allee effect to address limitations in existing rate-induced tipping (R-tipping) models, specifically the unecological resurgence phenomenon. By applying geometric singular perturbation theory to a ramped inverse prey carrying capacity, the authors demonstrate how bistability and a folded critical manifold with a folded saddle singularity create a genuine attractor for extinction. The model yields explicit expressions for the critical rate separating tracking from tipping dynamics, establishing it as a minimal framework for R-tipping to extinction.
A minimal predator-prey model with an Allee effect precisely predicts extinction tipping, not just temporary collapse, by generating a folded saddle canard.
Rate-induced tipping ("R-tipping") in ecological modelling is characterised by too-rapid change of an environmental parameter causing collapse of a population without any bifurcation being crossed ("B-tipping"). A well-known example of a slow-fast predator-prey model in which R-tipping is observed [Vanselow, Wieczorek, Feudel, Journal of Theoretical Biology, 479, 64-72 (2019)] suffers from unecological"resurgence", whereby populations driven to functional extinction recover towards stable coexistence. We propose an analytically tractable model, incorporating a strong Allee effect into the prey dynamics in a slow-fast Lotka-Volterra-type predator-prey system, which induces bistability and thus renders the extinction state a genuine attractor. Applying geometric singular perturbation theory (GSPT), we describe the dynamics of the extended model that is obtained by"ramping"of the inverse prey carrying capacity. We show the presence of a parabolic-shaped, folded critical manifold which admits a folded saddle singularity, the strong canard of which separates solutions that track the moving coexistence state from those that tip to extinction. The analytical simplicity of our model allows us to derive explicit expressions for the critical rate that separates"tracking"from"tipping"dynamics. Finally, we argue that our system represents a minimal model for rate-induced tipping to extinction, in that it incorporates three essential ingredients: bistability, a folded critical manifold, and a folded-saddle-type canard.