Rate-induced tipping in a solvable model with the Allee effect

We present a novel exactly solvable ordinary differential equation model for rate-induced tipping: a dynamic phenomenon of dynamical systems where a time-dependent parameter triggers the transition of stability of a system. Our model contains an Allee effect that induces a saddle point and admits an explicit solution along with the extinction threshold of a time-dependent Allee parameter. More specifically, we derive an integral inequality that serves as a necessary condition for the occurrence of rate-induced tipping. A remarkable point in the proposed model is that it can handle population extinction such that the solution completely vanishes in a finite amount of time. An unconditionally stable cubature method suitable for our model is proposed, and its superiority over the classical forward Euler method is discussed. As a proof of concept, the model reproduces the rise and fall of the membership of inland fisheries cooperatives in Japan from 1963 to 2023. The proposed model serves as a tractable tool for studying R-tipping phenomena.