Rate-induced tipping in a solvable model with the Allee effect
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866917324029689856 |
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| author | Yoshioka, Hidekazu |
| author_facet | Yoshioka, Hidekazu |
| contents | 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. We also discuss a fisheries application where inland fisheries rose and fell from modern times to the present in Japan. The proposed model serves as a tractable mathematical tool for studying rate-induced tipping phenomena. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_20128 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Rate-induced tipping in a solvable model with the Allee effect Yoshioka, Hidekazu Dynamical Systems Numerical Analysis 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. We also discuss a fisheries application where inland fisheries rose and fell from modern times to the present in Japan. The proposed model serves as a tractable mathematical tool for studying rate-induced tipping phenomena. |
| title | Rate-induced tipping in a solvable model with the Allee effect |
| topic | Dynamical Systems Numerical Analysis |
| url | https://arxiv.org/abs/2601.20128 |