速率诱导崩溃在具有阿利效应的可解模型中
Rate-induced tipping in a solvable model with the Allee effect
AI总结：
本文提出一个可解模型研究速率诱导崩溃现象，包含阿利效应和灭绝阈值，通过积分不等式分析稳定性，并讨论其在渔业应用中的意义。
摘要：
我们提出了一个新颖的精确可解常微分方程模型，用于速率诱导崩溃：动态系统中的一种动态现象，其中时间依赖参数触发系统稳定性转换。我们的模型包含阿利效应，该效应引入了鞍点，并允许显式解以及时间依赖阿利参数的灭绝阈值。更具体地说，我们推导了一个积分不等式，作为速率诱导崩溃发生的一个必要条件。所提出模型的一个显著特点是能够处理种群灭绝，使得解在有限时间内完全消失。我们提出了一种无条件稳定的立方方法，适用于该模型，并讨论了其优于经典向前欧拉方法的优越性。我们还讨论了渔业应用，其中内陆渔业从现代时代到现在在日本起伏变化。所提出的模型为研究速率诱导崩溃现象提供了一个可操作的数学工具。
英文摘要：
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.
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