Tipping points have been shown to be ubiquitous, both in models and empirically in a range of physical and biological systems. The question of how tipping points cascade through systems has been less explored and is an important one. A study of noise-induced tipping, in particular, could provide key insights into tipping cascades. Here, we consider a specific example of a simple model system that could have cascading tipping points. This model consists of two interacting populations with underlying Allee effects and stochastic dynamics, in separate patches connected by dispersal, which can generate bistability. From an ecological standpoint, we look for rescue effects whereby one population can prevent the collapse of a second population. As a way to investigate the stochastic dynamics, we use an individual-based modeling approach rooted in chemical reaction network theory. Then, using continuous-time Markov chains and the theory of first passage times, we essentially approximate, or emulate, the original high-dimensional model by a Markov chain with just three states, where each state corresponds to a combination of population thresholds. Analysis of this reduced model shows when the system is likely to recover, as well as when tipping cascades through the whole system.

临界点已被证明是无处不在的，无论是在模型中还是在一系列物理和生物系统的经验中。临界点如何通过系统级联的问题很少被探索，但却是一个重要的问题。特别是对噪声引起的倾翻的研究可以提供对倾翻级联的关键见解。在这里，我们考虑一个可能具有级联临界点的简单模型系统的具体示例。该模型由两个相互作用的种群组成，它们具有潜在的 Allee 效应和随机动力学，位于通过分散连接的单独斑块中，可以产生双稳态。从生态的角度来看，我们寻找拯救效应，即一个种群可以防止第二个种群的崩溃。作为研究随机动力学的一种方法，我们使用植根于化学反应网络理论的基于个体的建模方法。然后，使用连续时间马尔可夫链和首次通过时间理论，我们基本上通过只有三个状态的马尔可夫链来近似或模拟原始高维模型，其中每个状态对应于人口阈值的组合。对这个简化模型的分析显示了系统何时可能恢复，以及在整个系统中发生级联反应。