  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 well studied 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 four 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.
