Bifurcation or state tipping: assessing transition type in a model trophic cascade

Ecosystems can experience sudden regime shifts due to a variety of mechanisms. Two of the ways a system can cross a tipping point include when a perturbation to the system state is large enough to push the system beyond the basin of attraction of one stable state and into that of another (state tipping), and alternately, when slow changes to some underlying parameter lead to a fold bifurcation that annihilates one of the stable states. The first mechanism does not generate the phenomenon of critical slowing down (CSD), whereas the latter does generate CSD, which has been postulated as a way to detect early warning signs ahead of a sudden shift. Yet distinguishing between the two mechanisms (s-tipping and b-tipping) is not always as straightforward as it might seem. The distinction between "state" and "parameter" that may seem self-evident in mathematical equations depends fundamentally on ecological details in model formulation. This distinction is particularly relevant when considering high-dimensional models involving trophic webs of interacting species, which can only be reduced to a one-dimensional model of a tipping point under appropriate consideration of both the mathematics and biology involved. Here we illustrate that process of dimension reduction and distinguishing between s- and b-tipping for a highly influential trophic cascade model used to demonstrate tipping points and test CSD predictions in silico, and later, in a natural lake ecosystem. Our analysis resolves a previously unclear issue as to the nature of the tipping point involved.