Rate-induced tipping points are all about relative rates: is the basin of attraction for an equilibrium state changing slowly enough that the system can track the change and remain in the basin? If so, the system can remain at this equilibrium, and if not we observe rate-induced tipping (r-tipping). Thus, in concept, we can understand r-tipping risk by comparing the rate of environmental change (which affects the basin) relative to the rate of ecological dynamics (which affects the ability to track the environment). The issue, however, is that neither of these – environmental change and ecological response – is comprised of a single process with a single rate. As the environment changes, basins of attraction move in phase space and change in shape. Birth, death, and interaction rates all influence a system’s ability to track environmental change. So, which rates should we be comparing to best understand r-tipping risk?
Results/Conclusions Using classic two-species models for ecological systems, we compare standard measures of environmental change and species response to measures that can be computed from a quasi-potential. The quasi-potential is a generalization of a potential function that can be constructed for non-gradient systems (i.e. most multispecies ecological models) and visualizes ecological dynamics as a landscape with peaks (unstable equilibria) and valleys (stable equilibria surrounded by their basin of attraction). Standard measures of environmental change include the rate at which the position of an equilibrium point moves in phase space; quasi-potential-derived measures include the rates at which the basin of attraction’s edges move. Standard measures of ecological response include the demographic rates themselves and the local rate of return to equilibrium (dominant eigenvalue); quasi-potential-derived measures include non-local return rates given by the depth and steepness of the basin. We conclude that r-tipping is best understood by comparing rates that are derived from the quasi-potential, as these provide a more complete mapping of dynamics onto the basin of attraction that is undergoing environmental change.
