Background/Question/Methods Cyclic systems can exhibit highly unexpected behaviour when subject to external forcing. Many predator-prey systems are inherently cyclic, and subject to climatic forcing that can also be dominated by certain frequencies. This coupling between two cyclic systems could mean that the observed predator-prey dynamics may be more dependent on the characteristics of the climatic time series than is generally suspected. In particular, we ask if the predicted changes to the colour and frequency of environmental noise could put cyclic predator-prey systems at risk. We investigate this question by studying two paradigmatic predator-prey models, the Rosenzweig-MacArthur and Leslie-Gower-May models, both with an Allee effect, under climatic forcing modeled via changes in prey productivity (prey growth and carrying capacity).
Results/Conclusions Our analysis of these models uncovers a counterintuitive behaviour, which we call phase tipping or P-tipping, where tipping to extinction does occur under climate variability, but only from certain phases of the predator-prey cycle, and only when there is a sudden drop in productivity. This tipping occurs even if the predator-prey cycle exists and is stable for all values of the climate parameters. Intuitively, P-tipping arises because a fixed drop in prey resources has distinctively different effects when applied during the phases of the oscillations with the fastest growth and the fastest decline of prey. We confirm the relevance of P-tipping to real ecosystems by using parameter values consistent with the snowshoe hare and Canada lynx predator-prey system and climate in the boreal and deciduous-boreal forests of North America.