  We identify the phase of a cycle as a new critical factor for tipping points
(critical transitions) in cyclic systems subject to time-varying external
conditions. As an example, we consider how contemporary climate variability
induces tipping from a predator-prey cycle to extinction in two paradigmatic
predator-prey models with an Allee effect. Our analysis of these examples
uncovers a counter-intuitive behaviour, which we call phase-sensitive tipping
or P-tipping, where tipping to extinction occurs only from certain phases of
the cycle. To explain this behaviour, we combine global dynamics with set
theory and introduce the concept of partial basin instability for limit cycles.
This concept provides a general framework to analyse and identify sufficient
criteria for the occurrence of phase-sensitive tipping in externally forced
systems.
