  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 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 attracting limit
cycles. This concept provides a general framework to analyse and identify
easily testable criteria for the occurrence of phase tipping in externally
forced systems, and can be extended to more complicated attractors.
