Abstract
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 counterintuitive 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.
Highlights
Tipping points or critical transitions are fascinating nonlinear phenomena that are known to occur in complex systems subject to changing external conditions or external inputs
Motivated by growing evidence that tipping points in the Earth system could be more likely than was thought [2,36,37], we show that P-tipping could occur in real ecosystems subject to contemporary climate change
Since the unexpected tipping transitions occur only from certain phases of predator–prey oscillations, we refer to this phenomenon as phase tipping or P-tipping
Summary
Tipping points or critical transitions are fascinating nonlinear phenomena that are known to occur in complex systems subject to changing external conditions or external inputs. — Phase tipping (partial tipping [26]) or P-tipping occurs when a too fast change or random fluctuations in the external input cause the system to tip to a different state, but only from certain phases (or certain parts) of the base state and its neighbourhood. The Canada lynx is endangered in parts of its southern range, and the snowshoe hare is a keystone species in the north, relied upon by almost all of the mammalian and avian predators there [49] These examples illustrate the ubiquitous nature of cyclic predator–prey interactions, and their significant economic and environmental importance.
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