On the effect of forcing on fold bifurcations and early-warning signals in population dynamics

Abstract

The classical fold bifurcation is a paradigmatic example of a critical transition. It has been used in a variety of contexts, including in particular ecology and climate science, to motivate the role of slow recovery rates and increased autocorrelations as early-warning signals of such transitions. We study the influence of external forcing on fold bifurcations and the respective early-warning signals. Thereby, our prime examples are single-species population dynamical models with Allee effect under the influence of either quasiperiodic forcing or bounded random noise. We show that the presence of these external factors may lead to so-called non-smooth fold bifurcations, and thereby has a significant impact on the behaviour of the Lyapunov exponents (and hence the recovery rates). In particular, it may lead to the absence of critical slowing down prior to population collapse. More precisely, unlike in the unforced case, the question whether slow recovery rates can be observed or detected prior to the transition crucially depends on the chosen time-scales and the size of the considered data set.