No early warning signals for stochastic transitions: insights from large deviation theory.

In the study of Boettiger & Hastings [1], we demonstrated that conditioning on observing a purely stochastic transition from one stable basin to another could generate time-series trajectories that could be mistaken for an early warning signal of a critical transition (such as might be due to a fold bifurcation [2]), when instead the shift is merely due to chance. While the goal was to highlight a potential danger in mining historical records for patterns showing sudden shifts when seeking to test early warning techniques, Drake [3] draws attention to a potentially more interesting consequence of our analysis. Drake argues that the bias observed could be used to forecast purely stochastic transitions—a task previously thought to be impossible [4]. We feel this interpretation is too generous. The pattern Drake points to arises in any large deviation, regardless of whether a system is or is not at elevated risk for a transition. We illustrate this pattern in systems with and without bistability, demonstrating that early warning signals do not exist for purely stochastic transitions. Here, we provide a numerical demonstration that the pattern in question for consideration of an early warning signal appears not only before purely stochastic transitions (as seen in [1]) but also during any large deviation. As large deviations can occur even in stochastic systems that have only a single stable point, these patterns cannot be considered indicators of stochastic transitions. We demonstrate this in two scenarios: first using the Allee model of alternative stable states considered in [1], (eqn 2.1–2.2 and fig. 2), and then in a simple Ornstein–Uhlenbeck (OU) model, which has only a single stable state. Rather than condition on a stochastic transition having occurred (as in [1]), we now condition on having merely observed a sufficiently large deviation (larger deviations will be rarer and show a more pronounced pattern). We pick values such that we obtain a sample of a few hundred large deviation events in a sample of 20 000 replicates. The OU model is defined by a stochastic differential equation in which there is only a single optimum whose strength is proportional to the displacement,