Anticipating critical transitions in multi-dimensional systems driven by time- and state-dependent noise

When approaching a one-parameter bifurcation, the feedbacks that stabilise the initial state weaken and eventually vanish; a process referred to as critical slowing down (CSD). This motivates the use of variance and lag-1 auto-correlation as indicators of CSD in order to anticipate bifurcation-induced critical transitions. Both indicators require a prior dimension reduction to a one-dimensional time series. The use of variance is further limited to time- and state-independent driving noise, strongly constraining its generality. Here, we propose a data-driven approach based on deriving a multi-dimensional Langevin equation to detect local stability changes and anticipate bifurcation-induced transitions in systems with generally time- and state-dependent noise. Our approach substantially generalizes the conditions underlying existing early warning indicators, which we showcase in the example of a two-dimensional predator-prey model. This reduces the risk of false and missed alarms significantly and allows for a more holistic understanding of the multi-dimensional system at hand.