Conditions for skilful spatial and temporal tipping point early warning signals

As tipping points are extremely difficult to predict using an initial value modelling approach, forewarning of bifurcation tipping points instead often depends on the analysis of observational data, using approaches that aim to detect reducing system resilience. The most commonly used Early Warning Indicators (EWIs) rely on the phenomenon of critical slowing down, which is the tendency for fluctuations of a state variable to get larger (increased variance) and longer lived (increased temporal autocorrelation), as the bifurcation is approached. This is measurable in low dimensional systems that remain close to a quasi-equilibrium state in the run-up to the bifurcation. However, in systems that are subject to rapid changes in external forcing, such as the contemporary Earth system, this condition is unlikely to be met and EWIs become less reliable. In addition, temporal EWIs require long observational time series. For these reasons, it makes sense to consider spatial EWIs that can make use of the spatial detail resolved by present-day observations, especially from remote sensing. In this paper, we explore the conditions under which spatial and temporal EWIs will each be reliable, using a simple spatially coupled model with a fold bifurcation. In the weak coupling limit, we find spatial EWIs give reliable warning of the bifurcation, even if the external forcing is changing rapidly. However, in the limit of strong spatial coupling, spatial early warning disappears. Under these conditions, we find temporal EWIs give reliable warning as long as the external forcing is not changing quickly compared to the characteristic timescale of the system. We conclude that spatial early warnings will be especially useful in systems that have relatively weak spatial coupling, but which are also subject to rapidly changing external forcing (e.g. forests under contemporary climate change).