Spatial methods

Spatial analogues of the time-based EWS above exist too, allowing spatial information about a system within a single time step to be used. This can be considered as a space-for-time substitution, and eliminates the need for enough long-term data to have a moving window to measure AR(1) or variance on. 

As a system approaches a tipping point, responding more sluggishly to external perturbations and sampling more of the state space, it is expected that there will be higher spatial autocorrelation and variance (Kéfi et al., 2014). This can be calculated, for instance, by Moran’s I (Kéfi et al., 2014) and spatial variance (Guttal and Jayaprakash, 2009). The change in skewness observed in time series data also has a spatially analogous statistic (Guttal and Jayaprakash, 2009), noting again that this is not specifically related to CSD.

Some ecosystems have a clear self-organised spatial structure (e.g. drylands, peatlands, salt marshes, mussel beds; Rietkerk and van de Koppel, 2008). The emergence of such spatial patterns is thought to increase their resilience (Von Hardenberg et al., 2001); and could even allow them to evade tipping points altogether (Rietkerk et al., 2021). The size and shape of these patterns have been shown to change in a consistent way along stress gradients and have been suggested to be good candidate indicators of ecosystem degradation (Von Hardenberg et al., 2001; Rietkerk et al., 2002; Kéfi et al., 2007). 

One of the most studied examples is the case of dryland ecosystems, where changes in the shape of the patch size distribution could inform us about the stress experienced by the ecosystem (Kéfi et al., 2007). As the stress level increases, the larger vegetation patches in the system fragment into smaller ones, which leads to a change in the shape of the patch size distribution from power law-like to a truncated power law (Kéfi et al., 2011). A number of metrics can be used to quantify the shape of the patch size distribution, such as the parameters of the best fit (e.g. the slope of the power law fit), the size of the largest patch in the system, or the power law range (Kéfi et al., 2014; Berdugo et al., 2017). We note that the use of spatial EWS is also dependent on some knowledge about the underlying system’s spatial feedbacks (Villa Martín et al., 2015).

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