Spatial correlation

Due to critical slowing down, neighboring units in space look more similar to each other when a system approaches a bifurcation point, i.e. they become increasingly correlated. This increasing similarity can be quantified by the spatial correlation function, or Moran’s I, between ecological states separated by a certain distance. The near-neighbor spatial correlation, the analog of autocorrelation at lag 1 for time series, is calculated for the distance between nearest neighboring units of the system.

Spectral reddening

Increased memory due to critical slowing down also manifests itself as spectral reddening, i.e. spatial variation becomes increasingly concentrated at low ‘spatial frequency’, or the number of times that a pattern is repeated in a unit of spatial length (wavenumber). We quantify this reddening by the Discrete Fourier Transform (DFT) that decomposes spatial data into components of sine and cosine waves of different wavenumbers.