Fig 1.
Systems with positive (left) and negative (right) feedback.
The right panels depict the continuous transition scenario, where positive feedback is absent or weak. An increase in the local vegetation density ρ is followed by a decrease in the local growth rate r due to enhanced competition and depletion of resources. Accordingly, a single stable state appears where r vanishes (upper panel, right). An increase in the stress decreases vegetation density at the equilibrium point (lower panel, right). The chance of a spatial cluster to grow decreases with its size [16] and changes sign at the equilibrium point, as illustrated by the arrows. On spatial domains, the transition (when the density reaches zero) belongs to the directed percolation universality class [17, 18]. The left panels illustrate the case where positive feedback mechanisms allow for local growth only above some critical density. Such a system admits two alternative steady states (marked green and red), separated by an unstable fixed point (black). When the stress parameter increases or decreases one of these states may lose its stability at a tipping point (purple) via a saddle-node bifurcation. On spatial domains the transition may be either continuous or discontinuous, depending on the strength of stochasticity [19]. Positive feedback implies that small patches shrink on average, while large patches grow [16].
Fig 2.
Temporal analysis of vegetation patterns.
Panel A shows the survey area, the Sahel region in Africa (taken from Sentinel-2 cloudless— https://s2maps.eu by EOX IT Services GmbH (Contains modified Copernicus Sentinel data 2016 and 2017)), together with (average) rainfall lines (taken from [26]). In panel B the local response curve, i.e., the normalized differences between the years 1999 and 2002 ((αρ2002 − ρ1999)/ρ1999, where ρ is the local vegetation density and the constant α was chosen such that the average growth is zero), is plotted against ρ1999 for all pixels between rainfall lines 400-700 mm/y (main plot) and for other precipitation regions (inset), showing what appears to be purely negative feedback (see Text C in S1 Appendix for details and errorbars). A spatial cluster is defined as a connected collection of elementary 30m squares all, above some threshold vegetation density ρth (see Methods). The chance of a cluster of a given spatial area A, , to grow between the years ’99 and ’02, normalized by the chance of growth of an elementary cluster,
, is plotted in panel C. Its increase with the size of the cluster, predicted in [16], appear to support the positive feedback hypothesis. Different lines correspond to different threshold density ρth used to define a “cluster”, making it clear that the positive response of small clusters is independent of this definition.
Fig 3.
LRC and SRC for specific rainfall lines.
The date for 350-400 mm/year region (panels A-C) exemplify the typical scenario, where the LRC (panel A and with error bars in panels B and E) decreases monotonously with density. On the other hands, in panel D one notices that the 1999-2002 line appears to indicate positive feedback. This is a rare exception (see S3 Fig), the apparent positive feedback disappears in the 2002-2015 interval. Panel C shows a typical SRC curve, while in panel F the curve is almost flat; this flat curve is, again, an exception, as seen in S1 and S2 Figs. The details of the smoothing and fitting procedures are given in the Text B in S1 Appendix.
Fig 4.
LRC and SRC from stochastic simulations.
The local response curve (LRC, main panel) and the spatial response curve (SRC, inset panel) as obtained from simulation of the model analyzed in [16, 19]. Eq (2) was simulated on a 100 × 100 lattice with a = 174.5, b = 40 c = 1.6 and ζ = 2. In this parameter regime the deterministic dynamics supports two alternative steady states (one of them absorbing) and the demographic stochasticity is relatively strong. The LRC is negative, while the SRC is positive, in agreement with the empirical result presented above.
Fig 5.
The relative frequency distribution of vegetation cover is plotted here for four levels of mean annual precipitation using the data of 1999 (blue) and 2002 (red).
The average vegetation cover decreased during this period, so the histograms of 2002 are shifted systematically to the left. The histograms for 650-700 mm/y (panel A) admit a clear single peak for both years, in the region 800-850 mm/y (panel B) one observes a crossover from a unimodal to bimodal distribution and for 550-600 mm/y (panel C) the histogram has a double peak in 1999 and a single peak in 2002, meaning that the modality of the histogram is not a robust feature of the system. The local response curve (LRC) in all cases has a negative slope (left inset). The spatial response curve (SRC) shows signs of positive feedback in most of the cases, but there are some rare exceptions, See S2–S13 Figs for more details.