Humans and climate affect ecosystems and their services 1 , which may involve continuous and discontinuous transitions from one stable state to another 2 . Discontinuous transitions are abrupt, irreversible and among the most catastrophic changes of ecosystems identified 1 . For terrestrial ecosystems, it has been hypothesized that vegetation patchiness could be used as a signature of imminent transitions 3,4 . Here, we analyse how vegetation patchiness changes in arid ecosystems with different grazing pressures, using both field data and a modelling approach. In the modelling approach, we extrapolated our analysis to even higher grazing pressures to investigate the vegetation patchiness when desertification is imminent. In three arid Mediterranean ecosystems in Spain, Greece and Morocco, we found that the patch-size distribution of the vegetation follows a power law. Using a stochastic cellular automaton model, we show that local positive interactions among plants can explain such power-law distributions. Furthermore, with increasing grazing pressure, the field data revealed consistent deviations from power laws. Increased grazing pressure leads to similar deviations in the model. When grazing was further increased in the model, we found that these deviations always and only occurred close to transition to desert, independent of the type of transition, and regardless of the vegetation cover. Therefore, we propose that patch-size distributions may be a warning signal for the onset of desertification.It is of the utmost importance to find early warning signals of transitions that can alter ecosystems' services in fundamental ways, causing losses of ecological and economic resources 2,4 . Determining proximity to transitions is especially important for arid ecosystems, which may convert into deserts 2,4,5 . According to the Millennium Ecosystem Assessment, increasing external pressures by human activities or climate change will lead to desertification, affecting the livelihood of more than 25% of the world's population 1 . A mechanism playing a dominant role in the functioning of arid ecosystems is local facilitation among plants [6][7][8][9] . Local facilitation is the biophysical ameliorative effect of sessile organisms, such as plants, on their neighbouring environment. Such local positive interactions induce vegetation patchiness 6,7,10 and determine the response of this patchiness to environmental change 3 .We investigated how the spatial organization of vegetation is influenced by the degree of external stress by combining modelling and field data from three grazed Mediterranean arid ecosystems in Spain, Greece and Morocco. In each of these ecosystems, we collected data on three sites that differed with respect to the livestock grazing pressure (Table 1; Methods). In each of the nine (3 3 3) sites, we analysed the number and the sizes of the vegetation patches (see Methods), and plotted the number of patches, N(S), as a function of their sizes, S. We fitted these patch-size distributions to two different mod...