Abstract: Arid ecosystems cover about 40% of Earth's land area and are home to over two billion people, yet they remain vulnerable to climate change and human actions. Using numerical simulations, and data from Mediterranean ecosystems in Spain, Morocco and Greece, Kefi et al. show that patch-size distribution of vegetation follows a power law. As grazing pressure increases, patch size deviates from the power law close to the transition to desert conditions. So patch-size distribution may be a useful early warning of desertification. The cover shows an arid landscape (top) in the El Planeron nature reserve in Belchite, Spain, and the lower panels show degradation in this landscape. In a separate paper, Scanlon et al. use satellite imagery to show that the size distribution of tree clusters in the Kalahari basin also follows a scale-free power law. This can be explained by positive feedback associated with preferential environments near existing trees. In News & Views Ricard Sole discusses both papers. COVER IMAGE Sonia & Michael Kefi/ Yolanda Pueyo/ Santiago Begueria Portugues This paper describes and models the effect of grazing on vegetation patchiness in three arid Mediterranean ecosystems. The patch size distribution of the vegetation in these ecosystems follows a power law, which can be explained by invoking local positive interactions among plants. Deviations from power laws occur when grazing pressure is high, and may be a harbinger of imminent desertification. Humans and climate affect ecosystems and their services1, which may involve continuous and discontinuous transitions from one stable state to another2. Discontinuous transitions are abrupt, irreversible and among the most catastrophic changes of ecosystems identified1. For terrestrial ecosystems, it has been hypothesized that vegetation patchiness could be used as a signature of imminent transitions3,4. 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.