Intermittent percolation and the scale-free distribution of vegetation clusters

Abstract

Understanding the causes and effects of spatial vegetation patterns is a fundamental problem in ecology, especially because these can be used as early predictors of catastrophic shifts such as desertification processes. Empirical studies of the vegetation cover in some areas such as drylands and semiarid regions have revealed the existence of vegetation patches of broadly diverse sizes. In particular, the probability distribution of patch sizes can be fitted by a power law, i.e. vegetation patches are approximately scale free up to some maximum size. Different explanatory mechanisms, such as plant-plant interactions and plant-water feedback loops have been proposed to rationalize the emergence of such scale-free patterns, yet a full understanding has not been reached. Using a simple model for vegetation dynamics, we show that environmental temporal variability -- a well-recognized feature of semiarid environments -- promotes in a robust way (i.e. for a wide range of parameter values) the emergence of vegetation patches with broadly distributed cluster sizes. Furthermore, this result is related to a percolation phenomenon that occurs in an intermittent or fluctuating way. The model also reveals that the power-law exponents fitting the tails of the probability distributions depend on the overall vegetation-cover density, in agreement with empirical observations. This supports the idea that environmental variability plays a key role in the formation of scale-free vegetation patterns. From a practical viewpoint, this may be of importance to predict the effects that changes in environmental conditions may have in real ecosystems. From a theoretical side, our study sheds new light on a novel type of percolation phenomena occurring under temporally-varying external conditions, that still needs further work to be fully characterized.