Power Laws in Arid Ecosystems

20Sep07

Recently in Nature: Two papers claiming to find power-law distributions in the patch sizes of vegetation. [1,2] Using field data from arid Mediterranean regions, Kefi et al found a power law distribution (, where N is the number of patches and S is the patch size) for areas with low grazing, and a truncated power law distribution () for areas with high grazing. They then constructed a computer model to reproduce the data and find out what kind of underlying dynamics could have created the power law distribution. Their model was a cellular automaton reminiscent of that in the Game of Life: vegetation was represented as cells that survived or died according to the status of their neighboring cells. Here are a few conflicting influences exerted on a ‘cell’ of vegetation by its neighbours:

Competition for local resources (e.g. soil nutrients and water)

Soil quality. Vegetation enhances soil quality. Unvegetated soil tends to erode easily and form a soil crust that is unfriendly to revegetation.

They also simulated the effects of grazing in the simplest possible way — as a higher mortality for cells. Grazing was found to upset the power law distribution and result in a distribution similar to that of the truncated power law found in their field data. In particular, grazing seemed to have the same effect as weakening the local positive interactions between cells (such as the increase in soil quality when you have neighbours). Unsurprisingly, they also found that above a certain level of grazing, local positive interactions are insufficient to stave off extinction. Grazing tended to upset the power law distribution just before the entire region transitioned into a desert. Thus they suggest that the lack of a power law distribution in patches of vegetation could be a flag for areas in serious ecological danger.

Kefi et al write that “As far as we know, power laws have not previously been described for patch-size distribution of vegetation in arid ecosystems.” It is quite a coincidence, then, that the other paper in Nature by Scanlon et al also studies an arid ecosystem and finds power laws for vegetation patch sizes in it. Scanlon et al hypothesize that the local positive interactions in the Kalahari region they studied are due to the retention of water in the soil by the shade from neighbouring vegetation. Exposed soil tends to dry up much faster and become inhospitable to seedlings. In short, even plants have a herd mentality, at least in areas with low rainfall.

1. Kéfi S, Rietkerk M, Alados CL, Pueyo Y, Papanastasis VP, Elaich A, de Ruiter PC. Spatial vegetation patterns and imminent desertification in Mediterranean arid ecosystems. Nature Vol. 449, No. 7159, pp. 213-217.
2. Scanlon TM, Caylor KK, Levin SA, Rodriguez-Iturbe I. Positive feedbacks promote power-law clustering of Kalahari vegetation. Nature Vol. 449, No. 7159 (2007), pp. 209-212.

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