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Abstract:

Full paper appears as Scaling: Rivers, and Transportation Networks. P. K. Haff 408, 159-160, 2000. Rivers and Blood

P. K. Haff Division of and Ocean Sciences Duke University Durham, North Carolina 27708 [email protected]

Recent (Banavar, West) on allometric scaling in has attempted to explain the ¾ scaling “law” of animal , whereby the whole-animal metabolic rate of a wide range of increases approximately as M .75 . Banavar et al. argue that their results are applicable to other, non- biological systems whose “metabolism” is sustained by a branching distribution network analogous to the cardiovascular . In particular, they extend their model to river systems. For organisms, a key feature of these models is the introduction of an “extra” factor of length L beyond the L3 factor representing Euclidean volume of the animal. This extra actor arises in one case (Banavar) from a measure of the volume of blood waiting in the arteries before it passes through the , where metabolic is delivered to the tissues. In the other model (West) it arises from an interpretation of the circulatory system as a fractal or space-filling network. The discussion below is based principally on the picture of Banavar et al. Both authors develop an effective fourth dimension, from which the ¾ scaling derives. West does so using the space-filling nature of fractal networks, and Banavar does so by noting that the animal volume enters into metabolism in two different fundamental ways at once, first by providing the locus of the instantaneously metabolizing itself, while at the same time storing blood before (and after) it gives up its oxygen during metabolization. The blood volume argument of these authors is essentially a geometrical model that by-passes detailed organismal dynamics (the same is true of the West et al approach), which is the reason for the claim of a near universal applicability of their approach, not just to scaling in but to scaling in many other fields as well, including river networks. Here I advance a somewhat different interpretation of the blood-flow-river-flow analogy. A strictly geometrical argument is inappropriate for river systems in this view. First, it is necessary to recapitulate the argument for M .75 metabolic scaling in animals. I then derive the analogous result for “metabolic” scaling of landscapes.