A stochastic theory of community food webs. III. Predicted and observed lengths of food chains

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Abstract
This paper offers a quantitative theory of the length of food chains. The theory derives from a mathematical model of community food webs called the cascade model. The paper tests the predictions against data from real webs. An exact formula for the expected number of chains of each length in a model web with any given finite number, S, of species is, to our knowledge, the first exactly derived theory of the length of food chains. Since the numbers of chains of different lengths are dependent in the cascade model, we evaluate the goodness of fit between the observed and predicted numbers of chains by a Monte Carlo method. Without fitting any free parameters, and using no direct information about chain lengths other than that implied by the total number of species and the total number of links in a web, we find that the cascade model describes acceptably the observed numbers of chains of each length in all but 16 or 17 of 113 webs. Of 62 webs previously used to test the cascade model, the cascade model describes acceptably the chain lengths in all but 11 or 12. With a fresh batch of 51 webs, we establish first that (apart from two outlying webs) the numbers of links are very nearly proportional to the numbers of species and that the constant of proportionality is consistent with that in the original 62 webs. This finding verifies the so-called species--link scaling law with new data. The cascade model describes acceptably the chain lengths of all but 5 of the 51 new webs. Most of the 16 or 17 webs with chain lengths described poorly by the cascade model have unusually large average chain lengths (greater than 4 links) or unusually small average chain lenghts (fewer than 2 links).

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