A Stochastic Theory of Community Food Webs. V. Intervality and Triangulation in the Trophic-Niche Overlap Graph

We report new empirical and theoretical information about the dimension of trophic-niche space and the structure of food webs, as measured by the frequency of intervality and triangulation of overlap graphs and resource graphs in community food webs. Briand assembled and edited 113 community food webs, and informed us of which of those webs have interval trophic-niche overlap graphs. In this collection of 113 webs, the overall proportion of webs that are interval is as high as, or higher than, the proportion of interval webs observed previously. However, the fraction of webs that are interval is strongly associated with the number of species in the webs. The fraction of interval webs declines from one for small webs (16 or fewer species) toward zero for large webs (33 or more species). According to new mathematical and numerical calculations presented here, the cascade model predicts, as observed, that the probability that a web is interval is near one for webs with fewer than 10 species, declines as the number of species increases from 10 to 30 or 40, and is very near zero for larger numbers of species. However, in the range of 10 to 40 species, the cascade model predicts a more rapid decline in the relative frequency of intervality than is observed. Using the predation matrices of the same 113 webs, we determined which webs have triangulated overlap graphs and triangulated resource graphs. The empirical, mathematical, and computational results on the relative frequency of triangulation parallel those on intervality. The broad ecological interpretation of our findings is that the larger the number of species in a community, the less likely it is that a single dimension suffices to describe the community's trophic-niche space and the less likely it is that there are no "homological holes" in the overlap graph and resource graph. Most reported webs with small numbers of species are incomplete descriptions of real communities. If future webs have larger numbers of species and are described in greater detail, we predict that those webs will have smaller relative frequencies of being interval and triangulated.