Universal Biomass and Energy Flow Distribution in Weighted Food Webs

Previous studies on food webs always neglect weight information of edges and nodes. However, as the empirical food webs collected by researchers indicate that not only topological structures, but also weight information of each edge and vertex is available, in which, edge weight is the energy flux between any two species, and vertex weight is the total biomass of a given species on the food web. We define two variables Fi and Bi for each species i representing the total energy flux through i and the total biomass of it respectively. Then we find following common patterns of these variables by investigating 20 empirical weighted food webs: (1) Fi and Bi all follow DGBD distribution (with two exponents a and b) which is a kind of deformed Zipf law; (2) A power law relationship Bi ∼ Fiτ with an exponent τ in [0.63, 1.75] is always held for all the empirical webs. This relationship can be viewed as the counterpart of the Kleiber's 3/4 allometric scaling law in the population level. Finally, several mathematical relationships among the exponents a,b in both distributions and τ are derived and tested against the empirical food webs.