Common Patterns of Energy Flow and Biomass Distribution on Weighted Food Webs

Abstract: Weights of edges and nodes on food webs which are available from the empirical data hide much information about energy flows and biomass distributions in ecosystem. We define a set of variables related to weights for each species $i$, including the throughflow $T_i$, the total biomass $X_i$, and the dissipated flow $D_i$ (output to the environment) to uncover the following common patterns in 19 empirical weighted food webs: (1) DGBD distributions (Discrete version of a Generalized Beta Distribution), a kind of deformed Zipf's law, of energy flow and storage biomass; (2) The allometric scaling law $T_i\propto X_i<sup>{\alpha}$, which can be viewed as the counterpart of the Kleiber's 3/4 law at the population level; (3) The dissipation law $D_i\propto T_i<sup>{\beta}$; and (4) The gravity law, including univariate version $f_{ij}\propto (T_iT_j)<sup>{\gamma}$ and bivariate approvement $f_{ij}\propto T_i<sup>{\gamma_1}T_j<sup>{\gamma_2}$. These patterns are very common and significant in all collected webs, as a result, some remarkable regularities are hidden in weights.