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

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 $T_i\propto X_i^{\alpha}$, which can be viewed as the counterpart of the Kleiber's 3/4 law at the population level; (3) The dissipation law $D_i\propto T_i^{\beta}$; and (4) The gravity law, including univariate version $f_{ij}\propto (T_iT_j)^{\gamma}$ and bivariate approvement $f_{ij}\propto T_i^{\gamma_1}T_j^{\gamma_2}$. These patterns are very common and significant in all collected webs, as a result, some remarkable regularities are hidden in weights.