On The Universal Scaling Relations In Food Webs

In the last three decades, researchers have tried to establish universal patterns about the structure of food webs. Recently was proposed that the exponent $\eta$ characterizing the efficiency of the energy transportation of the food web had a universal value ($\eta=1.13$). Here we establish a lower bound and an upper one for this exponent in a general spanning tree with the number of trophic species and the trophic levels fixed. When the number of species is large the lower and upper bounds are equal to 1, implying that the result $\eta=1.13$ is due to finite size effects. We also evaluate analytically and numerically the exponent $\eta$ for hierarchical and random networks. In all cases the exponent $\eta$ depends on the number of trophic species $K$ and when $K$ is large we have that $\eta\to 1$. Moreover, this result holds for any number $M$ of trophic levels. This means that food webs are very efficient resource transportation systems.