## PX2nn

The logarithm of p can also be a convenient value to measure AMI and ascendency ofthe system:

The concept of role establishes a relationship between number of nodes, connectivity, and information theory indices (AMI, A, and C).

Given the relation [42], when AMI of a system is known, the corresponding number of roles is deduced from the AMI power of e (Nepero number). As a consequence, with assigned ascendency and TST, the number of roles results e raised to the power of A/TST. Moreover, dividing the value of C by TST, it is also possible to infer the logarithm of roles when the maximum development is reached and, accordingly, the associated number of roles (Pc):

In Table 4, TST, AMI, ascendency, development capacity, and number of roles (corresponding to A and C) in real ecosystems are summarized.

For the 14 ecosystems of Table 4, when the number of roles is computed starting from the ascendency, through formula [45], it ranges between 3.465 583 (Crystal River Creek - delta temp.) and 9.492 356 (Charca de Maspalomas), with an average value of 6.405 319. These data confirm how the concept of role shows several relationships to that of TP, since both these indices are limited by the maximum number of passages that energy can experience from the ultimate source of energy (imports - outside environment).

Values of 10-20% are rather accepted as average living compartment efficiency in processing food, setting six to eight as the maximum number of energy steps (corresponding to TPs) in real networks. While TP is a property depending exclusively on compartment feeding activity, both prey items and predators affect the role that is assigned to each node. In ecosystems, even though procedures computing TPs and roles slightly differ, they are deeply settled by energy and efficiency constraints.

Therefore, it appears evident how the number of roles computed through the development capacity (eqn [46]) is overestimated (minimum = 22.645 125, maximum = 234.292 711, mean = 119.320 020), being associated to an ecologically unfeasible topology that does not consider constraints to energy-transfer efficiency in natural systems.

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