## Biomass Inclusive Ascendency

The above indices were calculated on the trophic flows between compartments. It is also possible to calculate a systems ascendency that embraces the connection between biomass stocks and the trophic flows. This biomass inclusive ascendency can be used as a theoretical basis to derive element limitations for compartments, to identify limiting nutrient linkages, and to quantify the successional trend to include larger species with slower turnover times.

Above, AMI was calculated as the difference between two flow probabilities, the unconstrained or a priori joint probability, and the constrained or a posteriori conditional probability. AMI can also be calculated between a biomass (unconstrained or joint) probability and the resulting flow (constrained or conditional) probability, thereby calculating a relationship between biomass and flows. From the principal of mass action, the joint probability of whether a quantum of biomass leaves compartment i (Bi/B) and enters compartment j (B/B) is Bfi/B2 This expression constitutes the unconstrained joint probability that a quantum flows from i to j. No constraining assumptions are made about this exchange, with the exception of the magnitudes of the stocks. The corresponding constrained distribution is taken as the conditional probability of the actual flow from i to j or Tj/T. This constraint is an addition to the probability calculated from the stocks only, and therefore, structure and function are tied together. The information gained is calculated as follows:

[15a

Redundancy

The fourth part of the overhead is that of internal transfers and represents the extent of pathway redundancy. There are disadvantages to the system in maintaining redundant, or parallel pathways. For one, there can be an increase in dissipations, whenever transfers occur not only along the most efficient route, but also along leakier pathways. Also, the resource transferred along different parallel pathways might not always end up at the right time at the consumer.

TjB2 T.B.Bj

Summing over all realized combinations of i and j and weighted by the joint probability of occurrence, one arrives at the biomass inclusive AMI, AMIB:

or or which is also called the Kullback-Leibler information. Scaling by the total system throughput gives the biomass inclusive ascendency, AB:

Ab = TST X TjTjB2

AB is sensitive to changes in biomass and can thus show the sensitivity of the whole system to changes in stock of a particular compartment.

The above term can be split into the following terms:

E^og

T,B TBi

The first term is exactly the same as in the above definition of the flow ascendency. Therefore, also the biomass inclusive ascendency rises with an increased number of compartments, increased specialization of flows, and increased throughput. The second and third terms become zero whenever the proportional flow through each compartment is the same as its proportion of the biomass. Only in this case would AB equal A. In all other cases, AB will exceed A.

Limiting elements in compartments and limiting flows

If one is interested in calculating a compartment's contribution to the ascendency of a particular element k (e.g., C, N, P, S, ...) during a certain time step l, then one has to substitute into above equation the element and the time step:

T ijklB1 T..B iklBjkl

where Tju is the flow from i to j of element k during time step l.

To show how the ascendency responds to turnover times of various elements, the differential of AB regarding compartment p is given as 