Following the lead of Tribus and McIrvine, as in Ulanowicz (1980), one may scale this information measure by the total activity (T..) to yield the storage-inclusive ascendency (Ulanowicz and Abarca-Arenas, 1997) as:
Biomass storage: Odum (1969) proposed 24 trends to be expected as ecosystems develop and mature. These could be grouped under increases in species richness, trophic specificity, cycling, and containment. It happens that, other things being equal, an increase in any of these attributes will result in an increase of systems ascendency. As a result, Ulanowicz (1980, 1986) proposed as a phenomenological principle describing ecosystem development, "in the absence of major perturbations, ecosystems exhibit a tendency to increase in ascendency". Those factors that lend to an increasing ascendency, therefore, should be considered as significant contributors to ecosystem development.
From a mathematical point of view, one can elucidate how a system gains in magnitude by calculating what contributes to positive gradients in ascendency. So, for example, if one wishes to know what changes in the Bk foster increases in ascendency, one would want to study the partial derivatives, (8A/8Bk). After rather tedious algebraic manipulation, the results reduce to:
This formula has a straightforward meaning. The first term in parentheses is the overall throughput rate. The second quotient is the average throughput rate for compartment k. That is, the sensitivity of the biomass-ascendency is proportional to the amount by which the overall throughput rate exceeds that of the compartment in question. If the throughput of compartment is smaller than the overall rate, ascendency is abetted. In other words, increasing ascendency is favored by slower passage (longer storage) of biomass through compartment k., i.e. biomass storage favors increased ascendency.
Proposition 2: When several elements flow through a compartment, that element flowing in the least proportion (as identified by Liebig (1840)) is the one with the longest retention time in the compartment.
We begin by letting Tijk be the amount of element k flowing from component i to component j. We then consider the hypothetical situation of ideally balanced growth (production). In perfectly balanced growth, the elements are presented to the population in exactly the proportions that are assimilated into the biomass. This can be stated in quantitative fashion: for any arbitrary combination of foodstuff elements, p and q, used by compartment j,
TJq Bjq where an asterisk is used to indicate a flow associated with balanced growth. Now we suppose that one and only one element, say p without loss of generality, enters j in excess of the proportion needed. That is, TJp = T*p + ep where ep represents the excess amount of p presented to j. Under these conditions we have the inequality
Multiplying both sides of this inequality by the ratio T*p /Bp yields
B jp Bjq
In words, this latter inequality says that the input rate of p into j is greater (faster) than that of any other element by the 'stoichiometric' amount ep/Bjp. Over a long enough interval, inputs and outputs must balance, and so we can speak about the input rate and throughput rate as being one and the same. (This does not weaken our argument, as there is an implied steady-state assumption in the Liebig statement as well.)
Now we suppose that only two of the elements flowing into j are supplied in excess. Again, without loss of generality, we call the second element q. It is immediately apparent that if ep/Bjp > eq/Bjq, then the throughput rate ofp exceeds that of q, and vice versa. That is, a slower throughput rate indicates that one is closer to stoichiometric proportions. This last result can be generalized by mathematical induction to conclude that the element having the slowest throughput rate is being presented in the least stoichiometric proportion, i.e. it is limiting in the sense of Liebig.
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