## The Liebigs law of the minimum at the light of ecosystem principles

The Liebig's Law of the minimum may be seen as a deductive consequence of the principle of increasing ascendency (Ulanowicz and Baird, 1999). Let us see why.

Increasing ascendency also implies greater exergy storage, as can be demonstrated by two propositions in sequence:

Proposition 1: Longer biomass retention times contribute to increasing ascendency.

Let Bt represent the amount of biomass stored in the ith compartment of the ecosystem. Similarly, let Ttj be the amount of biomass that is transferred from compartment i to compartment j within a unit of time.

Information is now the measure of change in a probability assignment (Tribus and Mclrvine, 1971). The two distributions in question are usually the a priori and a posteriori versions of a given probability, which in the present case is the probability that a quantum of biomass will flow from i to j. As the a priori estimate that a quantum of biomass will leave i during a given interval of time, one may use the analogy from the theory of mass-action that the probability can be estimated as (Bi /B.), where B. represents the sum of all the Bi. In strictly similar manner, the probability that a quantum enters some other compartment j should be proportional to the quotient (Bj/B.). If these two probabilities were completely independent, then the joint probability that a quantum flows from i to j would become proportional to the product (BiBj /B2).

Of course, the exit and entrance probabilities are usually coupled and not entirely independent. In such case the a posteriori probability might be measured by empirical means in terms of the Tir That is the quotient (Tj/T..) would be an estimate of the a posteriori joint probability that a quantum leaves i and enters j.

Kullback (1959) provides a measure of information that is revealed in passing from the a priori to the a posteriori. It is called the Kullback-Leibler information measure, which is given by:

where p(a) and p(b) are the a priori probabilities of event at and bj, respectively, and p(a, bj) is the a posteriori probability that at and bj happen jointly. Substituting the probabilities as estimated in the preceding paragraphs, one obtains the form for the Kullback-Leibler information of biomass flow in a network as:

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