and the expected time the substance of compartment j will reside in i before leaving the system, the future residence time, is determined by the correspondent element of the matrix where and

Past residence time

The fraction of gj that came from compartment j is q'v = fv /&

and the expected time the substance of compartment i has resided in j since entering the system, the past residence time, is determined by the matrix where

In the case of a two-compartmental model shown in Figure 1, i=1

r |
f21 = 21 |
z2 = 9 | ||

x1 = 30 |
x2 = 90 | |||

f|,2 = 27 | ||||

1 |
f |

Figure 1 A two-compartmental static model of annual nitrogen balance in a hypothetical ecosystem: x1 is the stock in the pool of living organic matter and x2 is the stock in the pool of nonliving organic matter, in g m~2.

Input-Output Models Output environ

Output environ is a set of compartments that are eventually reachable by a portion of substance entering the system through the given compartment. The quantitative characteristic of the output environ of compartment k is given by the matrix

where f"K) jis a flow from jto i induced by a unit input to k given by the vector ek = (O;..^ l, O;...^ )T

The matrices quantify the relationship between inputs and intercompartmental flows as follows n

Outputs of an input

The effect of inputs on outputs are tracked by the matrix

where wi0k is the fraction of substance entering ecosystem as an input to compartment k that will leave it as an output from compartment i w"k = v - ; ' =k j=i )=i n n

Input environ

Input environ is a set of compartments that were 'visited' by a portion of substance leaving the system through a given compartment. The quantitative characteristic of the input environ of compartment k is given by the matrix

where (fk9)ij is a flow from jto i that is necessary to provide a unit output from k.

The matrices quantify the relationship between outputs and intercompartmental flows as follows:

Inputs of an output

The input contributions to outputs are disclosed by the matrix

where w'k is the fraction of substance leaving ecosystem as an output from compartment k that has entered ecosystem as an input to compartment i:

w9' = £ (fk' )j, - £ (f') v ' ' =k )=i )=i w' = i +£(A; -¿2 (A )=i )=i

In the case of the model mentioned above

W"

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