ENVIRONMENTAL CONSEQUENCES in rocky mountain coniferous ...

DISCUSSION
It is important to note that for the unlikely situation wherein the change in
state of a particular component, Xi, during any time interval, At, is unaffected by
either its own state or the state of any other environmental component (i
b = 0; j = 1 ,... ,n) equation 12 merely predicts that
i j
.em, that
or that the change in Xi during the interval from t to t+At equals the change in Xi
during the previous interval from t-At to t--not an unreasonable outcome.
Note also that, for the situation wherein the change in state of the ith component
is affected only by its current state, and not by the state of any other
component, equation 12 reduces to
An example of this latter situation would be one where the rate of growth in
population of a particular organism is either retarded or accelerated only by the
level of its current popul ation--the exponential growth function.
Finally, we should elaborate on the meaning of the biits, and the corresponding
challenge to researchers and staff special i sts. ~eferrin~- to figure 14, note that
b is the instantaneous slope of dXi/dt under the condition that nothing in the
ij
environment is changing except X and dXi/dt. Similarily bix is the instantaneous
j
slope of dXi/dt given that nothing is changing except Xk and dX 1 ./dt. When using a
1 inear approximation as indicated in figure 13, however, we are actually defining
b as the average slope of the dXi'/dt vs. X . relationship during the time interval
i j 3
At. The units of byj are the dimensions of dXi/dt divided by the dimensions of X j'
Geometrical interpretation of first partial derivatives
Figure 14. --&ornetria2 inteqpretation of
first partial deriva+iues.