..The Verhulst Function is a good function to use for fitting minerals-depletion data. It is used to model the fact that their is usually an exponential growth in the rate of extraction a mineral from the Earth, followed by a peak, after which the extraction rate declines exponentially. The function is
..
is the amount to be eventually extracted, 
is the rising exponential time constant, n is the falling exponential time constant and t1/2 is the time at which the resource is one-half depleted. The parameter n determines the amount of skewing at large times. For n = 1 the extraction curve is symmetrical and the peak occurs at t1/2. The deviation of the peak time from t1/2 is negative for n > 1 (skewed toward large times) and is positive for n < 1 (skewed toward small times).
The maximum of P(t) occurs at 
, which yields 
- Note that, for the symmetric case (n=1): 
and 
.
When a peak is symmetrical, the Verhulst function simplifies to
.
The asymmetry parameter, n, must be greater than 0. For the case of n=0, the Verhulst function becomes the Gompertz function:
.
The amount left to be extracted at time t is
.
The following graph shows the Verhulst function for = 100, t1/2 = 1950 and = 5 with 6 different values of n:
The area under all the curves is = 100.
The following graph shows the amount-left Verhulst function for = 100, t1/2 = 1950 and = 5 with 6 different values of n:
Of course, the amount already extracted at time t is -Q(t).
It is useful to define a "duration" for the extraction of a mineral by the difference in the times when (f-1)/f of it has been extracted and when 1/f of it has been extracted:
.
This is derived from 
.
For the symmetric case (n = 1) 
.
A good choice for f is 10; then the duration would be the time interval for extracting the middle 80% of .
L. David Roper interdisciplinary studies
L. David Roper, roperld@vt.edu
20-dec-12