. 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 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 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:
L. David Roper, roperld@vt.edu
20-mar-10