Verhulst Function for Modeling Minerals Depletion

Extraction

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

.

The maximum of P(t) occurs at , which yields .

is the amount to be eventually extracted, is the rising exponential time constant, n is the falling exponential time constant and t_1/2 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 t_1/2. The deviation of the peak time from t_1/2 is negative for n > 1 (skewed toward large times) and is positive for n < 1 (skewed toward small times).

Note that, for the symmetric case (n=1): and .

The asymmetry parameter, n, must be greater than 0. For the case of n=0, the Verhulst function becomes the Gompertz function:

.

Amount Left to be Extracted

The amount left to be extracted at time t is

.

The following graph shows the Verhulst function for = 100, t_1/2 = 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, t_1/2 = 1950 and = 5 with 6 different values of n:

Of course, the amount already extracted at time t is -Q(t).

Duration of Extraction

It is useful to define a "duration" for the extraction by the difference in the times when (f-1)/f has been extracted and when 1/f has been extracted:

.

A good choice for f is 10; then the duration would be the time interval for extracting the middle 80% of .

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Minerals Depletion

L. David Roper, roperld@vt.edu
24-jan-12