The Gamma Distribution
In typical use the oil shock model does not give a closed form solution. Because the input stimuli (normally provided by a set of discovery delta functions) needs to come from collected data and therefore displays a degree of randomness, we really should not expect anything approaching the symmetric simplicity of the Hubbert/logistic function :
Yet, under a set of idealized conditions, a variant of the oil shock model does revert to a fairly simple representation, that of the gamma distribution, which involves the repeated convolution of an exponential curve with itself N times total. I mentioned this first in the micro peak oil model and it makes sense to repeat it again to close the loop. Normalized, the gamma distribution looks like this:
dP/dt = k*r*e-r*T/(k+e-r*T)2
Plotted below with N=6, the gamma (in red) shows a distinct asymmetry with longer tails than the Hubbert curve (in yellow).
I chose N=6 to mimic a set of discoveries (the first 2 exponentials convolved together) convolved with the remaining four exponentials representing the fallow, build, maturity, and extraction phases of the conventional oil shock model.
I wouldn't typically use the gamma if I had discovery data available, but it does have the nice property of ease of use in data fitting applications and it has enough similarities to the Hubbert/logistic curve to serve as a replacement in traditional analyses. Plus its derivation rests on realistic first principles -- something in which the logistic function falls short.