An alternative model of consumption was developed by M.King Hubbert in the 50’s, using the derivative of the population ‘S’ curve, which relies on the math of hyperbolic cosine (Cosh(x)). With it, he was able to predict the US fuel crisis from 20 years out, and be accurate to within 6 months. He predicted the world fuel crisis would occur in about 2000, 50 years away. He was not aware of the impact of North Sea supplies, and he was predicting only ‘conventional oil’, so oil shale and tarsands don’t factor into his calculation. Reality showed the peak for conventional oil occurred in 2008, with a maximum only slightly higher than the peak production Hubbert had predicted, but with a significantly higher overall recovery.
The math looks suspiciously like the sum of resistances in parallel, one increasing exponentially with time, and one decreasing exponentially with time. The resistance to production comes from a lack of technological investment and a scarcity of product. The first declines with time, while the second increases. It can be shown to work very well for any renewable resource that is being extracted significantly faster than it renews.
The simplest Hubbert curves are in the form of y=1/cosh(x), which is y=1/(1/e^x + 1/e^-x). This produces a symmetric curve. A point of inflection occurs on either side of the peak, at 1/sqrt(2) of the peak production, at a finite time period before and after (x= -0.88137). The slope of the curve at inflection determines the timing to the peak. At 2%/year, the peak occurs 43 years later. Peak World EF should therefore occur around 2026, were we wise enough.
These curves can be both multi-cyclic and asymmetrical, so the actual math can be much more sophisticated. By changing the technology curve, we can increase the rate of consumption significantly, at a cost of moving the peak to occur more quickly. The total volume under the curve doesn’t change significantly, since the total magnitude of ecological capital is finite at the beginning of the process.