How to generate convenience
To add some context to the preceding post on matching the dynamics of a Logistic curve, I will step-by-step deconstruct the math behind the governing Verhulst equation. Although deceptively simple in both form and final result, you will quickly see how a "fudge" factor gets added to a well-understood amd arguably acceptable 2nd-order differential equation just so we end up with a convenient closed-form expression. This convenience of result essentially robs the Peak Oil community of a deeper understanding of the fundamental oil depletion dynamics. I consider that a shame, as not only have we wasted many hours fitting to an empirical curve, we also never gave the alternatives a chance -- something that in this age of computing we should never condone.
If we consider the discovery rate dynamics in terms of a proportional growth model, we can easily derive a 2nd-order differential equation whereby the damping term gets supplied by an accumulated discovery term. The latter term signifies a maximum discovery (or carrying) capacity that serves to eventually limit growth.
Now, if we refactor the Logistic/Verhulst equation to mimic the 2nd-order differential equation in appearance, it appears very similar apart from a conspicuous non-linear damping term shown at the lower right above.
That non-linear term in any realistic setting makes absolutely no sense. The counter-argument rests on a "you can't have a cake and eat it at the same time" finding. On the one hand, we assume an exponential growth rate based on the amount of instantaneous discoveries made. But on the other hand, believers in the Logistic model immediately want to turn around and modulate the proportional growth characteristics with what amounts to a non-linear "fudge" factor. This happens to just slow the contrived exponential growth with another contrived feedback term. Given the potential chaotic nature of most non-linear phenomena, we should feel lucky that we have a concise result. And to top it off, the fudge factor leads to a shape that becomes symmetric on both sides of the peak since it modulates the proportional growth equally around dD/dt=0, with an equal and opposite sign. As the Church lady would say, "How convenient!". Yet, we all know that the downside regime has to have a different characteristic than the upside (see the post on cubic growth for the explanation, and why the exponential growth law may not prove adequate in the first place).
Unfortunately, this "deep" analysis gets completely lost on the users of the Logistic curve. They simply like the fact that the easily solvable final result looks simple and gives them some convenience. Even though they have no idea of the underlying fundamentals, they remain happy as clams -- and I and people like R2 and monkeygrinder as angry as squids.
Labels: logistic math