### The real Logistic model derivation

Classical oil depletion modelers occupy a no-man's land situated between islands of poor data and a huge immovable object known as the Logistic curve. I always considered the Logistic formulation an empirical fit, widely used only because it generated a convenient, concise closed-form solution.

Wishing to get rid of empiricism at every turn, I think I came up with an analytical model that behaves much like the Logistic, but actually stems from much more understandable first-principles logic. It essentially branches off from the premise of the quadratic and cubic discovery models. Keeping it simple, I switch the power-law dependence of discovery growth to an exponential law:

dDiscovery(t)/dt = b*Discovery(t) - c*Integral(Discovery(t))This has the property of the rate of discovery increase tracking the current instantaneous rate of discoveries. Although arguable in the validity of its premise, it has a basis in human nature that nothing attracts success like success, which translates into a "gold-rush" mentality for the growth in discoveries. The decline comes about as a finite supply of discoveries accumulate and provide the negative feedback in the integral term. This turns into a classic 2

^{nd}-order differential equation.

D" - bD' + cD = 0I used an online differential equation solver to seek out the regime which corresponds to the classic growth and decline in discoveries:

This appears close in shape to the cubic growth model, but showing meatier tails and a sharper profile. It also needs an initial condition to kick in, as the solution degenerates to Discovery(t) = 0 without an initial discovery stimulus. D(0) and D'(0) provide the initial "gold rush" stimulus.

Similar to the cubic model, the backside part of the curve probably needs modification -- reflected by what I consider a different growth regime governed by a change in human dynamics:

dDiscovery(t)/dt = b0 - c*Integral(Discovery(t))I would justify this by suggesting that once a permanent decline kicks in by the relentlessly diminishing resources available for discovery, the incentive to discover turns into a constant (i.e. no more bandwagon jumpers), giving a damped exponential beyond the sharp decline (see the cubic example of this behavior below).

In general the shape of this curve mimics the shape of the Logistic curve, an exponentially ramped up-slope and an exponentially damped down-slope. Of course, the solution soes not match the simplicity of the Logistics curve, but we never intended to generate a concise solution; in my mind latching onto a concise thought process remains the ultimate goal.

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