The Missing Link

As far as I can tell, few models exist for the discovery dynamics of valuable yet finite resources. From a previous post, I introduced the concept of quadratic growth. This kind of growth has an underlying mechanism of a constant acceleration term -- in other words the rate of growth itself increases linearly with time. To first order, this explains scenarios that involve a rapidly increasing uptake of resources, and particularly those that spread by word of mouth. The growth of wiki-words in Wikipedia provides the best current-day example of quadratic growth. Unfortunately wiki-words grow out of an almost endless supply of alpha-numeric strings, which shows no signs of declining. However, for non-infinite resources we all know that growth ultimately abates and (quite frequently) suddenly. I first review two prime examples of this kind of dynamics: the old-fashioned gold rush and the extinction of species.

First, if we consider the gold rushes that occurred in places like California and Alaska during the 1800's, we invariably witnessed an accelerated search for the mineral as prospectors swarmed to a region. This accelerating growth in claims never lasted for long though -- within a few years, the region became scoured clean and history usually records a decline typically more spectacular than the original rise. We pretty much have to agree that finite resources played an important role in this behavior, and numerous ghost towns remain the only concrete evidence that any activity even occurred.

The passenger pigeon extinction provides an even more dramatic example of accelerating growth followed by sudden decline. From historical accounts of the colonial days of the new country, a few settlers started realizing that pigeon populations provided an easy source of food. More important, other settlers joined in and discovered increasingly lethal ways of decimating the bird population. So this perhaps century long accelerating increase in harvest numbers formed a framework for a precipitous decline in pigeon population within a few decades, ultimately followed by extinction of the species. The pigeon population essentially became a finite resource as reproduction dynamics could not overcome decimation by the numbers (e.g. through creative uses of dynamite). Although I have found few reliable estimates of the numbers (here), no one argues that wild pigeons essentially became extinct within the span of a few years from the late 1800's into the first few years of the 20th century.

So I ask the question: can we create a model of this "gold-rush"-like discovery of resources which effectively matches those of gold, passenger pigeons, and ultimately oil? I call this a missing link, because although I have a fairly good understanding of oil depletion post-discovery, the oil shock model, I don't yet have a good handle on the dynamics of discovery. For example, I hacked on a contrived decline term to my previous model of quadratic growth.

Clearly, this artificial break in quadratic growth does not follow from any physical process and I did this mainly to match empirical observations.

I assert that the key to modeling a finite resource limited decline lies in combining a quadratic growth term with a first-order feedback term describing the constraint. The latter term adds a variant of exponential growth/decline to the quadratic term.

Quadratic Growth : d2Q(t)/dt2 = k
Exponential Growth : dQ(t)/dt = aQ(t)

The combined real equation looks like the following:

d2Discovery(t)/dt2 = c - a3*Integral(Discovery(t))

Which basically says that the acceleration in discoveries is proportional to a constant suppressed by a drag factor that increases as discoveries accumulate. The drag term essentially describes the finite resource. If, on the other hand, I switch the sign on the drag factor, it becomes an exponential growth term which eventually dominates the quadratic term, forming a type of positive feedback (which models population dynamics). However, for negative feedback, the acceleration eventually becomes negative and the discoveries get driven into the ground. You can see the behavior in the following figure, where I plot the square root so you can see the divergence from pure quadratic growth depending on the feedback sign:

We can use calculus and Laplace transforms¹ to come up with the solution to the quadratic/feedback differential equation:

Discovery(t) = InitialValue +
   c*(e-at-eat/2(cos(K*a*t/2)-K*sin(K*a*t/2))/(K*a)2

where K = square root of 3. The term a acts like a characteristic value to the solution of a third order differential equation, while the value for c sets the amplitude.

I decided fit the model to historical estimates of oil discoveries, based on seeing the following kind of data reported:

Note that this figure shows a histogram of numbers of world discoveries which does not include the individual size of the discoveries; I consider this reasonable as the size forms a stochastically independent variable to the number of discoveries and random fluctuations would certainly modulate this profile -- but only in a statistical sense.

Initially, I decided to look at USA data, as the discovery estimates provided by Laherrere and the production numbers by Staniford at TOD generate a good dynamic range (for links see here).


I used the same oil shock model from the earlier post, and plotted the results for the USA below. It easily betters the Gaussian model as it accounts for the causality in the initial discoveries by Drake in Titusville, PA in 1859 (i.e. something has to start out the "gold rush"). It also models the out-years quite effectively, as the decline will become much less steep than a Gaussian will predict, something that Staniford would also likely admit to if he extended the profile much beyond the year 2010.


Also notice when I plot the model on a linear scale (see below), it becomes nearly as symmetric as a Gaussian! I can easily explain this as the right-heavy asymmetry of the quadratic feedback model gets balanced by the left-heavy asymmetry of the gamma distributions that form the basis of the oil shock model. The convolution of the two models effectively cancels out the left/right asymmetries and a fairly symmetric model results. With this revelation, I suggest we should seriously think about throwing out the Logistic and Gaussian formulations of the Hubbert peak models. We finally have a combined model which springs forth from first principles -- something that neither the logistic nor the Gaussian formulations can deliver. In other words, with the missing link uncovered, we have very little need for unfounded heuristics in modeling oil depletion or any kind of finite resource depletion (barring the role of economics of course).



Since the quadratic/feedback formulation shows self-scaling similarities similar to that of trig functions (i.e. period and amplitude), we can describe certain characteristics which depend on the a and c constants.


So we can easily estimate the terms for quadratic/feedback growth by simply overlaying the scaled profile on potential discovery curves. The figure to the right should match world discoveries to first order.

I will definitely pursue this analysis further, as the quadratic/feedback formulation looks promising. What to call it though; I have a feeling the particular equation I use exists with a fancy name somewhere in the literature. In the meantime, perhaps the "resource collapse" model would provide a fitting name?

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¹ The Laplace transform of the quadratic/feedback differential equation:
P(s) = c/(s³ + a³)