Seeing the Elephant
There’s an old parable about blind men trying to figure out what an elephant looks like. One man feels its trunk and says an elephant is like a snake. Another feels the leg and says an elephant is like a tree. And so on.
Likewise, a lot of different behavior patterns are commonplace in the world. There’s Exponential Growth and the S-Shaped Plateau, “Peak” Behavior (like Peak Oil) and slow Stagnation. Economists and ecologists and epidemiologists, etc, have come up with mathematical descriptions for each separately.
All of these, though, I think can be thought of manifestations of the same underlying process – different behaviors of the same elephant. This is what I think that elephant looks like: A Non-Renewable Resource gets converted into a Usable Good, which can then be used up or destroyed. Visually, this can be represented as:
Think about ‘Crude Oil’ being converted into ‘Gasoline’, or ‘Potential Buyers’ of a product being converted into ‘Current Owners’.
There are actually several ways I have found that S-shaped growth can happen – and ways it can look like Exponential Growth, Stagnation and so forth – and all are derivatives of the diagram above.
When the Non-Renewable Resource is available in essentially unlimited supply and the growth is restricted by some ‘carrying capacity’, then the structure above reduces to the following:
One way to represent ‘effect of remaining capacity’ is simply ‘1 – (Yeast Cells / max Yeast Cells)’. The growth rate is some constant (say, 0.05) multiplied by this effect. Thus, when the number of yeast cells is small, the growth rate is highest (near 5%, in this case) and as the population grows, the growth rate declines.
If cells can die, and the death rate goes up as the population goes up, this also gives S-shaped growth.
This is similar in concept to the birth rate declining as the population grows, but instead affecting the death rate.
A delay in a system can sometimes be thought of as similar to riding on a conveyor belt. You get on the conveyor, ride for a while, and then get off. For example, a house is built in one year, exists for 30 or 40 or 50 years, and is then pulled down to be replaced by a bigger, newer building.
If there is growth in the number of houses being built, it will take 30 or 40 or 50 years for the number of houses being replaced to catch up. In the intervening time, the number of houses in existence can be an S-shaped profile.
Finally, when there is a Non-renewable Resource and the Good it generates can decay, the system can exhibit and S-shaped profile. In the example below, the rate of buying (the rate at which Nonowners become Owners is simply ‘f * Nonowners * Owners’, where ‘f’ is a small fraction.
If Owners can never become former Owners, then the number of Owners will show a S-shaped profile. If a small portion of Owners disappear each time period, then then S-shaped profile will be followed by a long period of perpetual stagnation, like what is shown in the figure below – a simulation of the model above with f = 0.00005 and ‘becoming former Owners’ being 0.001 of the number of Owners.
In this graph, we can see all of the parts of the elephant: initially Exponential Growth, followed by S-Shaped Growth, followed by a peak, followed by a long period of Stagnation. Modeling the behavior of the system simply a matter of finding the variation of the models above that most closely resemble the structure of the real system. I think that many physical, social and economic patterns fit this profile in the long run – stock prices of companies, populations of countries – but we often only see a small selection of the overall behavior profile and mistake the elephant for being just a trunk or a leg.