Recently I was teaching a class about exponential growth. I gave a lot of the standard examples: bacteria, yeast cells in bread or wine, human populations. Then I gave some non-standard examples – the number of atoms that have undergone fission t picoseconds into the explosion of a nuclear device was memorable. As I often do, I then commented that no real-world process can continue to undergo exponential growth forever…
The bacteria eventually consume their host – or their growth is limited by the host’s immune system. Either way, the exponential character of the growth comes to an end. The plutonium atoms get vaporized and separated (so that fission no longer occurs) before more than a fraction of them give up their little iota of energy as they split. The yeast cells run out of sugars to consume or drown in their own waste products, carbon dioxide and ethanol.
So, in the real-world, exponential growth always “runs out of steam.” At first, the graph of such a quantity is both increasing and upward curving, but later, while the graph will continue to grow higher the curvature begins to trend downward. This is known as logistic growth.
The spot, right in the middle where the curvature changes from bending upward to bending downward is the point of inflection. That name literally means that there is no curvature at that point. (Imagine that red curve is a bird’s eye view of a road you are driving on. At first your steering wheel would be slightly turned to the left; later it will be turned to the right. At the point of inflection your steering wheel is actually pointed straight ahead – but only for an instant!)
Anyway, this one class, this one particular time, I also wondered aloud about whether we (the human population of the Earth) might be at the point of inflection? Or perhaps we passed the inflection point within the last few decades? This is certainly not an original thought, people have known for a long time about logistic growth, and certainly no one believes the Earth can hold infinitely many of us. There must be a limit! In descriptions of logistic growth they call it the carrying capacity. There are many competing theories about what the Earth’s carrying capacity is and whether we have reached the inflection point or not. Whether or not we’re there already, I’m convinced we’ll be moving into that downward curving portion of the logistic curve soon.
This is bad news.
One can refer to the carbon dioxide as “tiny bubbles” and the ethanol as “the water of life”, but the last few yeast cells in the fermenting wine just see it as choking to death in a vat that’s filled with their own waste.
Sorry.
Nutmeg Math 