Bacteria in Bottles – an example of Exponential Growth

Professor Albert Bartlett famously said:

The greatest shortcoming of the human race is our inability to understand the exponential function.

A short while ago I published a wibblette that tried to explain the exponential function.
Dr Bob Rich added a comment:

I agree, many of humanity’s historical problems are because people think in straight lines not exponentials. My usual analogy is the one water weed seed landing in a pond. It produces a plant in a day. That plant only produces 2 new ones, on the next day. On the 3rd day, each of those 2 plants duplicate. Right. Eventually, half the pond is covered. How long before all of it is?
I used to describe this when teaching statistics to students who took an Arts course to get away from mathematics.
Most of them COULDN’T WORK IT OUT.
🙂

This reminded me of a part of Professor Bartlett’s lecture ‘Arithmetic, population and energy‘ where he gives a good example of the exponential function. So here’s his example — all three minutes of it, with a transcript below if you’d rather read than watch and listen.

Let’s look now at what happens when we have this kind of steady growth in a finite environment. Bacteria grow by doubling, and one bacterium divides to become two; the two divide to become four; the four become eight, sixteen and so on.

Suppose we had bacteria that doubled in number in this way every minute. Suppose we put one of these bacteria in an empty bottle at eleven in the morning and then observe that the bottle is full at twelve noon.

Now, there’s our case of just ordinary, steady growth: it has a doubling time of one minute; it’s in the finite environment of one bottle. I want to ask you three questions:

Number one: At what time was the bottle half full?

… Well, would you believe 11:59, one minute before twelve, because they double in number every minute.

And the second question: If you were an average bacterium in that bottle, at what time would you first realise that you were running out of space?

… Now, think about this: this kind of steady growth is the centrepiece of the national [US] economy and of the entire global economy. Think about it.

Well, let’s just look at the last minutes in the bottle:

At twelve noon it’s full
One minute before, it’s half full
Two minutes before, it’s a quarter full
… then an eighth, then a sixteenth.

Let me ask you: at five minutes before twelve, when the bottle is only 3% full, and is 97% open space just yearning for development: how many of you would realise there was a problem?

Now, in the ongoing controversy over growth in Boulder [Colorado], someone wrote to the newspaper some years ago and said, “Look, there isn’t any problem with population growth in Boulder because,” the writer said, “we have fifteen times as much open space as we’ve already used.” So let me ask you what time was it in Boulder when the open space was fifteen times the amount of space we’d already used? And the answer is: it was four minutes before twelve in Boulder Valley.

Well, suppose that at two minutes before twelve some of the bacteria realise that they’re running out of space, so they launch a great search for new bottles. And they search offshore, on the outer continental shelf, in the overthrust belt and in the Arctic — and they find three new bottles. Now that is a colossal discovery; that discovery is three times the amount of resource they ever knew about before. They now have four bottles. Before the discovery there was only one! Now, surely, this will give them a sustainable society. Won’t it?

Well, you know what the third question is: How long can the growth continue as a result of this magnificent discovery?

Well, let’s look at the score. At twelve noon one bottle is filled; there are three to go. 12:01 two bottles are filled; there are two to go. At 12:02 all four are filled — and that’s the end of the line.