The program exhibited -- unlimited population growth

The program exhibited in the upper window of the main page is a difference equation model representing the growth of bacteria doubling every hour without restraint. It is an illustration from Chapter 3 of the Origin, but instead of using elephants, this version is simplified to consider bacteria.

The question examined in the program is this: Consider a bacterium that would divide every hour on the hour. That is a simplification of reality, and it represents the idea that every model is a simplified view of something more complex. The question is, how fast will the bacterial culture grow if growth were unlimited.

Suppose you start a culture plate of bacteria with plenty of food on Monday morning, then leave the lab until five full days later—5 days of 24 hours being 5x24=120 hours later. If there are no constraints on growth, how many bacteria will there be at the end of the week? Will they completely fill the culture plate? Will they overflow the culture plate and spill onto the floor of the lab? Or will they go beyond that? That is what the basic difference equation on the main page is to examine.

The program in the upper window, reproduced below, is written in Hopper and it can be read step-by-step. When the first line says Set t to 0, that is like starting a stopwatch running, and when it says Set N to 1, that means to set the number of bacteria to one, just a single bacterium to start things out.

    |  Set t to 0 and N to 1
    |  Repeat the following until t reaches 120
    |    Display t and N as text
    |    Display t and N in a graph
    |    Increase t by 1
    |    Double the value of N
    |  End

Then the next line says to repeat the lines that follow until t reaches 120, representing the number of hours in five days, 5x24=120. That means the following indented lines are repeated, the first one saying to increase t, the stopwatch time, by one, which in our case means one hour. The next line says to double the value of N. That means that the bacterium divides in half and the total number of bacteria doubles in that time step. The final line says Display t and N, which shows what the stopwatch says and what the number of bacteria is at that point in the process. The line labeled End closes the program (and is optional). But with the Hopper language, all of the statements are flexible. Instead of End if you prefer something like That's all, folks!, you could set it up that way instead. More about that in subsequent discussions.

All computer programs start with some source code and reduce that to a lower level that is closer to the language that the computing machine itself uses. In this case the Hopper code is converted to a C-family language, in fact to JavaScript, which runs efficiently on your browser. When you press "Compile", that accomplish the conversion and you see the converted program in JavaScript format in the window below Hopper program. It is not necessary to understand that more cryptic format, at first or even later, but here it is shown for those who are familiar with such computer languages. They can be rather confusing at the beginning because of their reliance on delicately placed semicolons and braces and parentheses, which Hopper is able to avoid.

The next step is to press "Run" near the top of the page, and that takes the Javascript program, converts it down to the machine code needed, and runs it. Then the results appear in a window below the Hopper program. If you scroll through the results, you will see the bacteria doubling every time the stopwatch t increases by one. At the end of 120 hours, there are about 10 to the 36th bacteria—that is more than 1,000,000,000,000,000,000,000,000,000,000,000,000 bacteria—if if they were able to grow without restraint.

So the question is, how many bacteria are 10 to the 36th? Will they fill the culture plate? Will they overflow the culture plate? Actually, even with very small bacteria, one micron on the side (1/10,000 centimeters, 0.000254 inches CHECK), 10 to the 36th bacteria would fill all the world's oceans from the bottom to top!

So what is the lesson here? It illustrates an inviolable law of biology, that there can be no unrestrained growth of populations. That is the conclusion in chapter 3 of the Origin, one can see that the consequences of that inviolable law led to all of the complexity of life on earth, including leading forward to events of today.

[To be continued]