|UBC Calculus Online Course Notes|
Exponential Growth: The Andromeda Strain
"The mathematics of uncontrolled growth are frightening. A single cell of the bacterium E. coli would, under ideal circumstances, divide every twenty minutes. That is not particularly disturbing until you think about it, but the fact is that bacteria multiply geometrically: one becomes two, two become four, four become eight, and so on. In this way it can be shown that in a single day, one cell of E. coli could produce a super-colony equal in size and weight to the entire planet Earth."
The frightening statement in Crichton's book may seem like science fiction, and if we take the pragmatic approach of a biologist, it sounds implausible and unrealistic. However, with the assumptions of "ideal circumstances", the mathematics behind this statement is surprisingly robust. We will actually determine its accuracy below, but our main focus will be not just to see if Crichton exaggerates, but also to understand the nature of so-called geometric growth. Better known to us in more recent terminology as exponential growth, this process hardly fails to surprise us with its potential for rapid increase.
For your consideration:
To appreciate the alarming growth, we might tabulate the first few values of the exponential expression for n=0,1,2...
Chosing a convenient time unit for the problem
Without actually thinking about it, you may have already "converted" the time units of minutes, hours, or days to time expressed in terms of multiples of , i.e. multiple of 20 minutes. Of course, this makes it much simpler to "count the number of doublings" and thus calculate how big the population would be after some time. This step, trivial as it might seem, is actually an example of a fundamental scientific method: the idea is that many natural processes have some kind of "intrinsic" timescale associated with them. If we look at these processes in their "own set of units", they look simplest.
To give a more convincing argument for why this might be useful, consider the fact that many kinds of species and populations could, in "ideal circumstances" undergo explosive (geometric) growth. Even though they may reproduce in more complex ways (not just by splitting into two copies), the population might also be characterized by a doubling time - for example, the population of rabbits in Australia doubled every few years after they were first introduced.
If we know the doubling time associated with the growth of some particular species, and we convert our calendar units (days, weeks, years, etc) to multiples of the doubling time, then the process of growth (on that scale) would look identical to the one we already studied for bacteria in this example. This means that by looking at just one case, and making the correct reinterpretation to various species, we have said all there is to say about the phenomenon of uncontrolled growth. This shows the power of mathematical reasoning. But it can also lead to a pitfall if we are not prepared to carefully evaluate the implications of our results.
For your consideration:
Shortcuts and recycling your envelopes!
Many students think that mathematics is all about "getting the right answer" and being accurate. This is vital in some cases (when we are designing a bridge or administering powerful medication) but superfluous in others. Applied mathematicians are famous for their "back of the envelope calculations" (sometimes scribbled on napkins at the diner, or on the back of ticket stubs at the opera). These are supposed to be rough estimates that get the answer in right ball park, without going overboard or getting overly bored with detailed calculations.
You might want to notice from our powers of 2 table that
This is a useful little conversion from powers of 2 to powers of 10. It is not exact, but it helps establish an "order of magnitude" approximation. Now that you know this handy fact, go back to the problem about human crowding on planet earth and see if the back of an envelope will do.