Purposes:
To study the historical data on human population growth;
To compare the "natural" and "coalition" differential equation models as possible descriptions of the growth pattern.
Prerequisites:
The concept of slope field for a first-order differential equation;
Log-log and semilog graphing and their uses in deciding whether data fit a given model;
Estimation of rates of change via difference quotients;
The power rule for integration;
The separation of variables technique for solving a differential equation.
This module works well as the last lab exercise in a first-semester calculus course, and it can also be used early in a second-semester course to stimulate interest as a striking example of very simple formal integration. Either way, there are strong messages about the importance of correct calculation, reinforced by obviously absurd answers that result from common errors. In particular, students find themselves in untenable positions if
a. they think that integrating "1 over something" always means take the log of "something," or,
b. they forget the constant of integration.
There are several modeling messages here as well, not all of which are made explicit in the module.
One has to make decisions about the relative "softness" of very old data, as compared to data gathered in modern times.
Even though there is a definite number of people in the world at any give instant, no one can possibly know that number -- or even be very sure about an estimate -- in spite of the linked Census Bureau page that appears to give actual counts.
The "standard" model of biological growth is not universally applicable.
Even a somewhat quixotic and oversimplified model can be remarkably successful -- the coalition model was proposed in 1960 and outperformed more sophisticated predictors for the rest of the 20th century. It leads to an absurd conclusion about a generation from now, and we are just beginning to see a slowing of the unsustainable growth rate.
A model that "blows up" in a physically impossible way may nevertheless tell us something important about the time just before the blow-up.
For students who do not encounter simple differential equations in their calculus course, the module could also be used early in a differential equations course -- to emphasize a modeling point of view and to review important calculus topics.
In addition to the references in the module, you will find more about the coalition model in David Smith's paper “Human Population Growth: Stability or Explosion?,” Mathematics Magazine 50 (1977), 186-197.
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