Population Growth Models

Part 4.1: Introduction

So far we have seen two models of unconstrained growth, i.e., models in which the populations increase in size without bound. In Part 2 we considered the exponential growth model governed by a differential equation of the form

For this model the productivity rate

is constant. As we have seen, the model population increases to infinity as time goes to infinity.

In Part 3 we looked at the coalition model for world population growth governed by a differential equation of the form

where h was a positive constant. In this case the the productivity rate

is itself increasing, and the model population increases to infinity in finite time. We'll refer to this type of growth as "superexponential."

Of course, most populations are constrained by limitations on resources -- even in the short run -- and none is unconstrained forever. The following figure shows three possible courses for growth of a population, the brown curve displaying superexponential growth and approaching a vertical asymptote (the dashed line), the green curve following an exponential growth pattern, and the blue curve constrained so that the population is always less than some number K. When the population is small relative to K, the patterns are virtually identical -- in particular, the constraint doesn't make much difference. But as P becomes a significant fraction of K, the curves begin to diverge, and, in the constrained case, as P gets close to K, the growth rate drops to 0.

We may account for the growth rate declining to 0 by including in the exponential model a factor of K - P -- which is close to 1 (i.e., has no effect) when P is much smaller than K, and which is close to 0 when P is close to K. The resulting model,

is called the logistic growth model or the Verhulst model. The constant K is called the carrying capacity or the maximum supportable population.

The word "logistic" has no particular meaning in this context, except that it is commonly accepted. The second name honors P. F. Verhulst, a Belgian mathematician who studied this idea in the 19th century. Using data from the first five U.S. censuses, he made a prediction in 1840 of the U.S. population in 1940 -- and was off by less than 1%.

This table shows the data available to Verhulst:

The following figure shows a plot of these data (blue points) together with a possible logistic curve fit (red) -- that is, the graph of a solution of the logistic growth model.

The next figure shows the same logistic curve together with the actual U.S. census data through 1940. This emphasizes the remarkable predictive ability of the model during an extended period of time in which the modest assumptions of the model were at least approximately true.

1. On the other hand, when we add census data from the most recent half-century, we see that the model loses its predictive ability. What (if anything) do you see in the data that might reflect significant events in U.S. history?

For more on limited and unlimited growth models, visit the University of British Columbia.

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modules at math.duke.edu

Copyright CCP and the author(s), 1999