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A biological population with plenty of food, space to grow, and no threat from predators, tends to grow at a rate that is *proportional to the population* -- that is, in each unit of time, a certain percentage of the individuals produce new individuals. If reproduction takes place more or less continuously, then this growth rate is represented by

where **P** is the population as a function of time **t**, and **r** is the proportionality constant. We know that all solutions of this natural-growth equation have the form

where **P _{0}** is the population at time

Of course, most populations are constrained by limitations on resources -- even in the short run -- and none is unconstrained forever. The following figure shows two possible courses for growth of a population, the green curve following an exponential (unconstrained) pattern, the blue curve constrained so that the population is always less than some number **K**. When the population is *small* relative to **K**, the two patterns are virtually identical -- that is, the constraint doesn't make much difference. But, for the second population, as **P** becomes a significant fraction of **K**, the curves begin to diverge, and 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 model a factor of **1 - P/K** -- 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 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:

Date(Years AD) |
Population(millions) |

1790 | 3.929 |

1800 | 5.308 |

1810 | 7.240 |

1820 | 9.638 |

1830 | 12.866 |

1840 | 17.069 |

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.

On the other hand, when we add census data from the most recent half-century (next figure), 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, such as the Civil War, the Great Depression, two World Wars?

For more on limited and unlimited growth models, visit the University of British Columbia. [Ed. Note: This link is not longer operable.]

Leonard Lipkin and David Smith, "Logistic Growth Model - Background: Logistic Modeling," *Loci* (December 2004)

Journal of Online Mathematics and its Applications