In the preceding sections we saw how the linear first-order differential equations lead to exponential solutions. Both the unbounded nature of the solution and the assumption of a constant growth rate indicate a modification that will give more realistic modeling of observed population growths.

**A.**A Nonlinear Equation

It seems reasonable to assume under that many conditions the growth
rate

Here

This is now a nonlinear first-order differential equation with several interesting features. The solution of Equation 2 can be shown to equal

for

This function is called a
*logistic *or *sigmoid, *and is illustrated below for several initial values of

An alternate form is

where

There are several very interesting features of this function.
For small initial populations, the initial increase is very much like
exponential.
This is obvious since the negative term in Equation 2 is small and the
equation looks linear.
However, as

It it interesting to note that it is possible to normalize the logistic into a "standard" form. If we scale both the amplitude and time by

then Equation 5 becomes

When plotted on semi-log paper, the logistic is an increasing straight line for small time, and becomes a horizontal straight line for large time.

The use of the logistic to model simple population growth with a limit is shown in Figures 4, 5, and 6. There have been many other applications of the logistic [1] [2] [3] [4] with some success and some failures. Unfortunately, if one tries to use this model to predict the limiting value while the system is still in the early stages of growth, a large error results since an exponential and a logistic look very similar in the early stages.

It is possible to manipulate the data so that a plot of it becomes a
straight line.
If the reciprocal of

and the logarithm is taken

we have a linear function.
Unfortunately, trial-and-error must be used to find