Logistic Equation

The logistics equation is a differential equation that models population growth. Often in practice a differential equation models some physical situtation, and you should read it'' as doing so.

Exponential growth:

This says that the relative (percentage) growth rate'' is constant. As we saw before, the solutions are

Note that this model only works for a little while. In everyday life the growth couldn't actually continue at this rate indefinitely. This exponential growth model ignores limitations on resources, disease, etc. Perhaps there is a better model?

Over time we expect the growth rate should level off, i.e., decrease to 0. What about

 (7.1)

where is some large constant called the carrying capacity, which is much bigger than at time 0. The carrying capacity is the maximum population that the environment can support. Note that if , then so the population declines. The differential equation () is called the logistic model (or logistic differential equation). There are, of course, other models one could use, e.g., the Gompertz equation.

First question: are there any equilibrium solutions to (), i.e., solutions with , i.e., constant solutions? In order that then , so the two equilibrium solutions are and .

The logistic differential equation () is separable, so you can separate the variables with one variable on one side of the equality and one on the other. This means we can easily solve the equation by integrating. We rewrite the equation as

Now separate:

and integrate both sides

On the left side we get

Thus

so

Now exponentiate both sides:

where $A=e^c$

Thus

so

Note that also makes sense and gives an equilibrium solution. In general we have . In any particular case we can determine as a function of by using that

so

William Stein 2006-03-15