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.
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
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
and integrate both sides
On the left side we get
Now exponentiate both sides:
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