The virtue of having a single, first-order equation representing yeast
dynamics is that we can solve this equation using integration techniques
from calculus. First we separate variables in (3),

annd then we apply partial fractions to the left-hand-side:

Now we can integrate both sides directly, using the facts that

and

Putting these three integrals together, relabelling constants , and using gives

or, exponentiating both sides,

where . Note that when we can see that

Now we can solve for ,

This is the general form of the solution to the logistic equation, (3). If we want to see explicitly how the initial conditions for the yeast population figure in we can substitute and to get

The behavior of typical solutions is plotted in Figure 1.

2000-07-31