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Figure 1:
Behavior of typical solutions to the logistic equation. All
solutions approach the carrying capacity, , as time tends to infinity
at a rate depending on , the intrinsic growth rate.
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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
|
(4) |
The behavior of typical solutions is plotted in Figure 1.
Next: Estimating Parameters
Up: The Logistic Model
Previous: Reduction to a Single
James Powell
2000-07-31