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Visualization of the model

There are many ways to visualize logistic growth aside from simple plotting on a linear scale. It is possible to define a change of variables that normalizes a logistic curve and renders it as a straight line (see Figure 2). This view is known as the Fisher-Pry3 Transform:

 \begin{displaymath}
FP(t) = \left( \frac{F(t)}{1 - F(t)}\right),
\hbox{ where } F(t) = \frac{N(t)}{\kappa}
\end{displaymath} (3)


  
Figure 2: The logistic growth of a bacteria colony plotted using the Fisher-Pry transform that renders the logistic linear.
\resizebox{4in}{!}{\includegraphics{bacfp.eps}}

Note that

 \begin{displaymath}
\ln (FP(t)) = \alpha t + \beta,
\end{displaymath} (4)

so if FP(t) is plotted on a logarithmic scale, the S-shaped logistic is rendered linear. We observe that the time in which the range is between 10-1 and 101 is equal to $\Delta t$, and the time at 100 is the point of inflection (tm). Moreover, because the Fisher-Pry transform normalizes each curve to the carrying capacity $\kappa $, more than one logistic can be plotted on the same scale for comparison. As we will see, this becomes particularly useful when we analyze more complex growth behaviors. Figure 2 shows the Fisher-Pry transform of the bacteria example in figure 2. On the right axis we label the corresponding percent of saturation ( 100 * F) at each order of magnitude from 10-2 to 1-2 rounded to the nearest percent.


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Next: Bi- and multi-logistic curves Up: The Component Logistic Model Previous: The Component Logistic Model
Perrin S Meyer
1998-07-14