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The Component Logistic Model

The logistic growth model assumes that a population N(t) of individuals, cells, or inanimate objects grows or diffuses at an exponential rate $\alpha$ until the approach of a limit or capacity $\kappa $slows the growth, producing the familiar symmetrical S-shaped curve. This model can be expressed mathematically by the following ordinary differential equation (ODE) which specifies the growth rate $\frac{dN(t)}{dt}$ as a nonlinear function of N(t):

\frac{dN(t)}{dt} = \alpha\,\, N(t) \underbrace{\left( 1 -\frac{N(t)}{\kappa}
\right)}_{\text{feedback term}}
\end{displaymath} (1)

For values of $N(t) \ll \kappa$, equation (1) closely resembles exponential growth. As $N(t) \to \kappa$, the feedback term slows the growth to zero, producing the S-shaped curve 2. It is easy to solve the logistic ODE to find the function N(t) which satisfies equation (1):

N(t) = \frac{\kappa}{1 + e^{-\alpha t - \beta}}
\end{displaymath} (2)

where $\alpha$ is the growth rate; $\beta$ is the location parameter which shifts the curve in time but does not affect the its shape; and $\kappa $ is the saturation level at which growth stops.

Figure: The logistic growth of a bacteria colony. The three logistic parameters, $\kappa $, $\Delta t$, and tm, are labeled. Source of data: [19]

While $\kappa $ can be easily seen in a graph, $\alpha$ and $\beta$cannot. Accordingly, we replace them with two related metrics, the midpoint and growth time. We define the growth time, $\Delta t$, as the length of the interval during which growth progresses from 10% to 90% of the limit $\kappa $. Through simple algebra, the growth time is $\Delta t = \frac{\ln(81)}{\alpha}$. We define the midpoint as the time tm where $N(t_M) = \frac{\kappa}{2}$. Again simple algebra shows $t_m = - \frac{\beta}{\alpha}$, which is also the point of inflection of N(t), the time of most rapid growth, the maximum of $\frac{dN(t)}{dt}$.

The three parameters $\kappa $, $\Delta t$, and tm define the parameterization of the logistic model used as the basic building block for Loglet analysis

\begin{displaymath}N(t)= \frac{\kappa}{1 +
\text{exp} \left[ {-\frac{\ln(81)}{\Delta t}}(t -t_{m}) \right] }.

The parameters chosen have real physical interpretations when we graph them. In Figure 2, we show these parameters on a curve fit to the growth of a bacteria colony in a petri dish.

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Next: Visualization of the model Up: Logistic Growth and Substitution: The Mathematics of the Previous: Introduction
Perrin S Meyer