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 $N(t)$:

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

$$N(t) = \frac{\kappa}{1 + e^{-\alpha t - \beta}}$$ (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 1: The logistic growth of a bacteria colony. The three logistic parameters, $\kappa$, $\Delta t$, and $t_m$, 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 $t_m$ 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 $t_m$ define the parameterization of the logistic model used as the basic building block for Loglet analysis

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

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.