Introduction

Logistic Growth and Substitution: The Mathematics of the

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
until the approach of a limit or capacity 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
as a nonlinear function of *N*(*t*):

For values of , equation (1) closely resembles exponential growth. As , the feedback term slows the growth to zero, producing the S-shaped curve

where is the growth rate; is the location parameter which shifts the curve in time but does not affect the its shape; and is the

While
can be easily seen in a graph,
and cannot. Accordingly, we replace them with two related metrics, the
midpoint and growth time. We define the **growth time**, ,
as the length of the interval during which growth progresses
from 10% to 90% of the limit .
Through simple algebra, the
growth time is
.
We define the
**midpoint** as the time *t*_{m} where
.
Again simple algebra shows
,
which is also the
point of inflection of *N*(*t*), the time of most rapid growth, the
maximum of
.

The three parameters ,
,
and *t*_{m} define the
parameterization of the logistic model used as the basic building
block for Loglet analysis

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.

- Visualization of the model
- Bi- and multi-logistic curves
- Taxonomy of bi-logistic curves
- Generalization of the Bi-logistic model
- Implementation in Loglet Lab