Visualization of the model
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):

 (1)

For values of , equation (1) closely resembles exponential growth. As , 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):

 (2)

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 saturation level at which growth stops.

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 tm 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 tm 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.

Next: Visualization of the model Up: Logistic Growth and Substitution: The Mathematics of the Previous: Introduction
Perrin S Meyer
1998-07-14