Confidence Intervals on the Estimated Parameters: The Bootstrap

Logistic Growth and Substitution: The Mathematics of the

The Logistic Substitution Model

In our Introduction we promised that loglets could analyze the rise, leveling and fall of competitors substituting for one another. The ``species'' do not have to be organisms in an ecosystem; rather, they can be technologies and products competing in a market. For example, we can think of different modes of transportation (horses, trains, cars, airplanes, etc.) as competing in the same market. Our discussion will focus on competing technologies rather than species.

The **logistic substitution model** describes the fraction of the niche or
market share of the competitors. The life cycle of a competitor can be
partitioned into three distinct phases: growth, saturation and decline.
The growth and decline phases represent logistic growth processes, which
as we will see, influences the saturation phase.

The assumptions behind the logistic substitution model, as developed by Nakicenovic and Marchetti[11,14], are:

- New technologies enter the market and grow at logistic rates.
- Only one technology saturates the market at any given time.
- A technology in saturation follows a non-logistic path that connects the period of growth to its subsequent period of decline.
- Declining technologies fade away steadily at logistic rates uninfluenced by competition by new technologies.

The first assumption implies that growth can be modeled with an S-shaped logistic. The fourth also implies that the decline phase can also be modeled with a logistic with a negative . The second and third allows us to determine saturation behavior by competition from emerging technologies.

As an example, we apply the logistic substitution model to the
American recording media market. (This example is also featured in the
Tutorial; these data are included with the Loglet Lab software, along
with other data which illustrate concepts of logistic analysis.)
The shift in market dominance from vinyl records (LP's) to cassettes
to compact discs (CD's) proved to be a remarkably orderly
substitution, as can be seen in Figure 9. Figure
9 plots the number of LP's, cassettes, and CD's sold from
1977 to 1996. Because the logistic substitution model uses market
shares as opposed to absolute shares, it is general enough to be
useful for illustrating *competition* of species, products, or
technologies in systems where the assumptions of logistic growth and
decline hold.

- Numerical methods for logistic substitution
- Assessing the possible impact of new technologies
- Implementation in Loglet Lab