Bruce Hannon, U of IL, Urbana, IL 61801, b-hannon@uiuc.edu, June 99.

At the beginning of the book we discussed the importance of positive and negative feedback processes for the dynamic behavior of systems. Much attention has been given by researchers to negative feedback processes, because negative feedback processes lead systems towards equilibrium states. In this chapter we turn our attention to positive feedback processes. Positive feedback reinforces a given tendency of a system and can lead a system away from equilibrium states, possibly causing quite unexpected results.

Here is a simple model of a common process with a surprising result. To our knowledge, the first to point out this model was the economist Brian Arthur in his article on Positive Feedbacks In the Economy. Brian Arthur cites positive feedback processes as driving forces in determining which of two (or more) alternative new technologies will come to dominate a market. A classic example is the development of the two video technologies VHS and Beta. Both systems entered the market at roughly the same time and, early on, maintained almost the same market shares. It was not clear which of the two technologies would ultimately dominate the market. However, if one succeeded to increase its market share slightly over the other, its increased presence would make it more attractive for households to buy that technology. In response, video rental businesses and stores would carry a slightly larger amount of tapes and recorders for the respective system, which, in turn, makes it more attractive for new buyers, thereby increasing its attractiveness to prospective buyers even further.

The forces which determine what technology increases its market share during the first periods are often small and may appear to be random. Initially small differences in market shares ultimately increase until one system is almost entirely eradicated from the market. The video-technology, indeed, followed that trend, with VHS being the market-dominating technology today. Similar developments can be observed for a number of other technologies or products that have close substitutes which enter the market at the same time.

A potential application of positive feedback in ecological systems can be found in a study on The Impact of Preemption on the Zonation of Two Typha Species Along Lakeshores. Here we have the competition between two shore-water oriented plants which spread through root growth. If either of the two are the only plants in an area, that plant will become the sole plant growing in that area, however if the plants are equally present after a clear cut disturbance, either plant may dominate but most likely, however, the plant which does the best in shallow water will take over that area while the other will take the slightly deeper water area adjacent to its competitor. So we have here a mixture of positive feedback and adaptive advantage.

Brian Arthur illustrates the process of positive feedback with a simple model. The idea here is that balls of two different colors are chosen at a time and placed on a table. We start with one ball of each color on the table. The rule for choosing the next color is given by a function of the fraction of balls already on the table. If the fraction of balls of a given color is larger than a random variable which lies between zero and one then another ball of that color is added onto the table. By adding another ball of that color its fraction is increased. As a result, for the next drawing it is even more likely to draw another ball of the very same color... Positive feedback prevails.

In our model, the balls are either blue or red. We start with one of each color on the table. The system is described by two state variables which are represented by the stocks BLUE and RED. From these stocks we can calculate the fraction of each color. The STELLA II model is shown in the following diagram and the details of the calculation are listed at the end of this section in the model equations. Before you look them up, try to model the two-color positive feedback model yourself.

The fraction of each color oscillates wildly and then settles down to a particular value. However, this value is different on consecutive runs! Such a phenomena is thought by Arthur to be similar to a specific technology which comes to dominate (or disappear from) the market. The process is a matter of luck at least in part.

Set up a sensitivity run for the blue fraction, 20 runs with the initial values of the BLUE and RED always the same, at one. Do you see a trend? Now do the whole analysis over with the initial value in both colors set at ten. Now do you see a trend? What is going on here? Note that the initial ratio is the mean value of the fixed fractions, and the number of initial balls sets the standard deviation on the distribution of the fixed fractions. 

BLUE(t) = BLUE(t - dt) + (ADD_BLUE) * dt
INIT BLUE = 1 {Numbers of Balls}

INFLOWS:
ADD_BLUE = IF RANDOM_NUMBER < BLUE_FRACTION THEN 1 ELSE 0
{< means positive feedback with many fixed points of red/blue fraction.
> means negative feedback and the fractions always go to 0.5.
measured in Balls per Draw}
RED(t) = RED(t - dt) + (ADD_RED) * dt
INIT RED = 1 {Numbers of Balls}