Whenever two or more different items are required for a process, the relationship is shown with the interaction symbol. The output is a new product and the process is called production. Increasing the amount of any one of the required inputs increases the output, unless the rate of supply of another factor is limited. As shown in Figure II-6a, the two required factors are indicated by the letters: I and N. The amount of N available to the production process depends on the supply provided by flow J. If one increases the source pressure 1, the production increases rapidly at first, but as I is increased further, the other factor N is used up faster than it can be supplied by inflow J. In this situation N is the limiting factor because of the fixed, limited rate of inflow J. The flow of J is the limiting factor keeping production from increasing further.

Whereas most of our simulations graph quantity with time, the graphs in Figure II-6b show production as the one factor I is increased. The limiting factor graphs shown are found in all fields of science and called diminishing returns in economics. There are thousands of experimental studies with graphs like this.

Examples of External Limiting Factors

An example is the production of grass in a field. I is light intensity (insolation), J is inflow of nutrients from rain, and N is nutrients in the soil next to the plant roots. When light first increases, more grass is produced. But, soon production is limited by nutrients: even with more and more sunlight, there can be no increase in production. The amount of production depends on the supply of available nutrients in this case from rain.

Another example is production of electricity in a coal power plant where the inflow of coal is limited because there are a fixed number of available miners. Coal is the limiting factor. No matter how many buildings and furnaces are built, production of electricity is controlled by the supply of coal.

"What if" Experiments

1. In the external limiting program what is the effect of increasing the inflow of limiting factor J?
   After one run if you type CONT, the program increase J and runs again. The result is a similar curve that goes a little higher because the limiting inflow J is larger. The program does this several times making family of curves each representing the way productions is limited by J when J in increased. (On the Macintosh, to type CONT, the Command window at the bottom of screen must be active. If it is not, pull down the Windows menu to Show Command.)
   


2. Think of a different example. What are J, I and P in your example?
   Run the original program with the CONT runs and explain the results in terms of your example.
   


3. How and why would the graph be different if the production process was more efficient?
   Increase K1 and run the program.

COMPUTER MINIMODELS AND SIMULATION EXERCISES FOR SCIENCE AND SOCIAL STUDIES

Howard T. Odum* and Elisabeth C. Odum+
* Dept. of Environmental Engineering Sciences, UF
+ Santa Fe Community College, Gainesville

Center for Environmental Policy, 424 Black Hall
University of Florida, Gainesville, FL, 32611
Copyright 1994

Autorização concedida gentilmente pelos autores para publicação na Internet
Laboratório de Engenharia Ecológica e Informática Aplicada - LEIA - Unicamp
Enrique Ortega
Mileine Furlanetti de Lima Zanghetin
Campinas, SP, 20 de julho de 2007