3. Growth on a Nonrenewable Source (NONRENEW)

A nonrenewable source is based in a storage, a limited quantity of energy. If we use the example of a dead log, it is a storage of wood available to be eaten by beetles. The population of beetles will increase exponentially as they eat the log and the quantity of log will decrease. As the log is used up, the quantity of beetles decreases too.

In the diagram (Figure II-3a) E is the nonrenewable source, the dead log. The amount the log decreases is proportional to the quantity of log (E) and the quantity of beetles (Q); the equation for the decrease in log is:

DE = - K*E*Q

In a period of time, like a week, the amount of log will be the amount started with plus the change:

E = E + DE

The change in quantity of beetles is also proportional to the quantity of log (E) and the quantity of beetles (Q). The net increase of beetles (K1*E*Q) is their growth and reproduction (K2*E*Q) minus the effort used to get their food (K3*E*Q). The decrease in beetles, their death rate, is proportional to their quantity (K4*Q). Therefore, the change in quantity of beetles (Q) over time is their growth minus their death. The equation is:

DQ = K1*E*Q - K4*Q

At the end of a week the number of beetles will be the number at the beginning of the week plus the change:

Q = Q + DQ.

You get a graph (Figure II-3b) when you run the program. We have graphed both the change in the log (E) and the change in beetles (Q). Notice that the log does not decrease much until the population of beetles increases fast and then the log gets used up quickly.

Examples of Nonrenewable Models

This model represents any system with a limited source, which is used up. An economic example is a mining town living on a limited source of gold ore. As the ore is mined, the town grows on its revenues from the gold. When the mine runs out, the town economy decreases until everyone leaves and it becomes a "ghost town."

A person given an inheritance could fit this model. He uses the money to go on more and more trips. As the money runs out, his trips decrease and then cease.

"What if" Experiments

  1. What would happen to the beetle population if the log, which fell in the storm, were a bigger one? Would the population become larger or would it live longer or both? Write down what you think would happen, and then try it by changing statement 40 to: E = 180.
    Consider the mining town. Does this mean that with a larger quantity of ore the economy would grow faster and bigger or would the miners mine the ore slowly to last longer? If this model is general, what would you predict our world economy would do if large new deposits of fossil fuels were found? Would we use them up quickly to develop more, or would we conserve them to keep a steady economy longer?

  2. What would happen to the log (E) and beetles (Q) if you started with 100 times more beetles? Try changing statement 50 to: Q = 10. Explain this experiment using the mining town example.

  3. Try a different species of beetles, which have a more efficient growth rate. Change K1 to 0.0015. What happens to Q? And what to E? Then discover another beetle species that is less efficient: change K1 to 0.0005.

  4. What would you change to increase death rate of the beetles? How would E and Q change? Try it and explain your results.


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

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