A population which has all the food it needs will increase faster and faster because the more individuals, the more mouths, the more food consumed and the more growth of the whole population. This is called exponential growth. Exponential growth is a constant percent increase in each time period.

An example is a small population of laboratory mice given food pellets in containers, which are kept filled no matter how much the mice eat. The more they consume the more is supplied, and the faster the population increases. Each week there are even more mice. As long as the necessities are available, the number of mice will increase exponentially. Since this kind of unlimited food supply is not possible indefinitely, eventually the population of mice would stop growing so fast. Then we would have to use a different model to fit the new situation of a limited food supply.

In the diagram (Figure II-a); E is a source that keeps a constant concentration, no matter how much is used from it; it is relatively unlimited. Q is the quantity which is using E. In our example, E is the continuous supply of food and Q is the mice.

The interaction symbol (the fat arrow with an * in it) shows that the mice are eating the food to produce more mice. Because the increase of mice is dependent both on the food source (E) and on the number of mice we already have (Q), the more mice you have the more they eat and produce baby mice.

The equation for the increase in Q is K1*E*Q.

K1 is the proportion of Q*E which becomes new mice each week; it is the growth rate coefficient. K1 is a combination of two coefficients, K2 and K3. The increase in Quantity of mice depends on their growth and reproduction (K2*E*Q) minus the effort they expend in getting their food and water (K3*E*Q). K1*E*Q is the net growth (K1 = K2 - K3).

K4 is the death rate coefficient, the proportion of Q which dies. Q is the number of mice who die each week, the death rate.

Therefore, the change in quantity of mice over time (DQ) is the increase (KI*E*Q) minus the decrease (K4*Q):

The quantity of mice, (Q) after a week is the number we started with plus the change:

You get a graph (Figure II-b) when you run the program showing changes in Q; the population (Q) increases slowly and then more and more quickly.

Examples of Exponential Models

This is a model that correctly describes the growth of populations of plants or animals with unrestricted resources. During the early states of population growth, when the demand for food is small compared to the amount available, almost any population of plants or animals will grow exponentially. World human population growth has been exponential until recently and still is in some countries.

Industries, like oil and mining, have grown exponentially when new oil fields and sources of gold were found. We could even point to the United States, from the early 1800´s to 1900's as an economy which was grown on of natural abundant resources and newly discovered fossil fuels.

"What if" Experiments

Let's make some changes in the life of our population of laboratory.

1. If you double the concentration of food, what will happen to the growth of the mice population? Predict and then try it. You will type 60 E-2. Now cut the concentration of food (E) in half - each pellet has half the nutrition. What happens to the mouse population?
   


2. What if you increase the growth rate of the mice? Perhaps the researcher has found another kind of mice that eats more efficiently. What will the graph of Q look like? Try it. Type 70 K1 = 0.08. Then try a population of mice who are less efficient eaters. You will change statement again, making K1 less than the original 0.07. What does that graph look like?
   


3. Another change, which might occur is in the death rate. What would happen to the growth of the mice population if the mice caught a virus increasing the death rate? To test your hypothesis, would you increase or decrease K2 ? Show what happens to the growth of the population. Then make them more healthy than the original population by changing K2 in the other direction.
   


4. If E is 1 and you make K1 equal to K3, what will happen to the population? Try it.

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