3.7 Case Study: Population Modeling



An important purpose of this Case Study in Calculus is to give first-hand experience doing mathematics in an experimental setting. As such, the background to keep in mind is the Scientific Method which should form the context for the experiments. The U.S. population will be modeled on two different assumptions, called Malthus and Verhulst. The corresponding differential equations will be solved both numerically (approximately) and analytically (which is possible in these cases). Then the two models will be compared and a judgment will be made about which is the more realistic.

By the time you complete the CSC, you should know:


Suppose we want to model the growth of a bacteria colony. We know that the rate of change of the population (P) is directly proportional to the population. Early observations suggest that they are in fact equal. If the initial population is 100, use Euler's method with stepsize = 0.1 to predict the population in 6 hours.

After three hours, we observe the bacteria colony and find the population is now 1600. Use this information to solve the differential equation in the previous example, and find a more accurate constant of proportionality. How does this model differ from the previous one? Again, predict the population at t = 6 hours and compare with the previous estimate.

More research indicates that the growth of the bacteria colony is proportionally dependent on the amount of oxygen available. In the controlled environment where our sample resides, all the oxygen will be exhausted in 3 days (72 hours). Thus, our new differential equation modeling the bacteria growth is

Determine the population at time t = 6 hours, and describe the long term behavior of the bacteria colony.


Euler Population Predictions


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Interesting Application

The Cambrian period was 540 to 500 million years before the present. Question: Was diversity (measured as number of invertebrate genera) in the Cambrian period logistic?

3.6 Optimization Table of Contents 4.1 Modeling Accumulations

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Copyright © 2005 Donald L. Kreider, C. Dwight Lahr, Susan J. Diesel