## 3.7 Case Study: Population Modeling

### Summary

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:

• What the Scientific Method is.
• The assumptions of the Malthus model and the resulting differential equation.
• The assumptions of the Verhulst model and the resulting differential equation.
• Using actual U.S. population census data from 1790 to 1990, how to find the approximate value of the solution curve by matching the population in 1990 (for Malthus and Verhulst, in turn).
• How to solve the Malthus and Verhulst differential equations analytically and thereby find the solution curves exactly.
• How to compare and interpret the predictive results of the approximate and exact solutions.
• How to compare and interpret the predictive results of the Malthus and Verhulst models.
• How to write a CSC report, especially the Interpretation and Summary.
• This CSC as another example of applying mathematics you have learned to solving a real-world problem.

### Examples

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.

### Applets

Euler Population Predictions

### Videos

See short videos of worked problems for this section.

### Exercises

See Exercises for 3.7 Case Study: Population Modeling (PDF).

Work online to solve the exercises for this section, or for any other section of the textbook.

#### 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?

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