Mathematics & Science Learning Center
Computer Laboratory

Applications of Differential Equations

Population Dynamics

An Exercise for the Student

(continued from last page...)

A very wise man by the name of Leonard of Quirm was once overheard making the statement: "To say that the population of Sto Helit breeds like rabbits is to impugn the reproductive restraint of our fluffy woodland friends!" You're about to get an opportunity now to analyze the very population he was referring to.

Sto Helit Reality

Census data for the city-state of Sto Helit was gathered by the Burgermeister's Executive Council from the year 1400 until the dissolution of the Council in 1480. The census data is given below in ordered pairs of the form {year, population in thousands}:

{{1400, 29.63}, {1410, 32.88}, {1420, 36.90},
{1430, 40.32}, {1440, 45.99}, {1450, 48.09},
{1460, 52.91}, {1470, 58.79}, {1480, 65.41}}

Your job is to analyze the Sto Helit population in exactly the same way that we analyzed the United States population while I was helping you. You may need to look back fairly frequently at how we made certain steps when we did the example. To jump back you can use either the bullets in front of the step summary or the Table of Contents for this lab, accessed, as usual, by clicking on the compass button at the bottom of the page.

Your primary goal is to finish up with the three graphs, reality, Malthusian model, and Verhulst model, all displayed on the same coordinate system in different colors so that you can get a visual impression of how well the two models did in fitting the actual population data. Obviously getting as far as making this picture requires a multitude of preliminary steps. A quick summary of the main steps follows:

Summary of Main Steps in the Exercise

Click on the bullet preceding each step to go to the corresponding step
in the U.S. population example.

1. Reality

   	Copy and Paste the data from here into Mathematica, and assign it a variable name. I'd suggest a name like SHreality since you have already used the name reality in this session, and it would be all too easy to get things muddled. (In fact I'd suggest a similar naming convention for all of the variables you are about to introduce.)
   	Normalize the matrix that you entered by reducing all of the year data so that the first year becomes time zero, etc.
   	Create a named plot of this new matrix in an appropriate color.

2. Malthus

   	Solve the Malthusian initial value problem using the new data's first entry as an initial condition. Read the result into an appropriately named variable, and strip away any extra information using the usual trick.
   	Find the value of the remaining parameter by solving an appropriate equation using the FindRoot command.
   	Substitute the result back into the previous step and read the result back into the variable representing the Malthus solution.
   	Create a named plot of this new solution in an appropriate color.

3. Verhulst

   	Solve the Verhulst initial value problem using the new data's first entry as an initial condition. Read the result into an appropriately named variable, and strip away any extra information using the usual trick.
   	Find the values of the remaining parameters by solving an appropriate pair of equations using the FindRoot command.
   	Substitute the result back into the previous step and read the result back into the variable representing the Verhulst solution.
   	Create a named plot of this new solution in an appropriate color.

4. Comparison

   Display the graphs of reality and both models all on the same picture using the Show command.

Once you have the final comparative picture you can decide upon the answers to questions such as which model seems to fit reality better, and why did Leonard of Quirm make the statement that he did? As a final exercise you should use both models to predict the population of Sto Helit in the year 1500.

You're now finished with this laboratory. You can now quit altogether, go back and start over, or go to the general table of contents for all of the laboratories.

If you're lost, impatient, want an overview of this laboratory assignment, or maybe even all three, you can click on the compass button on the left to go to the table of contents for this laboratory assignment.