III.7  Exercises

Exercise 1 Optional

Starting in 1790, the U. S. population has been measured every 10 years.  The following data have been collected.

Year    Pop      Year    Pop     Year    Pop
1790    3.9      1860   31.4     1930   122.8
1800    5.3      1870   38.6     1940   131.7
1810    7.2      1880   50.2     1950   150.7
1820    9.6      1890   62.9     1960   179.3
1830   12.9      1900   76.0     1970   203.3
1840   17.1      1920   92.0     1980   226.5
1850   23.2      1920  105.7     1990   248.7

We want to find a model for the U. S. population growth.

Part A

Find an exponential model for the above data.  First, use only the data from the period 1790 through 1860.  Once you have this exponential model built, examine how well it conforms to the data after 1860.  Comment.  Graph the data points and the exponential curve on the same set of axes for comparison.

Next, build an exponential model that gives a best fit for all the data.  Again, graph the data and the curve on one set of axes for comparison.  Comment.

You may find the Fit function in the package Statistics`LinearRegression` helpful.  You can begin by building a list of data points.  One way to do this is

Part B

Build a logistic model for the population data, .  In order to determine the values of the constants and , you can use the fact that is a linear function of P in this model.  You can approximate for any interior value of t by computing the difference quotients .  Then you can get a least squares fit to the difference quotients.  Graph the data and the model on one set of axes.  Comment on the suitability of this model.

Exercise 2 REQUIRED

The equation gives a logistic type model of a population with natural growth rate , carrying capacity and constant harvesting rate ( positive).

Part A

The value is called the critical harvesting rate.  Show that the population becomes extinct if H exceeds this critical harvesting rate.

Part B

Plot portraits of solution curves in the quadrant for three cases: no harvesting (); subcritical harvesting (use ); and supercritical harvesting (use ).  Let , 0, and choose various values for the initial population.

Exercise 3 REQUIRED (part B is optional)

Part A

Now consider the differential equations

which were discussed briefly in the lab material.  How many equilibrium points does this system have?  Starting at the point (1,2), graph the solution curve as t ranges from 0 to 12.  Try a few other starting points and do the same thing (modify the upper bound on t if needed).  What do you conclude?  Can you think of a real-world situation which would make the term plausible (perhaps not!)?  Explain.

Part B (OPTIONAL)

Now consider a system in which the prey species is limited with a logistic growth model, say

)

Draw solution curves beginning at several different starting points and explain what is going on in the system.