Preliminaries. We shall study the population model with harvesting .

Example 1. First, enter the D..E. into Mathematica and solve it.

Example 2. Second, enter the characteristic equation and find the roots.

There are three possibilities, equal real roots, distinct real roots, and complex roots.
We desire that the D. E. has one or two constant solutions, with a real constant.

Example 3. Case (i) One critical point. Suppose that .

Show that there is one root of the characteristic equation , which is

Show that there is one constant solution to the D. E. which is ,

and that .

Example 4. Solve the population model with harvesting
using the constants a = 2, b = 1, k = 1, and explore this situation.

The constant solution is:

The general solution is:

Example 5. Plot some solutions to this D. E.
The constants for solutions with the initial condition x[0] = 2, 3, 4, 5, 6 are:

The constants for solutions with the initial condition are:

Example 6. Discuss the graphs in the above plot.
What are the vertical lines ?
What are the curves that lie below x = 1.
What are the curves that lie above x = 1 ? What use are they ?

In order to clear things up, it is necessary to specify the individual domain,
for each of the solutions, then plot all the curves on the same graph.

Delete the output graphs given above and report the following composite graph for your report.

Example 7. Case (ii) Two critical points. Suppose that .

Then there are two real roots of the characteristic equation ,
they are .

Example 8. Solve the population model with harvesting
using the constants a = 4, b = 1, k = 3, and explore this situation.

The constant solutions are:

The general solution is:

Example 9. Plot some solutions to this D. E.
The constants for solutions with the initial condition x[0] = 4, 5, 6, 7, 8 are:

The constants for solutions with the initial condition are:

You need to get the above list of functions. If you can't seem to do it, then type them in !

The solutions with the initial condition are:

You need to get the above list of functions. If you can't seem to do it, then type them in !

Discuss the graphs in the above plot.
What are the vertical lines ?
What are the curves that lie below x = 1.
What are the curves that lie above x = 3 ? What use are they ?

In order to clear things up, it is necessary to specify the individual domain for
some of the solutions, then plot all the curves on the same graph.

Delete the output graphs given above and report the following composite graph for your report.

Example 10. Look at the above graph and summarize what happens for the various initial conditions x[0]>0.

(c) John H. Mathews, 1998