Module for the Harvesting Model

Numerical Methods for O. D. E.  using Mathematica. (c) John H. Mathews, 2003

Preliminaries.

We now study the population model with harvesting  .  

Example 1.  Use Mathematica to solve the D. E.  .  

Example 2.  Find the roots of   .
They will be the equilibrium solutions or constant solutions to the D. E.    in example 1.
       and   .  

The three cases.

    There are three possibilities, equal real roots, distinct real roots, and complex roots.

We require that the D. E. has one or two constant solutions, with a real constant.
    
Case (i) One critical point.  Suppose that  .

There is one root of the characteristic equation  ,  which is  .  

Case (ii)  Two critical points.  Suppose that  .  

There are two real roots of the characteristic equation  ,  they are .  

Case (iii)  Complex solutions.  Suppose that  .  

This case does not fit the real world situation at hand.

Example 3.  Case (i) One critical point.  Suppose that  .  
Then there is one root of the characteristic equation  ,  which is  .  
We can replace in the D. E.   and obtain

    

The solution    has the property that  .  

The following example illustrates the situation in example 3 with  .  

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

Example 5.  Plot the solutions to the D. E.    in example 4
that have the following initial conditions  x[0] = 2, 3, 4, 5, 6.

Example 6.  Plot some more solutions to the D. E.    in example 4
Plot those that have the following initial conditions  .

Example 7.  Discuss the graphs in the plot in example 6.
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.

Example 2.  Case (ii)  Two critical points.  Suppose that  .

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

They will be the equilibrium solutions or constant solutions to the D. E.    in example 1.
       and   .  

Example 9. Solve the population model with harvesting  .  
using the constants  a = 4,  b = 1,  and  k = 3,  and explore this situation.

Example 10.  Plot the solutions to the D. E.    in example 9
that have the following initial conditions  x[0] = 4, 5, 6, 7, 8.

Example 11.  Plot the solutions to the D. E.    in example 9
that have the following initial conditions  .

Example 12.  Plot the solutions to the D. E.    in example 9
that have the following initial conditions  .

Example 13.  Discuss the graphs in the plot in example 12.
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 each of the solutions, then plot all the curves on the same graph.

Example 14.  Look at the above graph and summarize what happens for the various initial conditions  x[0] > 0.

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