Background

    The exponential model    is used to study uninhibited population growth and solution is the exponential function  .  When the term   is added we obtain the logistic differential equation which is used to model inhibited population growth or bounded population growth.  The logistic differential equation is

            .  

One form of the solution is  

            .  
            
The terms have been carefully determined so that the initial condition is  

            .

The limiting value  L  of  y(t)  is given by  
    
            .

The graph is the so called "S-shaped" curve. The choice of parameters    creates the curve shown below.  

Proof  Logistic Differential Equation  Logistic Differential Equation  

Harvesting a Logistic Population

    When the harvesting term  -k  is incorporated into into bounded population model we have
    
            .  

There are three solution forms for this differential equation, and they correspond to the nature of the stationary solutions  ( x(t) = c).  

Definition(Stationary Points)  The stationary points of the D. E.    are solutions where and are the roots of the characteristic equation  

            .

The roots are known to be  ,  and the stationary solutions are  .  

Remark.  Since  x(t)  is a real function, there are no stationary solutions when  .  

Case (i) If     there is one stationary solution .

    When  ,  the differential equation has the form    and the solution is  

            .  

The solution with the initial condition    is

            .

If    then   .

If    then function  x(t)  has a vertical asymptote at  

and the population   x(t)  becomes extinct at some time    (where ),  i. e.  

            .

Proof  Harvesting Model  Harvesting Model  

Case (ii)  If   there are two stationary solutions    and  .

    When  ,  the differential equation has the form    and the solution is  

            .       

The two real roots of the characteristic equation  ,  are  .  

The solution with the initial condition    is

    .

If      then   .  
    
If      then the population   x(t)  becomes extinct at some time  ,  i. e.   .  

Proof  Harvesting Model  Harvesting Model  

Case (iii)  If   there are no stationary solutions.

When  ,  the differential equation has the form    and the solution is  

            .  

The solution with the initial condition    is

            .

The function  x(t)  has a vertical asymptote at   so the population   x(t)  becomes extinct at some time    (where .), i.e.  

            .

Proof  Harvesting Model  Harvesting Model  

Numerical Solutions

    If only the graph of the solution is required, and the formula is not needed, then an efficient way to solve the differential equation is with a numerical method such as Modified Euler's method or the Runge-Kutta method.  The choice of method depends on the accuracy required.  If an accurate table of numerical values is required then the Runge-Kutta method should be used.  

Computer Programs  Harvesting Model  Harvesting Model  

Program (Modified Euler's Method)  To compute a numerical approximation for the solution of the initial value problem with over at a discrete set of points using the formula .

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

Example 2.  Find the roots of  ,  and equilibrium solutions of the D. E.  .
Solution 2.

Case (i) If     there is one stationary solution .

Example 3. Solve the population model with harvesting  ,
using the constants  a = 2,  b = 1,  and  k = 1,  and explore this situation.
Solution 3.

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

Example 5.  Plot some more solutions to the D. E.    in Example 3
Plot those that have the following initial conditions  .
Solution 5.

Example 6.  Discuss the graphs in the plot in Example 5.
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 ?
Solution 6.

Example 7.   Solve the D. E.    using the formula with the initial condition  .  
Solution 7.

Example 8.   Use the Modified Euler Method to find numerical solutions to the D. E.  .  
Solution 8.

Case (ii)  If   there are two stationary solutions    and  .

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

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.
Solution 10.

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

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

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 ?
Solution 13.

Example 14.   Solve the D. E.    using the formula with the initial condition  .  
Solution 14.

Example 15.   Use the Modified Euler Method to find numerical solutions to the D. E.  .  
Solution 15.

Case (iii)  If   there are no stationary solutions.

Example 16.  Solve the population model with harvesting  ,  
using the constants  a = 3,  b = 3,  and  k = 3.
Solution 16.

Example 17.  Plot the solutions to the D. E.    in Example 17
that have the following initial conditions  x[0] = 1, 2, 3, 4, 5, 6.
Solution 17.

Example 18.   Solve the D. E.    using the formula with the initial condition  .  
Solution 18.

Example 19.   Use the Modified Euler Method to find numerical solutions to the D. E.  .  
Solution 19.

Research Experience for Undergraduates

The Logistic Curve  The Logistic Curve  Internet hyperlinks to web sites and a bibliography of articles.  

Harvesting Model  Harvesting Model  Internet hyperlinks to web sites and a bibliography of articles.  

Download this Mathematica Notebook The Harvesting Model

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(c) John H. Mathews 2004