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 (i).

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

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

            . 

  The solution with the initial condition    is

    .

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

Proof (ii).

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 (iii).

 Return to Numerical Methods - Numerical Analysis

(c) John H. Mathews 2004