Harvesting Model

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.  

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.  

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.   
.  

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.  

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.  

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
.