Sequences -- Logistic Models -- Stretch Your Understanding

You should use one of the computer algebra systems below with this module. Click on the appropriate icon for your preferred CAS and then arrange your screen so that you can easily move back-and-forth between this window and your CAS window. Click on the appropriate help button for help.

1. The behavior of a logistic model of the form

   p(n) = a [1 - .001 p(n - 1)] p(n - 1)

   changes very dramatically as the constant a changes. Try a number of different values of a in the range 0 <= a <= 4 and describe the behavior of these models.

2. Mathematicians like to study the models

   p(n) = a [1 - .001 p(n - 1)] p(n - 1)

   that are based on replacing the constant R in the exponential model by the the function

   R(p) = a(1 - .001 p)

   because these functions R(p) are easy to work with. Using a CAS system, however, it is easy to work with more realistic families of models -- for example, models based on the function

            a,                     if p < 500
   R(p) =  { 
            a - a(p - 500)/1500,   if 500 <= p.
   
   

   

   Experiment with models of this form trying different values of the constant a. Describe the results of your experimentation and compare these results with similar experiments based on logistic models.