Discrete Logistic Models

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.

Exponential models of the form

p(n) = R * p(n - 1)

make unrealistic predictions because they say that the growth rate from generation to generation is constant. In reality many factors affect the growth rate -- most importantly, the size of the population. When the population is small there is plenty of food, water, and shelter and the population will grow rapidly but when the population is large there will not be enough food, water, and shelter and the population will grow more slowly or may even decline. Thus the constant R in the change equation

p(n) = R * p(n - 1)

should be replaced by a function R(p) that depends on the population. This gives us a model of the form

p(n) = R(p(n - 1)) * p(n - 1)

In general, the function R(p) will look something like the function below.

When the population is low then R > 1 and the population will rise for the next generation. When the population is high then R < 1 and the population will fall for the next generation. Notice that in this particular example R(p) is constant when the population is relatively small. This is quite common. A very small population in a large habitat will reproduce as fast as biologically possible until the population is high enough so that the effects of crowding begin to be felt.

Mathematically, the simplest functions R(p) are linear functions like the function shown below.

Linear functions have the form

R(p) = a(1 - b p)

leading to models of the form

p(n) = a[1 - b p(n - 1)] p(n - 1)

These models are called discrete logistic models and are very rich and interesting. They appear to be very simple and yet they exhibit some extraordinarily complex behavior. The figures below show several examples.

Check your Understanding

For each of the following logistic models calculate

p(1), p(2), ... p(20)

Note that these are the same models shown in the graphs above.

1. p(1) = 10

   p(n) = 2 [1 - 0.001 p(n - 1)] p(n - 1)

   answer

2. p(1) = 10

   p(n) = 2.8 [1 - 0.001 p(n - 1)] p(n - 1)

   answer

3. p(1) = 10

   p(n) = 3.4 [1 - 0.001 p(n - 1)] p(n - 1)

   answer

4. p(1) = 10

   p(n) = 3.54 [1 - 0.001 p(n - 1)] p(n - 1)

   answer

Stretch Your Understanding