Logistic Models and Cobweb Diagrams

In a logistic model the population multiplier is a linear function of the form

Graphically this function looks like

Notice that when the population is zero the population multiplier is a This number represents the largest possible population multiplier -- how fast the population would multiply when there was no competition for the available food, water, and other resources. Notice also that when p = C the population multiplier is zero and the population would crash abruptly. If p > C then the function says the population multiplier is negative, which of course makes no biological sense. When p > C the population multiplier is really zero.

A logistic model looks like

We can write this as

where

Click here to open a new window with a Java applet. Arrange these two windows so you can easily move back-and-forth between them. When you are done working on this module you should close the window with the applet.

If your browser is Java enabled you should see a Java applet in the new window. Otherwise you will just see two horizontal bars. This Java applet shows the function f(p) described above with a = 2.8 on the green graph paper. Both scales on the green graph paper run from zero to C. The scale on the bottom of this applet enables you to change the value of a. If you click on the far left of this scale a will be changed to 2. If you click on the far right of this scale then a will be changed to 4. Clicking on intermediate points on the scale sets a to values between 2 and 4. Each time you change the value of a the graph on the green graph paper will be redrawn. Experiment with different values of a to see the effect on the graph of f(p)

Notice the dot on the bottom of the green graph. It indicates the initial value p₁ of a model. If you click anywhere along the bottom edge of the green graph this dot will move to the point at which you click.

Notice that each time the initial value is changed a vertical line is drawn from the marked position of the initial value to the diagonal line. This is the start of a cobweb diagram. At the same time a dot is placed at the same point on the left side of the blue graph. This is the start of a graph of the sequence

To draw this graph and the cobweb diagram one step at a time click anyplace in the upper half of the applet. Experiment with different values of a and different initial values. You will seem some unexpected behavior.