Dynamical Systems

Worksheet 9--Classwork

Introduction to the Logistic Model

So far, we have looked at the simplest type of dynamical system, with a linear recursion rule. Now we will study a family of systems known as "the logistic model." These systems are often used to model animal population growth. There are two parameters, a and b, and the recursion rule is quadratic rather than linear.

P_n+1 = a*P_n(1 - b*P_n)

Until further notice, keep parameter b set at .001, so the rule will look like this:

P_n+1 = a*P_n(1 - .001*P_n)

You only need to worry about the effect of parameter a on the behavior of the system. For openers, the value of paramter a will be 1.8:

P_n+1 = 1.8*P_n(1 - .001*P_n)

1) With the value of parameter a set at 1.8 and then assign different seeds (P₁) to different group members. Run some iterations and share your data. Do all seeds have the same ultimate fate? Make sure you try a wide variety of types of numbers.

NOTE: It is possible to use the graphing calculator to get these numbers fairly easily. Type in some number as your seed and hit enter. Then you type in the expression carefully, using the parentheses where indicated and hittng the 2nd-ANS key in place of the variable P_n in the formula. After that, you can just hit enter to get repeated iterations.

2) You should have found in #1 that some seeds will lead to a sequence that grows without bound, whereas others will produce a convergent sequence. If you didn't find that, then you were leaping to conclusions without enough data! Those seeds that lead to a sequence without bound can be referred to as members of the escape set. Try to define carefully the full range of the escape set. Which seeds belong to the escape set and which ones don't?

3) Do you believe that escape set will still be the same if you change the value of a? Say why or why not. Pay attention to the arithmetic involved in the recursion rule.

4) Try a few different a values in your group, and check to see if the same seeds are in the escape set.

5) You may not have considered the possibility of parameter a itself being a negative number. See if that changes your argument for #3.

6) From now on, you should avoid the escape set and stick with P1 values between zero and a thousand. Also, keep the value of parameter a positive. Try a=.8; sprinkle several different seed values out among your group members, and see what kind of sequences you get. Would you say that the system with a=.8 behaves like the system you saw with a=1.8?

7) Now have each group member experiment with some value of parameter a that's between 0 and 1. Do they all produce the same type of behavior? Is there any reason why they should?

8) Move on to positive a values greater than 1 but less than 4. Experiment. Figure out what types of things you'd like to check on when you go to the computer room later this week.