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.
The simplest realistic population models are models like the logistic model
p(1) = 50
p(n + 1) = 2.8 [1 - .001 p(n)] p(n)
in which the population is "pulled in" or "attracted" to an equilibrium point. In this case the equilibrium point is
p = 1.8/.0028 = 642.86
When a sequence like this one "approaches a value" like 1.8/.0028 then we say that it has a limit, 1.8/.0028, and we write
Lim p(n) = 1.8/.0028 = 642.86 n --> oo
Notice that the two words limit and bound have different and very specific meanings to a mathematician, although nonmathematicians often use these words interchangably. In this section we use the word limit descriptively. There is a more precise and detailed discussion of this idea elsewhere.
For any reasonable discrete dynamical system
p(1) = _____
p(n + 1) = f(p(n))
we have the following theorem
Theorem:
If L is a limit of the sequence p(1), p(2), ... then L is an equilibrium point.