Consider the discrete equation

This is called a discrete logistic equation, or the logistic map. We can compare this with the continuous logistic equation in Example 5.19.

[This is useful to model population dynamics when we consider births at discrete time intervals only.]

We are only interested in . We can sketch the graph of against .

From the graph, if , then stay within .

The fixed points satisfy . Hence, the fixed points are

However, be aware that the second fixed point only makes sense for (in order to be non-negative).

We can carry out perturbation analysis to determine their stability.

Therefore,

• For the fixed point at :

  • stable if ,
  • unstable if .

• For the fixed point at :

  • stable if ,
  • unstable if .

We can illustrate the behaviour of solutions using cobweb diagrams.

• Consider .

  

  This shows that is a stable fixed point: solutions for any initial condition in will converge to .