Question 23.91

image 61. (Requires programmable calculator, spreadsheet, BASIC programming, or preferably the use of software listed in the Suggested Websites section) The behavior of the logistic population model increases from 0 to 4, the system changes from one behavior to another through the following possible states:

  • The population simply dies out.
  • The population tends toward a nonzero equilibrium point.
  • The population oscillates between 2 points.
  • The population oscillates between 4 points, then 8 points, then 16 points, and so on.
  • The population oscillates between numbers of points that are not powers of 2, until at last the population oscillates between three values.
  • The population behaves chaotically.

Explore what happens for various values of between 0 and 4, trying to identify where the shifts in the system’s behavior take place.

61.

Period 2 begins at , period 4 at , period 8 at 3.544, period 3 at , and chaotic behavior onsets at about 3.57.