Objective:

Generalizing the population function of Lesson 1, we consider here the population model , where parameter r takes a value in between 0 and 4. In this lesson, we present the stable population level observed in the range r = (0, 3) and the remaining range r = (3, 4) will be explored in Lesson 10. First, all initial populations eventually die off when r1, so that the population model gives rise to total extinction x = 0 in the range r = (0, 1). However, it can sustain a growing population level for r1. This is depicted by the round curve that starts at r = 1 and increases to reach r= 2/3 at r = 3.

Over the entire range r = (0, 3) shown above, we can say the population level is stable because any initial population would settle down to x = 0 for r1 and for r1.

Scary words: Stable and unstable fixed point, graphical iteration, population extinction.

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