Stable Population growth in the range r = (1, 3)

Obviously, solution (4) is valid for any value of r which means that the round curve and diagonal always start out from the common origin x = 0. However, as soon as r becomes larger than 1, the round curve can intersect with diagonal at another point on the x-axis, which is given by solution (5). So, function (1) now has two fixed points x = 0 and for r 1. Although x = 0 was the stable fixed point for r 1, it has become an unstable fixed point as r is larger than 1, so that it pushes away iterates from approaching x = 0. Hence, is now a fixed point of function (1), as shown

First you evaluate in the second column entries of table 5. By Prog#9 you can determine the final iteration values in the third column, and also record the maximum iteration numbers in the last column of table 5. Did you use big enough iteration numbers to reach the final iterate of the second column? In particular, we must point out that a maximum of 14,000 iterations is not really large enough for r = 3, as evidenced by final iterates 0.66467 and 0.66866, which oscillate about the target value 0.66666.

Table 5. Stable fixed points for r = (1.2, 3)

[prev] [ 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7] [next]

back to the lesson list