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Summary of the population dynamics up to r = 3 We plot the results of tables 4 and 5 in figure 4 to summarize the population level for r = (0, 3). For r
Figure 4. Fixed-point diagram of the population model We now examine in detail population dynamics for the two values r = 0.5 and 2, which are indicated by a round dot in figure 4. First, for r = 0.5 we have already observed that three initial x0 = 0.3, 0.5, and 0.8 get iterated to zero population in table 4. This is typified in figure 5 by the successive iterates resting at x = 0, whereas the initial transient iterates are tentatively shown by a dotted line. One cannot help but notice a striking similarity between zero iterates in figure 5 and trailing decimal digits of 3/4 = 0.75000..., which are also zeros. Hence, figure 5 has a perfect analogy to the bouncing soccer ball that comes to rest at the center hollow of a concave floor (figure 4 of Lesson 8).
Figure 5. Successive iterates for r = 0.5 Next, in table 3 for r = 2 we have also observed the iterations eventually settle down to x = 0.5 from several of the initial x0, and this is schematized in figure 6 by a doted line connecting x0 = 0.3 to x = 0.5. Here, we compare successive iterates at x = 0.5 with the repeating decimal digits of 1/3 = 0.333... (figure 1 of Lesson 8). We, therefore, infer that successive iterates of figure 6 have period 1.
Figure 6. Successive iterates for r = 2 |
1 population model (1) cannot support life, so that any initial population eventually dies off to give zero population x = 0. However, as r becomes larger than one, there is a steady growth of population by 
, reaching x = 2/3 at r = 3 as shown in figure 4.
