How do we understand what is happening? To begin we rewrite Eq. (2.29) as

and investigate this as a map relating N(t + 1) to N(t). Clearly if N(t) = 0, then N(t + 1) = 0; also if N(t) = K(1 + r)/r, then N(t + 1) = N(t). In Figure 2.10, I have plotted this function, for three values of r, when K = 100. I have also plotted the 1:1 line. The three curves and the line intersect at the point (100, 100), or more generally at the point (K, K). Using this figure, we can read off how the population dynamics grow. Let us suppose that N(0) = 50, and r = 0.4. We can see then that N(1) = 60 (by reading where the line N = 50 intersects the curve). We then go back to the x-axis, for N(1) = 60, we see that N(2) = 69.6; we then go back to the x-axis for N(2) and obtain N(3). In this case, it is clear that the dynamics will be squeezed into the small region between the curve and the 1:1 line. This procedure is called cob-webbing.

What happens if N(0) = 50 and r = 2.3? Well, then N(1) = 107.5, but if we take that value back to the x-axis, we see that N(2) is about 89. We have jumped right across the steady state at 100. From N(2) = 89, we will go to N(3) about 111 and from there to N(4) about 82. The behavior is even more extreme for the case in which r = 3: starting at N(0) = 50, we go to 125 and from there to about 31; from 31 to about 95, and so forth.

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