So, let's review a bit. A simple iterative rule can lead to
stability or fluctuation or chaos, depending on what value one
chooses for a parameter of the iteration (in the present case,
k, the steepness of the inverted curve relating current to next
population size). The figure below gives you a summary of that,
one you can play with not only to review but also to discover
new things on your own. The left part of the figure is the
staircase sort of diagram you've seen before. The right part of
the figure is something new, a "bifurcation diagram", which
helps to visualize changes in the behavior of the system as the
value of k varies. The parameter k is plotted along the x-axis
of the bifurcation diagram (from 1 on the right to 4 on the
left). What the curve shows is the population behavior for every
k value when starting at a particular initial population size
(x). For small and medium k values (to the left) what the
bifurcation diagram shows is that (as we found earlier), the
population reaches a single, sta