Let's, for example, keep the inverted curve but make it steeper, as shown in the figure to the right. The "steepness" of the curve can be measured as "k", the slope of the tangent to the curve at the origin (the line labelled k in the figure, steeper here than in the earlier one), and regarded, as the maximum possible rate of population growth. For small values of k, order and stability are the norm. For larger values of k, things change in important ways. One example of the kinds of change that occurs with steeper curves in shown in the figure. The staircase no longer steps progressively to a recurring stable value. Instead, it hits a point on the curve which causes it to hit a second point on the curve which returns it to the original point which returns it to the second point which ... In short, with a slightly steeper curve relating population size in one generation to population size in the next, what results is not a fixed and stable final population, but rather a population which forever oscillates between two different values.
Is that REALLY true, that the same basic simple idea (that population size in one generation determines population size in the next according to an inverted U-function) can lead EITHER to stability or to fluctuation, depending on nothing more (or less) than one number, the k parameter, the steepness of the curve? Well, you certainly shouldn't take THAT assertion on faith, so you should again ask Serendip to do some plotting for you (you should click here even if you still have the extra window open from the previous page).
Notice that the curve is steeper than it was before. The k value is larger, 3.3 instead of 2 as in the previous example, as you can see by looking to the right of the lower scroll bar on the control panel (you can change k yourself with the scroll bar, and if you did before, that's why you didn't always see a simple staircase). Again, clicking below the horizontal axis will select a starting population size, and you can calculate subsequent population sizes by clicking "iterate" (and see them plotted above). Notice though that this time one doesn't get a simple staircase. Instead, after some initial values, one sees some kind of a box, meaning the population is bouncing back and forth between values in the vicinity of two distinct values (where the box intersects the inverted curve). You can see this as well as a permanent oscillation in the values of the population size plotted against time (above).
Depending on exactly what value you start with, it will take a while for the population oscillation to stabilize between exactly two values. Prior to this, the box may get larger or smaller for a while. But it eventually stabilizes (always with the same two values? Try it). If you're not sure whether it has stabilized, you can magnify parts of the figure (by click-dragging across a region you are interested in) and then keep iterating to see whether you get new lines or simply redrawing of the old ones). You can do this for smaller k values too, to verify that the population really does settle to one value in this k range, as we asserted earlier. And, if you get tired of clicking the "iterate" button so many times, you can tell Serendip how many iterations you want it to do by entering a number in the "Iterations" box and clicking "Go". By telling Serendip not to plot some of the initial iterations (enter number in the "Skip first" box), your plots may more clearly show the stable patterns that appear later.
Convinced? Stability OR fluctuation can result from the same basic set of rules, with small changes in one parameter? Interested? Want to know more? Be patient, there's one more step to go before you'll have seen how many DIFFERENT things this simple system can do, and we can begin to try and understand why. But before going even that one more step, let's pause and see what (again) we have already.