This is a much more complicated animation based on a slightly more complicated model for population rate of growth, but this slight mathematical complication will lead to great conceptual complications.

Our previous rate of population change was given by

DP / Dt = (800 - P)P/1600.

Now we add a new term, one that theorists devised to take account of predation, the idea that some of the little pink love spheres may in the course of time ... well,... to put it bluntly ... be eaten. The specific form of this new term I got out of a book, but the constants I devised to help illustrate certain ideas, and specifically the idea of population outbreak. Here's the new rate of growth:

DP / Dt = {(800 - P)/1600 - kP/(10000 + P2)}P.

Here k represents a constant that is to be set by the animation viewer - namely, you. You choose values of k by clicking on the little rectangles at the bottom of the animation. There are six of them, and the values they give are k = 6.25B for B=13,14,15,16,17,18, respectively. The default value is B = 15. It can be reset at any time, even in the middle of clicking through generations. In fact, that will be the point of all this.

In the box to the right of the rectangular buttons is the present population (if it seems to be different from what you are observing, that's maybe because the window got moved or covered, which can cause some of the pink love spheres to get erased). You are allowed 80 generations (that's how many times you can press the generations button (upper left)). As usual the lower right button will lead to a graph of the population over time. Oh, one last point: each time a pink love sphere is eaten a red-eyed leering thing appears, so these may be considered anti-pink-love-spheres. OH! One more last point: by clicking and holding the mouse while the cursor is in the middlish portion of the animation stage, you can arbitrarily (like a pink love sphere deity, if you will) increase the population. You'll get the hang of it, and we'll be showing this animation on future pages.

Well, we have a lot to talk about with this one, so take some time to play with it, and we'll go into some technical details in the coming pages.