### Valentine's Grouse

As Thomasina struggles with her new geometry, there is a parallel mathematical development taking place in the play. Valentine is trying to use ideas from chaos theory to explain the rise and fall of the population of grouse on the Sidley Park Estate. He knows the data about grouse kills on the estate for the past two hundred years, and he would like to extrapolate from this to predict the populations in the future. Curiously, he is using the exact same technique that Thomasina had experimented with years before. Well, not quite. As Valentine explains, "Actually I'm doing it from the other end. She started with an equation and turned it into a graph. I've got a graph --- real data --- and I'm trying to find the equation which would give you the graph if you used it the way she used hers. Iterated it. It's how you look at population changes in biology. Goldfish in a pond, say. This year there are x goldfish. Next year there'll be y goldfish. Some get born, some get eaten by herons, whatever. Nature manipulates the x and turns it into y. Then y goldfish is your starting population for the following year. Just like Thomasina. Your value for y becomes your next value for x. The question is: what is being done to x? What is the manipulation? Whatever it is, it can be written down in mathematics. It's called an algorithm."

One of the simplest such algorithms used by population biologists is the logistic equation given by Fk(x) = kx(1-x). Here x represents the population in question normalized so that x lies between 0 and 1. The constant k is a parameter; we would use one value of k for grouse, another for rabbits, and a third for elephants. Given an initial population x0 and a particular value of k, we can then iterate Fk to find the populations in successive years. For example, if k=1.5 and x0=0.2, then we find in succession

x0 = 0.2
x1 = 0.24
2 = 0.2736
x3 = 0.2981...
x4 = 0.3138...
.
.
.
x20 = 0.3333...
x21 = 0.3333...
so that the population has eventually stabilized at 0.3333.... If, on the other hand, we select k=3.2 and x0 = 0.2, then after several iterations we find that the population begins to cycle back and forth. One year the population is high; the next year it is low:
x0 = 0.2
.
.
.
x{20} = 0.7994 ...
x{21} = 0.5130 ...
x{22} = 0.7994 ...
x{23} = 0.5130 ...
Like Thomasina, we can turn this data into a graph by plotting the time series corresponding to this iteration. This is a plot of the iteration count versus the actual numerical values. The cycling behavior above is illustrated in Figure 7.

Fig. 7. A time series indicating cyclic behavior.

When we choose k=4 and x0 = 0.2, the results of iteration are anything but predictable (Figure 8). The time series for this iteration shows no pattern whatsoever. This is the phenomenon of chaos. Here we see that a very simple iterative scheme can yield results that are totally unpredictable.

Fig. 8. A time series indicating chaotic behavior.

Summary and References (Next Section)