Complexity_ A Guided Tour - Melanie Mitchell [13]
Linear versus Nonlinear Rabbits
Let’s now look more closely at sensitive dependence on initial conditions. How, precisely, does the huge magnification of initial uncertainties come about in chaotic systems? The key property is nonlinearity. A linear system is one you can understand by understanding its parts individually and then putting them together. When my two sons and I cook together, they like to take turns adding ingredients. Jake puts in two cups of flour. Then Nicky puts in a cup of sugar. The result? Three cups of flour/sugar mix. The whole is equal to the sum of the parts.
A nonlinear system is one in which the whole is different from the sum of the parts. Jake puts in two cups of baking soda. Nicky puts in a cup of vinegar. The whole thing explodes. (You can try this at home.) The result? More than three cups of vinegar-and-baking-soda-and-carbon-dioxide fizz.
The difference between the two examples is that in the first, the flour and sugar don’t really interact to create something new, whereas in the second, the vinegar and baking soda interact (rather violently) to create a lot of carbon dioxide.
Linearity is a reductionist’s dream, and nonlinearity can sometimes be a reductionist’s nightmare. Understanding the distinction between linearity and nonlinearity is very important and worthwhile. To get a better handle on this distinction, as well as on the phenomenon of chaos, let’s do a bit of very simple mathematical exploration, using a classic illustration of linear and nonlinear systems from the field of biological population dynamics.
Suppose you have a population of breeding rabbits in which every year all the rabbits pair up to mate, and each pair of rabbit parents has exactly four offspring and then dies. The population growth, starting from two rabbits, is illustrated in figure 2.1.
FIGURE 2.1. Rabbits with doubling population.
FIGURE 2.2. Rabbits with doubling population, split on two islands.
It is easy to see that the population doubles every year without limit (which means the rabbits would quickly take over the planet, solar system, and universe, but we won’t worry about that for now).
This is a linear system: the whole is equal to the sum of the parts. What do I mean by this? Let’s take a population of four rabbits and split them between two separate islands, two rabbits on each island. Then let the rabbits proceed with their reproduction. The population growth over two years is illustrated in figure 2.2.
Each of the two populations doubles each year. At each year, if you add the populations on the two islands together, you’ll get the same number of rabbits that you would have gotten had there been no separation—that is, had they all lived on one island.
If you make a plot with the current year’s population size on the horizontal axis and the next-year’s population size on the vertical axis, you get a straight line (figure 2.3). This is where the term linear system comes from.
But what happens when, more realistically, we consider limits to population growth? This requires us to make the growth rule nonlinear. Suppose that, as before, each year every pair of rabbits has four offspring and then dies. But now suppose that some of the offspring die before they reproduce because of overcrowding. Population biologists sometimes use an equation called the logistic model as a description of population growth in the presence of overcrowding. This sense of the word model means a mathematical formula that describes population growth in a simplified way.
FIGURE 2.3. A plot of how the population size next year depends on the population size this year for the linear model.
In order to use the logistic model to calculate the size of the next generation’s population, you need to input to the logistic model the current generation’s population size, the birth rate, the death rate (the probability