How To Tame a Function

If you grew up around animals, you probably know a bit about their reproductive cycles. Seeing as I did not, I have been slow to learn what I know about the Birds and the Bees of the birds and the bees. In the absence of really any first-hand contact with animal life, I have had to resort to books for my basic education in animal reproduction.

But what I’ve learned has deepened my understanding not just of the nitty-gritty biology facts but of the fairly abstract mathematics of chaos.

Here’s a bit of biology I didn’t know: the rules of mating for domesticated animals like dogs and cattle are different than they are for their wild cousins. The big difference is the timing. Wild animals often have a narrow reproductive window during a certain time of the year. But the biological changes that come along with tameness somehow also bend the rules of pairing off.

Here is how it’s described in the thrilling “How to Tame a Fox (and Build a Dog)”:

“All wild animals breed within a particular window of time each year, and only once a year. For some, that window is as narrow as a few days and for others it’s weeks or even months. Wolves, for example, breed between January and March. The window for foxes is from January to late February. This time of year corresponds to the optimal conditions for survival; the young are born when the temperature, the amount of light, and the abundance of food offer them the best odds for a successful launch into the world. With many domesticated species, by contract, mating can occur any time during the year and for many, more than once.”

This is of also course true of domesticated apes, i.e. us, the human people.

Imagine an experiment in population growth. We take a small group of wild animals to a protected island. These wild animals have abundant food and no predators on the island. A pretty sweet deal, all said. We set these wild beasts loose to eat, drink and … you know, have fun.

Well, they do have fun, and the population grows. But this is a wild species with an extremely narrow mating season. They can only reproduce once a year. But when they do, they give birth to big broods. This is a “nonoverlapping generation,” and its mathematics happens in nice, even steps. We can calculate the size of each generation one step at a time:

Pop_{n+1} = Pop_{n} \times r

But as Thomas Malthus pointed out way back at the turn of the nineteenth century, a good thing can’t last. If the population grows like this, it will quickly use up all of the resources in this island paradise. In which case, the population will be unable to continue to grow.

One of the first people to put Malthus’ ideas into math was Pierre Verhulst. He described “logistic growth” (intended to echo with unrestricted “logarithmic growth”) as a simple (if arbitrary) way to slow down overpopulation. The key is that the environment can only handle so much of a species — its “carrying capacity,” K — and each additional individual in the population slows the population down by an equal amount.

Pop_{n+1} = Pop_{n} \times r \times (1 - \frac{Pop_{n}}{K})

(Ben Orlin has a very clear exposition of the logistic in “Change Is The Only Constant.”)

You may be familiar with the logistic’s famous S-shape:

However, take care! The S-shape curve is not “wild” logistic growth, which happens in strictly nonoverlapping discrete generations. No, the S-curve is the tame, domesticated growth of Golden Retrievers, Angus Cattle, and American People who reproduce with more flexibility. You could even say, in the case of humans and other such species, that their populations are continuously increasing.

Ah, no worries though. Discrete functions are just like continuous functions, minus the continuity. They’re the dots without the lines. They aren’t meaningfully different, are they?

To be sure, sometimes the discrete function makes a nice smooth S, gliding into the carrying capacity without much fuss:

Thank you to the Desmos user who had the patience to write all those compositions of functions.

Then again, sometimes our island of wild animals with nonoverlapping generations ends up bouncing around the carrying capacity, each year their population crashing or rising above that set parameter:

It all depends on that growth rate. For some values of the growth rate, this oscillation actually converges on that carrying capacity, resulting in nice agreement between the discrete and continuous cases:

Ah, but pump that rate of increase up high enough and you get the real fun, which is utter chaos:


The population almost goes extinct for a minute there, but then experiences a nice bout of exponential growth. That’s chaos for you.

Animal species themselves can be wild or domesticated. It turns out that functions can be domesticated as well. In this case, it’s the continuous version of this function that is the tamer, better behaved variety.

There are indeed animals that are best modeled by the continuous function. There was a time when some people were very gung-ho about the logistic. In the 1920s Raymond Pearl declared the continuous, nice S-shaped logistic to be the “true law of population growth.”

It’s not. Robert May pointed out in the 1970s that the discrete case, though extremely simply, exhibits a huge range of behavior. “Their rich dynamical structure,” May writes, “and in particular the regime of apparent chaos wherein cycles of essentially arbitrary period are possible, is a fact of considerable mathematical and ecological interest, which deserves to be more widely appreciated.” The wildness of the discrete function may be a better fit for species that have nonoverlapping generations — many insect populations give birth in strict, discrete steps, like cicadas that emerge only ever thirteen years.

Chaos is in some ways a very abstract phenomenon, but learning more about ecology and population growth has made it very real for me. I’d seen the logistic map many times in the past, but I don’t think I’ve ever quite understood it until I connected it to its origins in animal populations. I’m left with a lesson for my teaching: there are some mathematical ideas that just work better when you learn them in their biological, natural setting.


Steven Strogatz, “Nonlinear dynamics and Chaos”

Charles J. Krebs, “Ecology: The Experimental Analysis of Distribution and Abundance”

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