Green Acres

Last month Mathalicious HQ packed up and headed to Austin, Texas, which now makes us a part of the single largest demographic group in the country.  No, not Texans.1  City folks.

Depending on how much faith you have in the Census, it turns out that something like 80% of all Americans live in urban areas.  For at least two reasons, that's crazy high.  First of all, it's almost impossible to get 80% of Americans to agree on anything, so it's borderline miraculous that so many people have agreed to share such a tiny fraction of the country's area.  Second of all, we've only been at this urbanization thing for a couple hundred years, and we started out extremely rural.

Back at the turn of the 19th Century, only about 6% of the U.S. population lived in anything resembling a city.2 Basically, we had a few ports on the eastern seaboard.  You know what?  Here's a fun, old-timey video that explains it nicely:

The point is, as soon as we got some decent cities underway, they started to expand.  And cities expand in an interesting way.  When urbanization is relatively low, cities grow roughly in proportion to their current size.  In other words, the larger the fraction of the total population that lives in cities, the more people3 end up moving to cities.  It's a positive-feedback loop.  Up to a point...

Because, no matter what else happens, one thing is certain: no more than 100% of the country's population can live in urban areas.  That means the percent of the total population in cities has an upper bound.  As urbanization approaches this bound, the rate of growth slows down.  All the things that seemed to entice people toward an urban life now seem to keep them in the country.

To recap: when the population of cities is small relative to total population, urban growth is fast.  But as cities get closer and closer to their carrying capacity, the rate of growth slows down.  If you look at the data over time, urban population (as a percent) has an S shape.

Populations that follow this kind of dynamic can be described by what's known as a logistic model.  It shows up all the time in ecology4, and it does a pretty impressive job of describing the history of U.S. urbanization.  Here (in blue) is 200 years' worth of census data showing the percentage of urban population.  And overlaid in orange is data generated by the logistic model.  Pretty good, right?

One of the reasons logistic models are interesting to mathematicians is that, even though they're extremely simple (notice there are only two terms on the right-hand side), they can exhibit a wide range of behaviors, some of which are really surprising.  You can vary the coefficients in the interactive below to see how they would affect future predictions for the urban population.

Of course these scenarios are purely hypothetical, but it's an informative exercise to look at different long-term outcomes and imagine what life in the America might look like.  If the current trend continues, what will society be like in the next century?  What if only half the population lived in cities?  What if cities took a nosedive altogether, and people moved en masse back to the country?

What kind of country (or city) would you like to live in?

Teachers, want to have this conversation in your class?  Check out the lesson materials for Green Acres on our site!


1.  Though I suspect that having everyone live in Texas is part of Texas's official five-year plan.

2.  The question of exactly what should constitute an urban area obviously wasn't quite so settled (!) back in 1800.  In fact,  the term "urban area" wasn't even officially defined until the 1880 census — and it's undergone some changes over the years — so a grain of salt is in order.  But you get the idea.

3.  Relatively.  As a percent of total population.

4.  If you took some higher-level math in college, I'll bet you $1 you modeled predator-prey dynamics with the logistic function in a Differential Equations course.


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