Topic 7.9: Logistic Models with Differential Equations

Why rumors slow down

Now let’s go back to the town and find the rumor’s rule, the thing that tells us how fast it spreads at any moment. Every text from a kid who knows to a kid who doesn’t makes one more kid who knows. A text between two kids who both already know does nothing at all, and it just fizzles.

Early on, almost every text reaches somebody new, but hardly anybody is sending them, so the rumor crawls. Late, it’s the other way around. Everybody’s texting, but nearly every text lands on someone who’s already heard, so the rumor crawls again.

That means the spreading rate depends on how many kids know and how many don’t. A text spreads it only when someone who knows texts someone who doesn’t, so if either group is tiny almost nothing spreads, and multiplying the two numbers does exactly that. If P is the number of kids who’ve heard, the number who haven’t is a thousand minus P. That gives the rumor’s rule, d P d t equals k times P, times a thousand minus P, for some positive constant k.

In our town k is one ten-thousandth. At six o’clock two hundred kids know and eight hundred don’t, and one ten-thousandth times two hundred times eight hundred is sixteen, exactly the rate the town showed us. Now try nine hundred kids. A thousand minus nine hundred is only one hundred, so nine hundred times one hundred, divided by ten thousand, is nine kids a minute, slower than the sixteen at six even though far more kids know.

We call this a logistic differential equation. The thousand is its carrying capacity, the value this population approaches as time goes on. AP questions describe this with a phrase worth knowing by heart, a rate that’s jointly proportional to P and to a thousand minus P. Jointly just means multiplied together, exactly the way we found it in the town. In general it’s d y d t equals k y times K minus y, where big K is the carrying capacity and little k is just a positive constant.

We can read a lot from this equation without ever solving it. When P is zero the rate is zero, because nobody knows, so nobody can tell anyone. Anywhere between zero and a thousand both factors are positive, so the rumor keeps spreading. Right at a thousand the rate is zero again, because there’s nobody left to tell.

That’s why, starting from two hundred kids, the number who’ve heard heads toward a thousand as time goes on, getting closer and closer without ever quite reaching it. On the exam you’d write that as the limit of P of t, as t goes to infinity, equals a thousand. Before I show you, I want your prediction for when the rumor is spreading fastest.

Let’s start small. With two kids who know and eight who don’t, each of the two could text any of the eight, so there are two times eight, sixteen texts that could spread it. Now scale it up to the whole school. Take a bar a thousand kids long, split it into the kids who know and the kids who don’t, and build a rectangle with those two lengths as its sides.

The area of that rectangle counts every possible pair made of one kid who knows and one kid who doesn’t, which is every text that could spread the rumor. Watch the rectangle as I slide the split. Lopsided splits give skinny rectangles, and the biggest area of all comes when it’s a square, which happens when exactly half the kids know.

Multiply that area by k and you get the spreading rate. That makes the graph of the rate against P a downward parabola, highest at five hundred kids, where the rumor spreads at twenty-five kids a minute.

If an exam question asks you to justify it, this sentence earns the point. The rate is a quadratic in P that opens downward, with zeros at zero and a thousand, so it’s greatest at P equals five hundred, halfway between them.

There’s a second way to see it, and it tells us the shape of the curve. Multiply out the rule, then differentiate with respect to time. The chain rule tacks a d P d t onto each P, so the second derivative is one ten-thousandth times a thousand minus two P, times d P d t.

Since d P d t is positive the whole way, the sign comes from a thousand minus two P, which is zero at five hundred. At two hundred kids that’s six hundred, positive, and between five hundred and a thousand it’s negative. That means the rumor’s curve is concave up until five hundred kids and concave down after, and that bend, a point of inflection, is the moment the rumor switches from speeding up to slowing down.

We’ve proved it happens at five hundred kids, but we still don’t know when. Drag the clock to the moment it spreads fastest, and watch the rate meter or just listen to the pings.

Now run an experiment of your own. Before you do, guess whether more starters would change when the rumor peaks, how many kids know at the peak, or both. Pick how many kids start the rumor at five oh five, run the morning, and see where it spreads fastest this time.

For our own rumor, about six fourteen came from watching the town, though, and on a test there’s no town to watch. We’ll get that time ourselves before the end, with nothing but the rule.

Where it ends

A rumor can’t start above its carrying capacity, but a fish population can, if someone overstocks a lake. Let’s switch to a lake where the population follows d P d t equals zero point eight P minus zero point zero zero two P squared. That doesn’t look logistic yet, so I’ll factor out zero point zero zero two P. It becomes zero point zero zero two P times four hundred minus P.

Zero point eight over zero point zero zero two is four hundred, so the carrying capacity is four hundred fish, not zero point eight. You’ll also see it written as zero point eight P times one minus P over four hundred. It’s the same equation, and the number dividing P is still the carrying capacity.

On its slope field the rows at zero and at four hundred are perfectly flat, because the rate is zero there, which makes them the two constant solutions. Suppose we stock the lake with six hundred fish, more than it can support. What happens to the population as time goes on?

Notice that the curve from six hundred is decreasing and concave up, so it isn’t an S at all, which is why you always check where the population starts before you sketch it. The last start to check is zero, because a lake with no fish stays empty forever. That’s why the limit depends on the initial condition, with every positive start heading to four hundred while zero stays zero.

That also tells us where the lake grows fastest. Starting from fifty fish, the rate peaks at half of four hundred, two hundred fish, and the lake really does pass through two hundred on its way up. Starting from six hundred, though, it never gets down to two hundred, so the answer always depends on where it starts.

Real forecasters have bet on this model. In 1920 two scientists at Johns Hopkins, Raymond Pearl and Lowell Reed, fit a logistic curve through the census counts of 1790, 1850 and 1910. They checked it against the censuses in between, and their curve leveled off at about a hundred ninety-seven million people. For a while it looked like a prophecy, because it called the 1930 census almost exactly.

Then the real country kept going, past their ceiling by 1967, and by July 2024 it had reached about three hundred forty million. A carrying capacity is only as good as the assumptions behind it.

Forecast the rumor

Now we have everything we need to do what we couldn’t do at the start, which is forecast the rumor ourselves, five minutes at a time. The first step is the forecast you already made, two hundred kids spreading it at sixteen a minute, which gives two hundred eighty at six oh five.

Next we re-check the rate at our new point, the way Euler’s boat re-checks the current. With two hundred eighty who know and seven hundred twenty who don’t, the rule gives twenty point one six kids a minute. Take the next step yourself, and drag the forecast to Euler’s estimate for six ten. The calculator’s there if you want it.

Let’s let it keep going. At six fifteen our estimate is about four hundred ninety-nine kids. Five minutes later it’s past six hundred. Our forecast crosses five hundred just after a quarter past six, so that’s our estimate of the fastest moment. The true curve crosses five hundred at about six fourteen, so with nothing but the rule and a few five-minute steps, we got within about a minute.

Now you decide how much work the forecast is worth. Pick the step size that gets six ten within ten kids of the model, using the fewest steps. Our forecast puts the fastest moment just after a quarter past six, and our proof explains why it happens at five hundred kids.

Snow day

Let’s put everything on one page. Euler’s method adds h times the slope at the old point, with h equal to the interval’s length divided by the number of steps. On our river, smaller steps gave better estimates every time. A logistic quantity heads to its carrying capacity, big K, from any positive start. A solution that climbs through half of K grows fastest right there, and the trap on the exam is answering with K itself.

As for our town, the sun’s coming up, almost every window is lit, and the snow is finally stopping. For the record, the district really did cancel school, and they made it official at six forty-five, when about ninety-six percent of the kids had already heard.