“One of the most widely used chemical compounds is zinc oxide. This policeman, this farmer, and this housewife don’t realize it, but they all depend on zinc oxide in their daily lives.” – Kentucky Fried Movie
The Alaska range at dawn. Not pictured: sexy farmers.
Thomas Malthus was a very, very gloomy guy – so much so that the term after his name, “Malthusian” has come to describe dismal, teeming masses of poor hungry people. He did the math and saw that food production produced arithmetically, and there were limits of how much food a single acre could produce, and a limit to how much food could be produced overall. People reproduced geometrically, and could reproduce much faster than the food supply. Probably because farming was and is much less fun than sex.
Looks like Pastor Malthus would be more of a party guy, and less of a “we’re all doomed” guy. Via Wikimedia
Pierre François Verhulst was modelling populations based on his reading of that gloomy Thomas Malthus, and (after a bit of tinkering in the math world by some other folks) they ended up with:
N(t)=K/1(+CKe^-rt)
Math sometimes solves multiple problems with the same solution. And one of those solutions is the S-Curve (or “Logistics Function”). Originally, Verhulst found it. What irritates me about Verhulst is that his middle name has that French curly-cue thingy hanging off the bottom of a perfectly useful “c”. So, we’ll just call him Pierre for the next sentence until we’re entirely done talking about him.
Here is Pierre – and he approved this picture, which kinda makes him look like offspring of a parrot and a serial killer. – Via Wikimedia
If you think back to an earlier (relatively popular) post (LINK) r is the rate of population growth, and K is the carrying capacity. If you maximum mating as your evolutionary strategy, you’re an r critter, like a rabbit. If you have a few offspring, and guard them like the crown jewels, you’re a K critter. If you go back to the post linked above, you’ll see how this equation determines the fate of nations . . . but this post is about more than that.
The equation above is (kinda sorta) what I graphed to make the following curve:
It’s called an S-curve (or sigmoid curve) because it looks like an “s” that’s been stretched out.
So, you can imagine that as a population of bunnies gets dropped on an unsuspecting continent with no natural bunny predators, the population will skyrocket, as happened when rabbits were introduced to Australia. In 1859, a dozen or so escaped from a hunting compound, and instead of forming the rabbit version of the A-Team they started reproducing, because rabbits like sex more than farming, too. That’s the beginning of the curve. Small growth, numbers wise, at first. A dozen rabbits, two dozen, a hundred, two hundred . . . .
As the numbers of rabbits increase, they reach a peak of maximum growth – they’re moving outward and taking over more and more territory. At the end? Growth slows as numbers peak.
In 1920, there were estimated to be 10,000,000,000 rabbits in Australia. Ten billion. In sixty years. Right now, it’s estimated that “only” 200,000,000 rabbits survive in Australia. The rabbit population growth followed the S-Curve until people figured out ways to, well, kill billions of rabbits. If they stopped killing rabbits, you’d see 10,000,000,000 in just a few years – the rabbits would shoot back up the curve.
Rabbit – it’s Australian for girlfriend, and these rabbits are drinking from the beer ponds of South Australia.
But it’s not just the population of Foster’s® drinking rabbits that this equation is used to predict.
Innovation
In many ways, the curve itself is a mathematical model of innovation or novelty. If you look at the adoption of a technology, for instance, it’s very well described by the curve. The adoption of the automobile, the Internet, (by population) television, radio, and even language elements are all explained by the S-curve.
My parents were, in many ways, really late stage adopters of stuff. Ma Wilder never had a microwave during when I lived at home – even though every one of my friends did. Video tape players? They got one when I was in college. They may have been the last “new” VCR purchasers.
Why? Don’t know. Pop Wilder had (generally) a really awesome income. It’s not like he was out of money – and he bought all the fancy stuff like VCRs and televisions with remote controls after he retired.
But this measure also applies to adoption of any new technology. And it shows that companies must continually innovate or their income streams will stagnate. Apple™ has made tons of money on the iPod© and the iPhone® and the iPad™ . . . but innovation has slowed, greatly. And nearly every phone is now an iPhone© or an Android™. When Jobs was a live, it really was the Steve-curve, rather than the S-curve. Now it’s the $-curve. Wonder how long that will last them? If they get in league with the forces of darkness and evil, maybe they can put together the NecrinomoPhone©, kinda like an iPhone™ but used to put you in league with Demons. Or Facebook®. But I repeat myself.
Construction
S-curves are tools used by construction companies to measure progress on construction jobs. When you think about it, construction starts slow. There isn’t a lot of work that can be done on a house until the foundation is in. And then framing can start, and then, once framing is complete and the building is sheathed? Lots of people can come and do their work at the same time – plumbers and electricians can do work with the drywall crew following closely behind. There is a great amount of work that takes place in a short time, provided there’s enough Copenhagen® and Bud Light™.
But finishing is hard – the last 5% often takes 20% of the project’s schedule. That’s because the available places for work drop off. And the last bits of work have to be done sequentially – you can’t put the carpet in until you’ve textured the ceiling, unless you like crunchy carpet. The S-curve is awesome at predicting the average construction time of a project.
Software Projects
Software projects are similar to building a house, except half of the houses completed would immediately burst in to explosive flame as soon as you tried to lock the front door. Oh, and you’d be locked inside. Inexplicably, every month your bedroom would mysteriously appear on the outside of the house, but in a different place each time. Sometimes you would flush the toilet and the light would turn off. Unless the switch for the fan was on, and then it would flush, but be refilled with goat’s blood.
We should be glad that contractors don’t hire software engineers. But the S-curve still defines the progress to the exploding houses that software engineers create.
Crop Response
If you don’t water a plant, it won’t grow. If a plant doesn’t have a vital nutrient, it won’t grow. If the farmer is having sex for reproductive purposes, well, the plant might grow if he remembered to water it and fertilize it.
But the responses to water and these vital nutrients is . . . an S-curve. Too little of that stuff? Low growth rates. Just right? High growth rates – but maybe you want to avoid maximum growth rates if the incremental fertilizer is expensive. Sometimes maximum isn’t optimum. Just ask Gary Busey.
But crops respond with that same S-curve response to the addition of a vital nutrient, or, if you gradually add in an inhibiting factor like salt, it forms an inverse S-curve – a little salt won’t hurt the wheat, but eventually it kills it and no production is possible.
There are other physical things that S-curves apply to – such as learning a foreign language (interesting), machine learning (complex) and tumor growth rates (ugh).
S-Curves also show up in seemingly unrelated things . . . like names and Chicken Pox. Source – XKCD.com
But Malthus has (at least for the last 225 years) been wrong. Food production increases have been amazing – we’ve gone from famines caused by crop failures to the only famines that currently exist are entirely political in nature. Food production has been increased through farming mechanization, nitrogen fertilizer production, food genetics, pesticide application, and better irrigation.
The continued S-curve of food innovation has saved billions of lives. And birthrates in most of the world are falling – leading to a real possibility that Malthus will be forever wrong, except about the “farmers like sex” thing.
