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Home >Opinion >Columns >Take this 20,000 Tyrannosaurus at a time

A familiar dilemma, again: in this time of a surging pandemic, with news by the hour of illness and tragedy, what’s a mathematics columnist to write about? The virus and the pandemic, even though the news is chock-full of that already? Or something else altogether, and is that being escapist?

I don’t have a good answer. So this is about something that intrigued me recently. I’ll leave it to you to search for parallels to the working of a virus.

You’ve heard of Tyrannosaurus Rex, the awe-inspiring dinosaurs that roamed the Earth a few years before you were born. Of course, they were not the only dinosaurs roaming, but the evidence shows that T-Rex was about the largest, certainly the most formidable. It was a carnivore, and it probably worked up a decent speed as it chased prey, so it’s likely other animals around it lived in fear for their lives.

But still, how much of a danger was T-Rex? I ask that in this sense: I would view an agitated crane as a possible threat. After all, it’s a big bird with a long and sharp beak it can use to damaging effect. But there are no cranes that I know of anywhere near me. Besides, their overall numbers planet-wide are too low to worry me; it’s unlikely that I ever will be accosted by a posse of aggressive cranes. Of ants, or humans, I couldn’t say the same. They exist everywhere I might choose to go, some of them aggressive.

Think of T-Rex in the same way. When it roamed the planet, was it like the crane, found only here and there? Or was it like ants and humans, found in plenty pretty much everywhere? Or let’s get down to numbers: at any given moment, how many T-Rex dinosaurs were there on Earth? Thousands, like cranes? Billions, like humans? Trillions, like ants?

Charles Marshall, a University of California (Berkeley) professor who led the team, described it this way: “When I hold a fossil in my hand, I can’t help wondering at the improbability that this very beast was alive millions of years ago, and here I am holding part of its skeleton—it seems so improbable. The question just kept popping into my head, ‘Just how improbable is it? Is it one in a thousand, one in a million, one in a billion?’ And then I began to realize that maybe we can actually estimate how many were alive and, thus, that I could answer that question."

But how do you estimate such numbers about an animal like T-Rex, that died out millennia ago, when all we have are fossils and skeletons of this remarkable creature? Still, there are over 40 of those. That’s actually enough to come to some reasonable conclusions about such things as T-Rex’s life expectancy, its weight and how widely it ranged.

The team used Damuth’s Law (1981), which relates a species’ body size, or weight, to how densely it populates its geographic range. Of course there are factors other than body weight that matter here. Still, think of Damuth’s Law as a general thumb-rule that tells us how many individuals of a given species a given area can support.

Broadly, the bigger the species, the lower its population density. Intuitively, this seems about right. Imagine a colony of ants: you’ll find them in roiling swarms. But have you ever seen elephants in similar swarms? At most, you can expect a small family-centric group of them, maybe 10 animals. Damuth’s Law is captured in a simple mathematical formula. It says that if you have an animal about 80 times as heavy as a second animal, its population will be about 1/27th as dense as the population of the first animal. If the first is about 10,000 times as heavy, its population density will be about 1,000 times less than the second.

Again, this seems about right, taking humans and ants. If you are about 10,000 times as heavy as an ant (you’re actually heavier than that), the ants make up for that avoirdupois deficit with sheer numbers. They’re found in collections that are certainly 1,000 times as dense as humans.

Back with T-Rex: from fossils and skeletons, the scientists estimated factors like the age at which the dinosaur reached sexual maturity and the proportion of the population that survived to a given age. These were useful in determining that an adult ("postjuvenile") T-Rex weighed between 3,700 kg and 6,900 kg, averaging about 5,200 kg. That number plugs directly into Damuth’s Law to produce a population density of about one T-Rex in every 110 square kilometres.

How much is that? Let’s say T-Rex had found its way to our subcontinent. This analysis says that there would be about 30,000 adults spread across India—2,800 in what is Maharashtra today, about 13 running around in what is now Delhi. Compare as well to estimates that this is about a sixth of the density of tigers. 200,000 tigers wandering about India in a time before hunting and all our “development" decimated their numbers? Sounds reasonable.

From the spread of the locations where T-Rex fossils have been found, the scientists estimated the geographical area that T-Rex actually occupied: about 2.4 million sq. km. Multiply by the dinosaur’s population density and we have the interesting figure I alluded to above: on a given day, there were about 20,000 T-Rex individuals alive. So really, going by those numbers, T-Rex was more like cranes than humans. You wouldn’t want to run into one, of course. But it was very unlikely that you would have anyway.

The scientists used this number to calculate another interesting one. Again from the fossil record, our best guess for how long T-Rex survived on Earth as a species—its “temporal duration"—is 2.4 million years. Now take the average “generation time" for an adult T-Rex. That’s not quite its lifespan, but a figure that takes into account when it became sexually mature, how many offspring it produced and its chance of surviving to a given age. Divide temporal duration by generation time and we have an estimate of how many generations of T-Rex actually lived on our Earth, and that number is about 125,000. So, if during each of those generations, there were 20,000 T-Rex individuals alive, that gives us the total number of T-Rex dinosaurs that ever lived on Earth: about 2.5 billion.

Which sounds like a lot, but isn’t really. For example, we estimate that about 108 billion humans have ever lived; and no, I don’t really want estimates for cranes and ants.

But finally, spare a thought for how dinosaurs are thought to have disappeared. One theory is that about 65 million years ago, a large celestial object smashed into the Earth, leaving the Chicxulub crater in the Yucatan Peninsula in Mexico. Of course this destroyed life in that region. But the dust from the collision spread around the world and lingered for years, causing lasting environmental change. That change is what killed any remaining dinosaurs.

20,000 T-Rex on Earth before that crash. None within a few years afterward. Think of that.

Once a computer scientist, Dilip D’Souza now lives in Mumbai and writes for his dinners. His Twitter handle is @DeathEndsFun

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