What is Exponential Decay?

A mathematical thought experiment. Warning — this is a rather gloomy post.

 Humans’ incapacity to understand the principle of exponential growth has caused us all kinds of grief of late (notably in our handling of the pandemic), and promises to cause us a great deal more soon (the end of cheap energy, climate and ecological collapse, interest rate shocks, and ‘peak everything’).

We tend to use the metaphor of a hockey stick, but it’s a poor one — that’s a model of sharp linear growth, not exponential growth at all. A better metaphor is the ‘rice on the chessboard’ reward story. The story goes that the inventor of chess, when asked by her ruler what she would like as a reward for coming up with the amusing game, asked for one grain of rice on the first square of the board, two on the second, four on the third, and doubling the amount on each square until all 64 squares were filled. The total amount of rice in this exponential reward turns out to be 1.5 trillion metric tonnes, more than all the grain ever produced on earth.

The human population chart above is one illustration of exponential growth. If you want an even more extreme one, a chart of global resource consumption would track the same curve until roughly 1800, and then would soar six times faster — imagine the chart stretched six times taller with the same line reaching to the upper right corner.

Population agencies at one point believed the human population would level off at eight billion. Then they changed it to nine billion. Now they’re saying eleven billion, at which point it will, supposedly by sheer collective human will, suddenly flatten out, about half a chart higher and one small dot to the right, and then stay there.

That is of course preposterous. Human population is no longer able to increase exponentially, but is still increasing linearly by 80 million per year — another billion every 12 years. The limit is not human will, but the capacity of the planet to — ever more wastefully — produce ever more human food and other resources. As long as we are producing more human food each year (at increasingly catastrophic costs) we will continue to produce more babies to consume it. Human population, and resource consumption (which is still increasing exponentially), will then suddenly stop increasing, as it did in the 14th century, but because of the current growth slope, it will happen much more abruptly. Since we’ve used more than half the world’s resources already and are using the rest up at an ever-accelerating rate, that point will certainly come before 2100, even if inevitable economic, climate and ecological collapse were not additional factors drawing this insanity to an imminent close.

And then what? Since there’s no safety valve this time, no frontier to escape to, what will immediately follow exponential growth of population and consumption will be something you hear much less about — exponential decay.

We’re starting to see how exponential decay works in the case and death counts for CoVid-19 in many countries. We saw a bit of it last summer, and politicians were delighted and prematurely declared the pandemic beaten. This time, despite the fact so many people have been propagandized to fear the vaccines, and despite the slowness and incompetence of the vaccine roll-out, we are seeing sharp drops in case rates in many countries that “mirror” the sharp increases we saw earlier, and this is happening long before herd immunity levels are reached.

Exponential decay is the precise “mirror” of exponential growth. When anything — a virus, locusts, mosquitos, the human species, the rate of consumption — reaches the limits to growth after a period of exponential growth, it will immediately experience an exponential decay that mirrors the exponential growth.

We will hit the wall of the limits to growth in resource production and consumption before we reach the limits to human population, and at that point, human population, with no additional food to sustain it, will likewise stop growing.

For the past 50 years, total global resource consumption (using the surrogate of global energy use) has been rising at an annual rate of about 6% per year. That doesn’t seem like that much until you consider that human population, depicted in the remarkable chart above, is only growing at just over 1% per year. A 6% annual (exponential) growth rate means a doubling every twelve years. In other words, we will use as much of earth’s resources in the next twelve years as we used in all of our previous history. And we’ve already used up half of all the known resources of the planet. Though we may dredge up some more, at ever-increasing incremental cost, it is not unreasonable to believe we will hit the wall about then — 2033. At that point human population will be about nine billion. Even without climate and ecological collapse, the shit will then hit the fan.

My guess is that we will stall off the peak for a few more years after that, at least for those in affluent nations. So let’s be generous and assume we can grind out the global human civilization experiment until 2040, with 9.5 billion humans.

It’s all downhill from there, and it will be steep. On the way up, we hit 4.7 billion humans in 1984, 56 years before the likely 2040 peak. So you might think we’ll be back to that population 56 years after peak, which is just before 2100.

But we’re still not realizing the impact of resource constraints, and just how steep the slope of exponential decay can be. Our population is constrained by resources available for consumption, which has been increasing six times faster than human numbers, and which will decrease six times faster after the peak. When we run out of affordable resources, we will face full-on global economic collapse, and our numbers will have to decrease to keep pace with the smaller amount of food, heat, and everything else we depend on in our vulnerable, prosthetic, human-made environments.

If you’ve been following the math, that means human population will have to track the pace of exponential decay of resources, which will be halved every twelve years just as it is doubling every twelve years now. By that logic, the decline from 9.4 billion humans to 4.7 billion will occur between 2040 and 2052. And then it will decline again by half to 2.4 billion humans by 2064 and to 1.2 billion humans (the population in 1850) by 2076. And to 600 million humans (the population in 1700) by 2088 and to 300 million humans (the population in 1300) by 2100.

Some of this decline will occur ‘naturally’, as, in a world of severe scarcity and precarity, people will simply stop having children. I’d like to say most of the decline will happen that way, but the numbers don’t lie: The first few decades after peak consumption are likely to be pretty terrible, even if we divert all economic activity to simple food production and stop procreating.

And there is no reason why that rate of decrease won’t continue into the 22nd century, but I think it would be foolish to speculate on what might be sustainable human numbers, without cheap energy or any energy-fuelled technology, and with a suddenly sparsely-populated (by humans) world with 97% of peak-civilization’s abandoned and scavengeable stuff (and nuclear and chemical wastes!) left behind. And we may have so desolated the planet — its biodiversity, its climate, its soils and waters — that even 300 million may be far beyond its carrying capacity for our kind.

My grandparents’ message to me, when they described the Great Depression, was how utterly unthinkable the reality of it was in the roaring 20s — how ghastly the suffering, how far people fell from their positions of privilege and seeming invulnerability, and, most of all, how quickly it overtook everyone. Exponential decline following exponential growth.

So here we are again in the roaring 20s — driverless cars, Mars missions, dreams of living forever, the “internet of things”. Enjoy it while we can. There is no soft landing on the other side of exponential growth; it’s a hard ride down. All civilizations, like all pandemics, end, usually with surprising speed in their last gasp as the limits to growth are reached and breached, and so will this one.

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7 Responses to What is Exponential Decay?

  1. Mitra Ardron says:

    I share your concerns, but I doubt it will be an exponential decline, more like a hard wall – because we have not only an exponential growth in population and consumption, but in change. The speed of change is increasing exponentially as well – and sooner or later we’ll hit a change that happens fast enough not to recover from. The pandemic was a wake up call to that, the speed it spread, but more importantly the resistance to taking action – the anti-vaxx/anti-mask/anti-lockdown stupidity for example. Just compare that to the political backlash when people are told (one way or another) that they can’t propogate, and we can see why action might be too slow to counter for example a mistake in a GMO that spreads a crop disease, or a computer virus that propogates through IoT devices and takes down all the systems that depend on it. I’m not arguing that any specific one of these will happen, but I’m suggesting that with systems approaching breaking point, ONE of these will happen so quick to crash the population, or maybe that will just be the first big fall of your exponential decline?

  2. ivo says:

    Mhh who would have thought of this before ???

    Luckily Thomas Robert Malthus, didn’t thought of this in 1798… Oh wait, he did.

    https://en.wikipedia.org/wiki/Malthusianism

    And then there is William Stanley Jevons, paradoxically, he saw the results of progress in 1865.

    https://en.wikipedia.org/wiki/Jevons_paradox

    In 1895 Svante Arrenius thought of CO2 as possible greenhouse agent.

    https://en.wikipedia.org/wiki/Svante_Arrhenius

    And more recent, in 1972 there was this report from Rome that created a few laughs.

    https://en.wikipedia.org/wiki/The_Limits_to_Growth

    Nothing to see here, move along…

  3. Brutus says:

    Two names you probably ought to have mentioned: Albert Bartlett and Ugo Bardi. As suggested above, the curves are unlikely to go smoothly up then down but rather result in a steep drop-off from a crest or plateau, which Bardi call the Seneca Cliff.

  4. Joe Clarkson says:

    Our population decline will indeed be exponential, but the base of the equation will be much lower than one, whereas during our increase the base was ‘only’ slightly more than one. The result, which is also found in the population dynamics of many species, is that the exponential decline is much faster than the rise.

    The rate of exponential decline gets greater and greater as carrying capacity is exceeded. On St Matthews island, the introduced reindeer population rapidly grew to about four times carrying capacity (to about 6000 peak population after 19 years of exponential growth vs an estimated 1670 population at the island’s original carrying capacity) and the population then plummeted to 42 reindeer in one year due to starvation. Carrying capacity had been reduced dramatically by the extent of over-population.

    Since the earth’s human carrying capacity is a complex interaction of our energy supply with the natural environment and since fossil fuels are a one-time source of energy, the fact that our population is about nine times higher than a fossil-fuel-free carrying capacity is daunting, as is the continuing degradation of that carrying capcity.

    I expect that we will use as much energy as we can find to slow the rate of population decline, which doesn’t bode well for the climate. Even with that last grasp of energy, the rate of population decline will be far steeper than its rise and the final leveling off will be a levels far below those of a few hundred years ago. And then there’s the effect of war.

  5. David Beckemeier says:

    Dang!!

  6. Kevin Hester says:

    Dave is a lot more ‘Hopeful’ than I am. I expect the Northern Hemisphere Grain season to fail this year, fail worse the next year and continue to fail miserably.
    I see collapse as the only realistic scenario.
    https://kevinhester.live/2019/09/05/collapse-the-only-realistic-scenario/

    We are just about to see the first “Blue Ocean Event’ in our species history. We are ‘Toast’.

    https://kevinhester.live/2019/08/27/cascading-consequences-of-the-loss-of-arctic-sea-ice/

  7. Philip says:

    An energy shock in the middle east due to war could happen anytime as well. The sooner there is disruption to food supply and further pandemics etc the play out of collapse will be extended. We are toast are in so many possible ways, when and how suggests the interconnections of the world are beyond our vision. Human ignorance is innate. Impermanence is certain. Broad post, Dave and interesting comments, good luck to all.

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