In a recent post, world population growth rate, I mentioned I’d have more to say on the question of the growth rate being exponential (or not) in the formal mathematical sense of the term as opposed to the everyday usage of exponential to mean really fast. Here is what I was thinking about:
The general form of an exponential function is
f(x) = Nacx
Since we usually talk about population growth in terms of an annual percentage increase, we can write
p(t) = s(1+r)t
where
- p(t) means the population at time t
- s = starting population (i.e. population at t0)
- r = rate of growth as annual percentage increase
- t = time in years
Using the same US Census Bureau data as in the previous post we can draw this graph:
The stars are actual data points reflecting the world population over 50 years of actual data. The red line is this exponential function:
pm(t) = 2,556,518,868 (1 + 0.018)(t – 1950)
where
- pm(t) means the population projected by this model at time t
- s = starting population (i.e. population at t0) = 2,556,518,868
- r = rate of growth as annual percentage increase = 1.8%
- t = time in years, t0 = 1950 up to t50 = 2000
which gives, for example, p(1959) = 3,001,781,000 — just 0.11% off the actual value of 2,998,566,971. The largest positive error is 2.71% at p(2000), the largest negative error -2.21% at p(1974). So, we can say this mathematical model of population growth is good within +/- 3% over the 50 years from 1950 to 2000.
Given the error range in the actual data (which I don’t know but may be as much as 10%) this exponential function is a fine model — which is confirmed by visual inspection of the graph. In the data set used, the annual growth rate in 1950 is 1.47%, it peaks in 1968 at 2.07%, and declines to 1.24% in 2000, so it’s not surprising using a 1.8% annual growth rate in the model produces a good fit to the 50 years of data seen here.
the rub
But this only shows that population growth over interval 1950..2000 is exponential. It says nothing about the nature of the growth beyond 2000.
For the 12 years from 1989 to 2000 the annual growth rate is monotonically decreasing. In fact, except for a some relatively small bumps, the rate has been going down since 1963, from a high of 2.19 to 1.24%.
This is a graph of the US Census Bureau growth rate data from 1950 to 2000. I haven’t dug into to it, but the dip around 1960 is likely the result of the famine in China just prior to that time. Notice the steady decline from 1963 peak onward.
A linear model of that decline projects the growth rate reaching zero at 2050.
Favoring more recent data, you can make the slope of that line steeper and achieve zero growth sooner or, being more conservative, make the slope more gentle and get to zero growth later. For growth to be exponential the blue line needs to be horizontal. (For the mathematicians: think 2nd derivative, the change in the change.)
This graph overlays absolute population numbers on top of the growth rate. From 1950 to 2000 the numbers are from actual data, beyond 2000 they are determined by the growth rate given by the blue line, a linear reduction in growth rate from the year 2000 forward. With this model the population tops out at 8.3 billion in 2050, after which, if the growth rate trend continues, the population begins to fall.
In this graph linear reduction in growth rate (in green) is compared with a constant growth rate (in purple). The constant (and by definition exponential) growth rate used in the purple line is 1.7%/year. The green dotted line is a growth rate that declines at a rate of 0.025 of a percentage point each year reaching zero at 2050. Up to 2000 the purple and green lines are forced to the actual values as presented by the US Census Burueau data set; thereafter they are determine from their corresponding growth rates.
The argument for the purple line lies in averaging the growth rates between 1950 and 2000 while the green line is derived from identifying the steady decline in growth rates from 1963 onward. To me, this 37-year trend (the green lines) wins out over even a weighted average of the growth rates (the purple lines) over 50 years. I see the world’s population stabilizing in the 8 to 9 billion range around 2050 as a pretty good projection.
However, a reason to be cautious about projections is that they depend on assumptions about future growth rates. It’s quite possible that the growth rate will continue to decline only to stablize at, say, 0.5% in 2030 as shown in this graph:
Notice the growth rates given by the dotted lines after the year 2000 and the corresponding absolute population curves.
Whenever growth rate becomes constant, absolute growth becomes (by definition) exponential — with each year producing more people than the previous one. Exponential growth is always problematic.
complexity theory footnote
The tools used by software developers come to mind in this discussion, in particular complexity theory which is used in computer science to compare how much memory or time programs (or parts of them) require. A software developer avoids code that requires exponential resources because the resulting software would be sure to fail when input values were high.
The Dictionary of Algorithms and Data Structures maintained by Paul Black at the US National Institue of Standards and Technology defines exponential like this:
In complexity theory, the measure of computation, m(n) (usually execution time or memory space), is bounded by an exponential function of the problem size, n. More formally if there exists k > 1 such that
Both of these conditions are met in this picture:
As discussed above, the model pm(t) is a good fix to the data within the 1950..2000 range. Being bounded by pa(t) and pb(t) it meets the conditions for Big Theta and Big Oh — conditions 1 and 2 in the preceding defintion. Thus, over this range, the model function pm(t) as well as the real-world function of the actual data are exponential. However, that is a trivial result since no one can dispute that past population change is exponential. The salient question is whether or not future population growth is exponential.
Consider the purple and red curves in this graph which we looked at previously:
While both the red and purple curves are exponential under any reasonable definition the issue is: are they good projections of the future? Or is the green curve, clearly not exponential, more appropriate?
Personally, given the finite resources of our planet I don’t see future population growth being exponential, but I also don’t think the current population of 6.5 billion or more is sustainable let alone a higher value. So I find no solace in the non-exponential nature of population growth seeing how we are fouling our environment to a stupid extent at the current population density. Adding 70+ million people per year (the current rate of growth) does not improve our or our children’s prospects.
conclusion
Considering all historical data, it’s fair to conclude that world population growth has been exponential in the formal, mathematical sense of the term. (See future curves.) When projecting that growth into the future, however, there are good reasons to think it will not be so. Since 1963, in the US Census Bureau data, the growth rate has been steadily declining which is not representive of an exponential function (in which the rate of growth rate is constant). Projecting that rate of decline into the future shows the world population topping out in the vincinity of 9 billion around 2050.
The most important questions for us today are:
- How do we ensure the growth rate reseaches zero? It is after all merely a projection, by no means a certainty.
- How do we ensure as much of our planet’s resources as possible are preserved for future generations?
mathematical footnote and resources
Some good on-line resources about Big O notation and complexity theory:
- For an excellent basic introduction see these class notes (PDF, 355KB) by Prof. Littman at Rutgers University.
- Mark Chu-Carroll, a superstar computer science & math blogger, has written about Big O notation and related topics in an introductory form here (be sure to read the comments) and in a more advanced way here.
- Naturally, Wikipedia has a page as well.
See also Human Population Growth, a nice summary of world population issues using real data.
Very well done. It’s an interesting issue, but I agree, as you know, that whether it’s exponential or not does not take away from the problem of continued population growth. One thought that makes me say, “Hmmm…” is your correct point that if the world population growth rate were to stabilize at, say, 0.5% in the future, that constant rate would mean the growth was once again exponential. That means that it could be exponential at a rate that that’s slower than the current non-exponential (or clearly arguably so) growth.
As you know, I still wonder what it means that some with strong backgrounds in this stuff (though perhaps not experts in precisely this math) say the growth is still exponential today. (e.g., arguments that it remains exponential as we ‘asymptotically approach logistic growth’ and stuff like that.
) But again, you’ve made the case well and shown too that any change from exponential to non-exponential gives us little comfort at this time.
I still wonder what it means that some with strong backgrounds in this stuff (though perhaps not experts in precisely this math) say the growth is still exponential today. (e.g., arguments that it remains exponential as we ‘asymptotically approach logistic growth’ and stuff like that.
)
Yeah, I wonder too. Looking at a 40 year decline in growth rates, I just don’t see the case for exponential growth any more. I’ve ask one math guru to comment. If you or anyone else knows a good demographer please point them here.
One thought that makes me say, “Hmmm…” is your correct point that if the world population growth rate were to stabilize at, say, 0.5% in the future, that constant rate would mean the growth was once again exponential. That means that it could be exponential at a rate that that’s slower than the current non-exponential (or clearly arguably so) growth.
And if the growth rate never goes negative we are going to being dealing with all the consumption-related problems related to a very large population that IMO is well beyond any reasonable safety margin.
I wonder if we had an Even Greener Revolution which completely solved all food production issues and were given a perfect source of clean energy and had the means to reduce the CO2 in the atmosphere — all of that — would we just continue to breed until every square inch of the planet was covered with people? Brings to mind that old movie Silent Running starring Bruce Dern. Or maybe Soylent Green with Heston. On the other hand, given that scenario perhaps the world as a whole would naturally experience negative growth like some parts of Europe do now.
But with both feet firmly planted back on the ground, I have to say getting to even 9 billion scares the begeezus out of me. Too crowded, too much likelihood of permanently altering Earth for the worse, too high a risk for resource wars.
“I wonder if we had an Even Greener Revolution which completely solved all food production issues and were given a perfect source of clean energy and had the means to reduce the CO2 in the atmosphere — all of that — would we just continue to breed until every square inch of the planet was covered with people?”
I wonder the same thing. Some people seem firmly to believe that if we truly solve the energy problem (I think they overlook food production), our troubles will be over. But it seems to me there are other impacts of ongoing population growth, separate from energy, which make it unsustainable. It would be well worth really cataloging those.
Most of the current growth in population seems to be in Africa. And in crude population density terms they do have more room than many other areas on the planet.
I think your green curve is close to reality.
I don’t think fearfulness serves us very well when it comes to policy formulation. Take a look at the war launched in 2003 in the name of fear if you do happen to think fear is a good foundation for policy.
“Most of the current growth in population seems to be in Africa.”
There’s still a great deal of growth happening elsewhere as well. India and the Middle East are examples. The bottom line is some of the major projections see world population increasing by about 40% by 2075.
http://www.un.org/esa/population/publications/longrange2/WorldPop2300final.pdf
As per capita consumption levels rise in many areas, population size and continued growth becomes increasingly problematic.
BTW, we really don’t know what will happen with world population growth over the next 50 or 100 years. The well known projections are merely plausible possibilities, not predictions.
“And in crude population density terms they do have more room than many other areas on the planet.”
Africa does have space. Yet space is of relatively little importance. We need to look not at the question of how many people can an area can hold, but at how many people an area (and the earth as a whole) can support. Many well informed estimates suggest we’re near or beyond the limits already. (see, e.g., the earth’s “ecological footprint.”)
I agree we shouldn’t base policy on fear.
For reference: The CIA’s 2006 figures for annual percentage growth by country are listed in order on this Wikipedia page along with a colored map. The only major countries not still growing are Russia, many of its former satellites in Eastern Europe, Germany, South African, and Botswana (the latter two due to HIV/AIDS).
I don’t think fearfulness serves us very well when it comes to policy formulation.
Neither does unbridled optimism. Both are blind to objective study of our world. However, this is a case where error on the side of more rather than less people has enormous and permanent negative consequences for future generations.
Just change the slope in that blue line a little or let it go from negative to positive and you have a much different outcome.
Growth rate depends not just on fertility but also lifespan. For important moral reasons I don’t think we should decrease lifespan to solve this problem. And I don’t think fertility is set to be a continuing problem. Fertility is low in all major populations centres except India and Africa. And both are trending downwards. India should hit 2.0 in about a decade given the long range trends.
I’m optimistic because of the data not in spite of the data. However I agree that excessive optimism can be as poor a guide to policy as excessive fear, but given human tendancies I think we should be more vigilent against fearfulness.
“And I don’t think fertility is set to be a continuing problem.”
As the authors of the UN report and the authors of the articles attached to it make clear, very minor changes in fertility rates can mean very large differences in population over time. And demographers are quite unable to make any sort of decent guesses about future social issues and other human variables which affect fertility rates.
Again, though, their medium scenario, which many take as reason for optimism because it shows population growth peaking around 2075 then eventually dipping back to 2050 levels, still has world population growing about 40% from current levels before peaking. That’s pretty massive, and is reason for concern.
Some, though, don’t think we’ll make it to such numbers because our clash with the earth’s human carrying capacity will mean nature will take over very harshly before that.
Trinifar — you said, “However, this is a case where error on the side of more rather than less people has enormous and permanent negative consequences for future generations.”
Yes, exactly.
Trinifar: “But with both feet firmly planted back on the ground, I have to say getting to even 9 billion scares the begeezus out of me. Too crowded, too much likelihood of permanently altering Earth for the worse, too high a risk for resource wars.”
- —
9 billion people shall have to consume much, much less than the 6.5 billion of people of today are doing. – I mean: isn’t that a most basic kind of fact?
What scares me the most, actually, is the fact that all peoples and nations of this world are pursuing a classic American Lifestyle. AND the fact that there really is no political, no economic, no industrial interest in curbing the world’s population’s fossil-fuels addiction. We are going to continue using oil until all deposits are depleted. And after that we are going to continue making use of coal, and simply to “pay our bad conscience away” by shopping for climate credits.
So: what are we going to get? A 30% world population growth within the next forty years: a world population of happy-go-lucky Americans, all overconsuming, overpolluting, overwasting at rates unheard of anywhere outside the USA?
It’ll be good for the world economy, for short. So, quite frankly: hmm? I guess this will be the case.
Magne,
Yep, it’s gloomy out there. One can excuse children for not being able to project the effects of their actions into the future. Where are the adults?
I just read some columnist saying that all the current 2008 USA presidential candidates were better on global warming and the environment in general than GWB. And that is true — and a very low bar to cross. Even when it is sincere, a new president will need a strong, massive, broad base of support to get any meaningful legislation through Congress against the incoming tide of corporate interests that say in effect “don’t harm my short term profits when future generations will be the ones paying the price for my actions now.”
We are the proxy votes for future generations and we need an informed electorate to make them count.
Magne – It is worth noting that GWB was also better on global warming before he was elected.
P.S. Or rather he advocated a more popular position.
How about this one?
http://www.lyricsmania.com/lyrics/bjork_lyrics_1732/greatest_hits_lyrics_8849/human_behaviour_lyrics_102142.html
Quote: “April 21, 1993: President Clinton announces that the United States will stabilize greenhouse gas emissions at 1990 levels by the year 2000.”
- —
http://www.mbe.doe.gov/me70/history/1991-2000.htm
Well, those lyrics are just fabulous, but hey: my intention was to post THIS link, and not that one.
- —
Thanks for that reference. I found it very amusing.
I think that any government promising to do X for the next 50 years in order to create certainty for the business community is spouting self serving rubbish or else they are seriously deluded.
Magne says, with tongue in cheek: “what are we going to get? A 30% world population growth within the next forty years: a world population of happy-go-lucky Americans, all overconsuming, overpolluting, overwasting at rates unheard of anywhere outside the USA? … It’ll be good for the world economy, for short. So, quite frankly: hmm? I guess this will be the case.”
BTW: Great post Trinifar.
My own perception is that unless we-the-World get serious about taking population/pollution/species loss etc. matters into our political/administrative governance structures we’ll find oursleves living this bit of ancient wisdom: “Those who give no thought to that that which is distant, will find sorrow close at hand.”
If we don’t crash-land our economic systems first, we’ll likely crash-land our ecological systems (and related “sources” and “sinks”) relative soon (maybe even as we speak).
Will any of it wake us up (in our public sense, not our corporate sense)? I dunno.
PS. It would be nice if corporate power-bases got wide-awake on this matter too, and found means to recognize public purpose as well as private gain.
I’m not holding my breath on either front. Although I’ll keep trying to do my bit-part to help keep the conversations moving, as do all the rest of you who have commented here.
With complexity theory as my worldview, there is alway hope re: “butterfly effects”, “tipping points”, etc.
I keep screwing up and not getting the correct “about” reference under my name, as above. This one ought to work, to keep my vanity alive for a nanosecond or two longer.
Dave, I repaired the vanity link.
With complexity theory as my worldview, there is alway hope re: “butterfly effects”, “tipping points”, etc.
Very nicely put.
Dave. – My own perception is that unless we-the-World get serious about taking population/pollution/species loss etc. matters into our political/administrative governance structures we’ll find oursleves living this bit of ancient wisdom: “Those who give no thought to that that which is distant, will find sorrow close at hand.”
- —
Very well put.
But the political systems of our times are all economy driven. You’re not likely to find any one political culture of our times, which isn’t all economy driven. And hey: from what I’m able to observe, every day, it’s become pretty clear to me, that “we, the people” are only in it for the money, too. Just like our political / administrative leaders are. And just as their various lobbyist groups are.
People simply do not want any change of lifestyles. To the contrary. People who already consume too much, want to consume even more. And people who don’t consume too much, quite naturally want to … you know … and who’s to blame them? …
As a matter of fact: I believe any political party which chooses to take environmental concerns very seriously is actually likely to lose a lot of voters. That’s how INSANE “we” – The Human Species – has become.
Magne, “People simply do not want any change of lifestyles.”
That’s too big a generalization for me. Lots of people want meaningful change and lots chose to live relatively lightly on the planet even when they do not have to. Sure we are a minority but all change begins that way.
Hm, okay. It’s a generalisation, of course. But then: from what I am seeing with my own eyes and hearing with my own ears, I understand that I live among people who fear change more than anything. People are very keen on pretending that everything’s in order and nothing serious is up, either with the weather patterns or the biosphere at large. – And I’m seeing no willingness to change. None, what-so-ever. I’ve been waiting for a positive reaction for a long time now, but I look around, and what do I see? – Zombies. Or living creatures – people -who have gone completely comatose.
http://news.uk.msn.com/green_barometer.aspx
Highlights:
“A recent study of 1,192 UK households, dubbed the Green Barometer, found that more than 80% of people believe climate change is having an impact on the UK right now. That is a very encouraging statistic. If four out of five people in the UK are taking the threat to the environment seriously, then we should be able to work together to stop the rot. However, the other main finding of the study, carried out by the Energy Saving Trust, was that 40% of us are doing absolutely nothing to reduce our energy use. This gulf between awareness and action is alarming.”
“The report highlights that it is strong measures that are not popular. Fewer people consider measures such as ‘green’ taxes (34%), road pricing (tolls & congestion charging) (30%) and carbon rationing (28%) to be as socially acceptable as same sex marriage (42%). Banning smoking in public places is twice as socially acceptable as any of these green measures.”
“If Dr David Lewis, a neuropsychologist, author and lecturer from The Mind Lab, is to be believed, time really is of the essence. He said: “Where climate change is concerned, the world is rapidly approaching a critical time. Once this has happened, making life style changes is significantly less likely to reverse or improve the outcome. We all have ownership of the world and need to face up to the urgent and important changes that need to be made immediately.”"
“Punishing people for not reducing their energy use and carbon footprint is not the answer. It would be a mistake to make people resent the environmental cause. Some form of incentive scheme, however, could well be the spur the UK needs to begin practicing what it is preaching.”
How about this?
People are changing. It’s fair to question whether the pace of change is fast enough, but the change is occuring.
[...] [update: see also part two] [...]
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