Malthusian Memes - An Exponentialist View
Viral Replicators - An Exponentialist View
Bacterial Replicators - An Exponentialist View
Cellular Replicators - An Exponentialist View
Rabbit Replicators - An Exponentialist View
Human Replicators - An Exponentialist View
Grey Goo - An Exponentialist View
Death By Replication
Exponential Assembly - An Exponentialist View

Population Doubling Mechanism
New Malthusian Scale
Human Global Ecophagy

External Links:
Population Set To Decline - International Institute for Applied Systems Analysis (IIASA)

Forecast Sees Halt to Population Growth by End of Century - from National Geographic News

World Population Growth, 1750 - 2150 : Population Reference Bureau

Total Midyear World Population for 1950 to 2050 AD - US Census Bureau

Historical Estimates Of World Population - US Census Bureau.

CIA World Fact Book

Natural Selection and Differential Reproduction - from Replicators: Evolutionary Powerhouses

Malthus, Thomas Robert, An Essay on the Principle of Population. John Murray. 1826. (6th edition) Library of Economics and Liberty.

Human Replicators - An Exponentialist View

"At present there are around 6 billion humans. In 40 years, if the doubling time stays constant, there will be 12 billion; in 80 years, 24 billion; in 120 years, 48 billion. . . . But few believe the Earth can support so many people. Because of the power of this exponential increase, dealing with global poverty now will be much cheaper and much more humane, it seems, than whatever solutions will be available to us many decades hence. Our job is to bring about a worldwide demographic transition and flatten out that exponential curve— by eliminating grinding poverty, making safe and effective birth control methods widely available, and extending real political power (executive, legislative, judicial, military, and in institutions influencing public opinion) to women. If we fail, some other process, less under our control, will do it for us." (Carl Sagan, 1997, p.20)


The last major milestone in global population was when our population reached 6 billion on 12th October, 1999. Of course, I doubt that anyone seriously thinks that we know for a fact that the 6 billionth person was born on that day. However, it was a reasonable day upon which to draw a line the sand and declare a particular baby the 6 billionth person. Another recent milestone (11th May, 2000) was that India reached 1 billion people.  

Various organisation around the world have made numerous projections regarding global population trends. In this article I will focus on two of the more recent such projections. The first is the Long-range World Population Projections (1998) from the Population Division of the United Nations. The second and more recent report (2001) is by International Institute for Applied Systems Analysis in Laxenburg, Austria. This report was the central feature of an article in Nature magazine, and predicted world population will begin to fall by 2100, which thus ensured that the report would make newspaper and TV headlines around the world. 

Long-range World Population Projections - United Nations Population Division (1998)

Replacement Rates:

The UN Population Division's projections are based on assumptions concerning the global replacement rate by 2050 or later:

The medium projection uses the popular definition of the replacement rate of 2.1 children per woman. This allows for the replacement of the parents, and any offspring who die before they reproduce. 

The projected populations (in millions) for 2150 are:

Although the emphasis in the UN's projections is on replacements rates, the UN also provide the annual growth rates at set intervals (1995, 2000, 2025, 2050, 2075, 2100, 2125 and 2150) for each scenario. They also take great care to explain the fact that a population can still experience positive growth for some time after the replacement rate is reached (page 3, Executive Summary). This natural lag effect is caused by the time it takes for parents and grand-parents to reach old age and start being counted in the death-rate. 

The calculation of replacement rates also depend upon the infant mortality rate. In countries like India, with a higher infant mortality rate, the replacement rate is in fact higher - around 2.4 -  than the figure of 2.1 children per woman. 

Also, it is clear that a population will grow faster if the mother consistently has her allotment of children younger. Here, I assume 2 female children per mother (if you prefer, assume 1 female per mother and halve my figures - my point is still proven):

Years 16 32 48 64 80 96
2 4 8 16 32 64
0 2   4   8

Table A. Young mothers will outbreed older mothers (all other factors being equal)

Although I can understand the appeal of the "replacement rate" (or fertility rate) model in popularising the concept of Zero Population Growth (each couple which has 0, 1 or 2 children can be said to have "done their bit" to stabilise population), it must be said that it is a very fuzzy concept. I find it typical of the generational approach of population modelling. That is, it focuses on the average number of offspring of the mother. Then it struggles to universally explain the resultant population growth! Also, it completely fails to explain the truly exponential nature of population growth. 

Recent Global Demographic History

Here is a brief history of global population growth, going back to when our population was estimated to have reached 1 billion:

Year 1800 1930 1960 1974 1987 1999
Billions 1 2 3 4 5 6
Years to add next billion 130 30 14 13 12

Table B. Linear presentation of global human population milestones

Our global rate of increase actually peaked at 2.1% per annum in the 1960's, and though still positive it has been in steady decline ever since. That's the good news. The bad news for us is that, the larger our population, then the less time it takes to add another billion. However, this is a typical linear view of population increase, and I am not fond of taking a linear view of population growth. 

Take another look at Table B - notice that it took 130 years to double our population from 1 billion to 2 billion, then only 44 years to double to 4 billion. Or you might have noticed that our population only took 39 years to double from 3 billion to 6 billion! So, constant population doubling (and variable doubling times) seems to be the pattern here. This has been noticed before:

"The immediate cause of the increase of population is the excess of the births above deaths; and the rate of increase, or the period of doubling, depends upon the proportion which the excess of the births above the deaths bears to the population." Malthus (1830) - A Summary View on The Principle Of Population

It is also possible to deduce the opposite case:

"The immediate cause of the decrease of population is the excess of the deaths above births; and the rate of decrease, or the period of halving, depends upon the proportion which the excess of the deaths above the births bears to the population."

Both of these statements are true, and present us with a simple yet accurate model of negative and positive population growth. Moreover, this model is universal in its application (it applies to all populations of replicators - see Replicators - An Exponentialist View for more).

New Malthusian Scale

Probably the main reason why Malthus' model is not consistently used for explanations of exponential growth is because the numbers in the sequence quickly get too large - such is the power of exponential growth. For example, adding the next 10 numbers to the doubling series (1, 2, 4, 8 etc) after 1024, we get 2048, 4096, 8192, 16384, 32768, 65536, 131072, 262144, 524288 and 1048576. 

However, with the introduction of my New Malthusian Scale, it is possible to take the entire 100,000-plus years of global exponential growth for humanity and represent it only using the numbers between 1 and 1024 (inclusive): 

Time 98,000BC                    
Kilopops 1 2 4 8 16 32 64 128 256 512 1024
Time       10000BC   1500BC 800BC 450BC 1000AD 1600  
Megapops 1 2 4 8 16 32 64 128 256 512 1024
Time 1805 1935 1977 2040?              
Gigapops 1 2 4 8 16 32 64 128 256 512 1024

Table C. The complete history of global human population doubling on the New Malthusian Scale

Note:  1Kilopop = 1024 individuals, 1Megapop = 1024 Kilopop, 1Gigapop = 1024 Megapop

This table can be used to demonstrate the demographic history of any human population, and could be extended to cover our demographic future. In this case I have gone back 100,000 years and assumed a population of 1 kilopop. Then I've added some estimated milestones using Historical Estimates Of World Population from the US Census Bureau and the "Atlas Of World Population History" by Colin McEvedy and Richard Jones (1978). The one thing that you can't fail to notice is that the population doubling times are variable. 

Surprisingly, modern demographers get confused when it comes to the exponential growth of human populations. Many assume that a constant rate of growth (and hence a constant doubling period) is required before a population can be said to grow exponentially. They naively assume that because an exponential curve is the result of a constant rate of growth then all exponential growth must require a constant rate of growth. 

In fact, variable growth rates lead to population doubling, which leads to exponential growth. Furthermore, variable doubling periods (not the same thing as variable rates of growth) result in exponential growth as surely as a regular doubling period. Read Paul R. Ehrlich and The Prophets of Doom - An Exponentialist View for more (including negative rates of growth and population halving).

UN Population Projections

Using the New Malthusian Scale, the UN's population projections for 2150 are:

To give you an idea of where we are now on this scale, our 1999 milestone of 6 billion (6,000,000,000) people is equal to 5.59 Gigapops. Or, looking at things the other way, how many individuals make up a Gigapop etc?

Population Individuals Year Of Milestone

1 Gigapop 



2 Gigapops 



4 Gigapops 



8 Gigapops 



 Table D. Approximate New Malthusian Scale milestones

In a very real sense then, if we assume that the UN's Medium projection is considered the most likely, then we are in the middle of a Malthusian doubling from 4 to 8 Gigapops. 

Malthusian Growth Rate Projections

Given that the idea of the UN's Medium projection is Zero Population Growth (even though they project a growth rate of 0.08% per annum in 2150!), let's make some projections of our own, but based around assumptions more suited to Malthusian population modelling. Taking a fairly realistic starting population of 6 Gigapops (6,442,450,944) in 2005, and a starting growth rate of 1%, here are my projections:

Projection (rate) Average Annual Growth Rate 2005 2075 2105 2145
Low 1.0% 6 3   1.5
Low-Medium 0.5% 6     3
Medium  1.0% - 0.0% 6 9.8 10.7 10.7
Medium-High 0.5% 6     12
High 1.0% 6 12   24

 Table E. New Malthusian Scale projections  in billions (red depicts negative growth, blue is ZPG)

As you can see, my crude projections aren't that far off the UN's projections. My projections are based on population doubling and halving times crudely calculated using the Rule Of 70 (divide the rate into 70 to get the doubling or halving time). 

I started with my High projection, and I assumed that the population growth rate would average 1% for the foreseeable future. Hence, every 70 years, the population doubles. The Low projection assumes an average negative growth rate of 1%, and so the population halves every 70 years. The Low-Medium and Medium-High projections assume average growth rates of 0.5% (negative and positive respectively), resulting in doubling and halving times of 140 years. For the Medium projection, I assumed Zero Population Growth is reached by 2105. This is consistent with the UN's assumption that an global replacement rate of 2.1 children (average) per woman is reached by 2050 (I am then allowing 50 years from 2055 for the lag effect of the deaths of the aging populace). Thus I have averaged out the growth for the Medium projection by assuming that the growth rate falls by one tenth of a percent per decade (from 1%) for the hundred years from 2005.

It might be deemed "unfair" to assume constant growth rates for all scenarios except the Medium projection, but that is similar to the UN assumption regarding replacement rates (which are only actually constant in all but the Medium projection).

I should hasten to add that I do not claim my projections are more accurate than the UN's. What I hope to show is that it is easy to make assumptions, and base projections upon those assumptions. What is difficult is to know whether you are right to make those assumptions in the first place. The UN Population Division assumes, in their Medium projection, that we hit replacement rates globally by 2050. Then, due to the natural lag effect explained above, Zero Population Growth is actually attained globally by around 2100. Ask yourself whether you believe that this is realistic and sustainable.

Of course, my other main reason for providing my own projections was to show that population modelling can illustrate true exponential growth if that is the message that we want to get across. My argument is that population doubling and halving can be used to explain the truly exponential nature of all population growth. It's really not that complicated, so why not help people to see it? I would prefer to see the UN's assumptions stated in terms of population growth rates and doubling times, and not the fuzzy concept of replacement rates. 

Population Set To Decline - Nature Magazine online (2001)

The International Institute for Applied Systems Analysis in Laxenburg, Austria, recently predicted an 85% chance that our global population will peak and then begin to decline by 2100. Here are the figures which prompted such headlines as "World population to fall by 500 million: scientists" (The Age, Melbourne, 2nd August 2001):

Region 2000 2075 2100
North America 314 441 454
South America 515 904 934
Western Europe 456 433 392
Eastern Europe 121 87 74
North Africa 173 336 333
Sub-Saharan Africa 611 1522 1500
European part of former USSR 236 159 141
Middle East 172 413 413
South Asia (incl. India) 1367 2242 1958
Central Asia 56 107 106
Pacific Asia 476 702 654
China region 1408 1422 1250
Pacific OECD 150 135 123
Total 6055 8951 8414

 Table F. Nature magazine population prediction 2nd August, 2001 (figures in millions)

I've highlighted North and South America in blue because the populations in those regions are the only ones expected to rise between 2075 and 2100. Oddly, one of the assumptions noted in the Nature magazine (2001) online article Population Set To Decline, is that fertility rates will drop to a figure between 1.5 and 2 children per woman as a society "modernises". Surely the USA, Mexico and Canada can be considered "modernised"? In fact, the same goes for most of South America too, doesn't it?

All three European regions, and Pacific OECD (all of which I've marked in red), are expected to experience sustained negative growth from now until 2100. In Australia (part of Pacific OECD) the fertility rate is said to have fallen to 1.7 children per woman. We are informed (The Age, 2nd August 2001) that our population will level off at 25 million in about 25 years (we're currently at around 19 million). The 813 million "Europeans" will slowly decline to a figure of 607 million by 2100. In the Middle East, a static population of 413 million by 2075 is predicted.

For the remaining regions we find populations rising until 2075 and then beginning to decline. 2075 is far enough away that the authors are unlikely to be around to see whether they are right or wrong. For me, the most dramatic reversal is their prediction for Sub-Saharan Africa, which currently sustains one of the highest growth rates on the planet (despite HIV / AIDS). South Asia (incl. India) is another significant reversal predicted by the team in Austria. 

In China the much-vaunted 1-child policy never had time to produce negative growth for China. This government policy was in fact relaxed to such an extent that the average family size in 1989 was 2.4 children (Ehrlich, 1990). Yet the advantage that Communist China still has over India and Africa is that they can enforce some sort of population control. This is harder in democratic India, and harder still in fractious Africa.

Clearly then, we come back to the hope for demographic transition - a modernised region naturally and voluntarily experiences a lowering of population growth rate. The evidence seems to be that because Europe has managed it, then the rest of the world can manage it too. Of course, one assumes that the rest of the world will be able to attain a standard of living comparable with that of Europe...and an equitable distribution of wealth...

A surprisingly successful factor in reducing world-wide fertility is said to have been contraception and family planning. Fertility in the developing world was reduced by 40% largely as hoped for, but the surprise factor was a similar reduction in the developed world. 


The first exponentialist, Malthus, was opposed to contraception on religious grounds. Nonetheless, fellow Englishman Francis Place (1771-1854) was so influenced by Malthus' essay that he wrote the first book advocating contraception in 1822. Fortunately for humanity Malthus' advice against contraception has been largely ignored, and Place's view has prevailed.

Unfortunately for humanity, many people seem to have forgotten that Malthus provided a sound model for population growth which is with us today. In fact, Malthus revealed a universal law of nature and used mathematics to explain it. He also provided convincing arguments to back his claim (see Malthus - An Exponentialist View for more):

"That the increase of population is necessarily limited by the means of subsistence."

"That population does invariably increase when the means of subsistence increase."

So, despite the optimism of the Nature article, I believe Malthus will be proven correct. It is the nature of all populations of replicators to grow exponentially whenever they can do so. Malthus recognised humanity's particularly powerful ability to encourage and harness the exponential growth of other replicators:

"The main peculiarity which distinguishes man from other animals, is the means of his support, is the power which he possesses of very greatly increasing these means."

From which Malthus concluded:

"The perpetual tendency of the race of man to increase beyond the means of subsistence is one of the general laws of animated nature, which we can have no reason to expect to change."  

Therefore, if any population of replicators on this planet is set to continue its growth, it is humanity. For an example of the sorts of technologies which will re-energise humanity's population growth in the future, read Drexler - An Exponentialist View. But nanotechnology will only continue what biotechnology, genetics, cloning, artificial intelligence, and still other scientific and technological advances will do in the shorter term. 

So, desirable though it may be, I still do not see a definite end to human population doubling on Earth in the next 100 to 200 years (though I expect our growth to continue to slow down for the moment). By 2100 our technology could quite possibly allow our population at the time to double yet again. I believe that the 15% margin of error allowed for in the Nature article is far too modest, and a serious underestimate of human capabilities.

But then, all assumptions in both the UN and Nature projections are based on our population staying on this planet. They would not even allow for a population of only 1 million people in space colonies within our solar system by 2150. Yet some scientists predict a population in the billions by this date. See my articles on two such scientists - Marshall T Savage and K. Eric Drexler. Myself, I would be happy to see any number from 1 million upwards living in space by 2150. This should be enough to secure a real future for our species.

Also, even if our population does lose 500 million between 2075 and 2100, that doesn't mean that global population doubling has ceased. If you look at the demographic history of places like China (64 million in both 200 AD and 100 AD) and Europe (64 million in both 1250 AD and 1450 AD) you will note dips in an otherwise fairly classical exponential growth curve (see Table B, Darwin's Views On Malthus). The one great illusion which I hope to dispel is that no population is forever stable. All populations of replicators are either growing or shrinking, all of the time. This great illusion is usually the result of dynamic equilibrium. The key word is dynamic, not equilibrium. Equilibrium is a state of balance, and all populations walk a Malthusian tightrope.

Before we leave Table F, here is what Carl Sagan had to say about minority populations which sustain exponential growth whilst the rest of the world are at Zero Population Growth (Sagan, 1997)

"...because exponentials are so powerful, if even a small fraction of the human community continues for some time to reproduce exponentially the situation is essentially the same - the world population increases exponentially, even if many nations are at ZPG."

This is also the theme of my article The Cassandra Prediction - Exploding The ZPG Myth. One thing, however, is clear. We cannot sustain positive rates of global human population growth whilst we remain on Earth - see Human Global Ecophagy for more.

Oddly, most Malthusians are pessimists...Cassandras predicting doom. Although people like to label me as a techno-optimist, I'm really a Malthusian realist - an exponentialist. Sometimes I do despair, but as an exponentialist I anticipate that human nature will prevail, and our species will find a way to continue the steady growth of its population, and perpetually push beyond the current means of subsistence. Each time we do so we will get knocked down, and each time we will get up again and push the limits some more. Here's another - rarely quoted - Malthusian prediction (Malthus, 1798):

"The germs of existence contained in this spot of earth, with ample food, and ample room to expand in, would fill millions of worlds in the course of a few thousand years."

I hope Malthus will be proven right.

Malthus, Thomas Robert, An Essay on the Principle of Population. John Murray. 1826. (6th edition) Library of Economics and Liberty.

Although Malthus wrote the 1st edition of "An Essay on the Principle Of Population" in 1798, he wrote a further 5 editions by 1826 and then wrote "A Summary View" in 1830. It is no straightforward matter to obtain copies of each edition, though Penguin Classics still carry a very affordable Edition 1, including "A Summary View", and an introductory commentary by Professor Frew (1970).

The 1st and 6th Editions are available online, and the show a remarkable transformation over time. The 1st Edition was really a response to the Utopian visions of various of Malthus' contemporaries, but the later editions of the book show a more structured analysis of global population data and trends. Malthus maintains his emphasis on variable population doubling times, and endeavours to understand why such differences exist between human populations. Irrespective of Malthus' own conclusions, the fact remains that such differences still exist today. A quick look at the latest report in Nature will confirm that. 

What Malthus so scientifically illustrated is something I call Malthusian Selection.

Malthusian Selection

To most people the word "evolution" is associated with Darwin and his successors. Many will have some understanding of Natural Selection, and genetics. Few will have heard of the principle of differential replication (see Natural Selection and Differential Reproduction - from Replicators: Evolutionary Powerhouses). Of those that have heard of differential replication (or the more commonly used term "differential reproduction"), most assume that differential replication is simply a direct result of natural selection (and some artificial selection by humanity). Further, differential replication is often seen (by those in the know) as purely related to population genetics.

So what name should we use to describe the principle whereby discrete populations of the same species, with only minor reproductive advantages that can be attributed to genetic factors, experience massive differences in their annual growth rates? In other words, in considering consider human replicators, why do some human populations grow faster than others?

The answer to both questions is Malthusian Selection. As Malthus put it (Malthus, 1798):

"Taking countries in general, there will necessarily be differences as to the natural healthiness in all gradations, from the most marshy habitable situations to the most pure and salubrious air. These differences will be further increased by the employments of the people, their habits of cleanliness, and their care in preventing the spread of epidemics. If in no country was there any difficulty in obtaining the means of subsistence, these different degrees of healthiness would make great difference in the progress of population;"

This is not to deny that genetic factors play a part. They clearly do, as some genetic diseases (e.g.. Sickle Cell anaemia, or even cancer) are fairly localised demographically. But how does this compare to non-instinctive human activity (war, medicine, education, politics etc) or environmental factors (fresh water availability, temperature, soil fertility, disasters etc)? These factors affect the reproductive success of all human populations (and environmental factors affect all life). 

Natural Selection, backed up with modern genetic knowledge, clearly explains the gradual evolution of species over geological time. In some cases, as with microbes, we don't have to wait geological ages to see evolutionary change. For humanity itself, evolutionary change is much, much slower. In fact, it's so slow that people often have difficulty believing in evolution.

Take another look at the figures in Table F, and you will find that Natural Selection works in slow motion compared to the principle of differential replication. This is evolution, too. This is the survival of the fittest, or the struggle for existence as Malthus called it. The problem with current evolutionary theory is the word "evolution" itself. Differential replication is typically seen as a result of Natural Selection. Whilst I am happy to agree that Natural Selection explains how species evolve, it clearly does not fully account for different rates of replication. Malthusian Selection also affects differential replication. Malthus already proved that, and so does Table F. Malthusian Selection thus becomes a part of evolution, even though it has nothing to do with how species evolve. This will give you some idea of the exponentialist view of evolution (see Evolution - An Exponentialist View for more).

It is fortunate for us that Malthus believed in the fixity of species...in Creation. For him, Natural Selection was unheard of, and yet he still saw reproductive advantages and disadvantages between rival human populations. He saw them because they existed, and they still do. 

Nonetheless, just as they did in Malthus' day (see the 6th edition of his essay), today's nations demonstrate the principle of Malthusian Selection nicely. Just take a look through the Long-range World Population Projections (1998) from the Population Division, United Nations.

At the time that Malthus and Darwin wrote, it was customary to write about the relative reproductive power of "savage" races compared with "civilised" races. Today, the acceptable terminology has shifted to "developed" and "developing" worlds. Or, we might refer to the West and the Third World. Whilst terminology can be offensive, it is irrelevant to the argument. Suffice to say that discrete human populations have always gained local and temporary advantage over rival populations in the Malthusian struggle for existence. Each population may have its moment in the Sun, only to fade into history as some new advantage allows a rival population to eclipse them. 


Malthus, Thomas Robert, An Essay on the Principle of Population. J. Johnson. 1798. (1st edition) Library of Economics and Liberty. 

Malthus, Thomas Robert. A Summary View on The Principle Of Population. 1830

Sagan, Carl. Billions and Billions: Thoughts on Life and Death at the Brink of the Millennium. Headline Publishing, 1997

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Copyright 2001 David A. Coutts
Last modified: 28 June, 2012