Evolution - An Exponentialist View
Famous Exponentialists
Replicators - An Exponentialist View
A New Malthusian Scale
The Scales Of 70
Population Growth Models
Understanding Compound Interest
The Exponential Method
The Myth Of The Exponential Phase
An Exponentialist Glossary
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Bacteria - An Exponentialist View

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Stephen Hawking
M. King Hubbert
Carl Sagan

The Myth Of The Exponential Phase


It is a commonly stated "fact" that the last two centuries (or some other timeframe) are somehow unique in terms of the exponential growth of our human population. This fact has grown to such an extent that it is worthwhile spending a short while stripping away the myth to reveal the actual facts. This article will reveal that, in fact, no law of nature has had to change to accommodate the prodigious increase in human numbers of the last two hundred years or the last ten thousand years. In fact, the same universal law of nature has applied to all population of any species across all time...and always will.

Part of the issue is that people get easily confused over what exponential growth actually means. I highlight some of that confusion in my articles Paul R. Ehrlich and The Prophets Of Doom, Albert Bartlett - An Exponentialist View and What Is Exponential?

For those expecting a discussion on the exponential phase of bacterial growth see Bacteria - An Exponentialist View.

Two Centuries Of Exponential Growth?

No less a person than Stephen Hawking is one of many luminaries which have mislead the public on this point (Hawking, 2001):

"In the last two hundred years, population growth has become exponential; that is, the population grows by the same percentage each year. Currently, the rate is 1.9 percent a year. That may not sound like very much, but it means that the world population doubles every forty years..."

Hawking clearly states that exponential growth means that a population "...grows by the same percentage each year...". Hence, pure logic must dictate that Hawking believes that our global population has been growing at 1.9% for the past 200 years. So if the population has been growing exponentially for 200 years then it must have been growing at the current annual rate of 1.9 percent. Has it? No!

Any demographic history of the world will reveal that, in fact, our global population has been growing at variable annual rates through time. Take, for example,, the U.S. Census Bureau's figures: Historical Estimates Of World Population.

A simple test of Hawking's assertion is to establish whether our global population has doubled 5 times in those 200 years (exactly every 40 years). The United Nations Population Fund issued Press Release OBV/53 POP/675 on 9 July 1998. In it they stated that it took until 1800 for humanity to reach 1 billion, and that by 1930 our population had doubled to 2 billion. Then it took only a further 30 years to reach 3 billion and fourteen years later, in 1974, we reached 4 billion. In 1987 we reached 5 billion, and 12 October 1999 was officially designated as "The Day Of Six Billion".

What is apparent is that we have only doubled twice (and not five times) in 200 years, and we're on our way (perhaps) to a third doubling (1, 2, 4...8?). Five population doublings would put our population at 32 billion! What is also clear is that the doubling period itself is not a constant 40 years (nor, indeed, any other constant period). The fact is that our population has been growing at variable annual rates, and doubling at varying periods as that rate speeds up or slows down. The only remarkable thing about human population growth in the last two centuries ha been our ability to sustain high rates of growth (up to around 2%).

I doubt that most people have a problem with Hawking's assertions on exponential growth - after all, he is a physicist not a demographer. The problem is, as you will come to understand if you read the articles on this website, that exponential growth is so poorly explained by demographers and scientists of all callings. I therefore find myself unwilling to let any such incorrect explanation stand unchallenged.  

Exponential Growth as a Transient Phenomenon in Human History

M. King Hubbert is well known for his predictions on resource scarcity, in particular what has become known as Peak Oil. Hubbert focussed on the power of human exponential population growth to overrun non-renewable resources.

At the time that America celebrated its 200 year anniversary since the Declaration of Independence in 1776, Hubbert claimed that those two hundred years represented  "...the emergence of an entirely new phase in human history" (Hubbert, 1976). This wasn't American hubris, but an acknowledgement that 200 years earlier most of our energy requirements had been met with renewable resources (human labour, animal labour, wind power and water power), and plants and animals also provided food and warmth. Those  non-renewable resources we used (principally coal, but also mineral products such as clay, lime, sand and metals) were used in small amounts (compared to the present day) and "...seemed inexhaustible" (Hubbert, 1976).

Hubbert, echoing the language of Malthus' Essay on the Principle of Population (1798), also acknowledges that human food supply (mostly plants and animals) also increases exponentially (Hubbert, 1976):

"Biologists discovered a couple of centuries ago that the population of any biological species, plant or animal, if given a favourable environment, will increase exponentially with time. That is, the population will double and redouble in in the successive rations of 1, 2, 4, 8, etc. during successive equal intervals of time. The period required for the population to double is different for different species. For elephants and humans the doubling period is a few decades, but for some bacteria it is as short as 20 minutes. Such a manner of growth obviously cannot continue indefinitely, but the significant question is this: About how many successive doublings on a finite earth are possible?" (my bolding)

Hubbert is correct to assert that the human food supply, based on populations of different animal and plant species, also grows exponentially. He also provides a useful hint that balancing exponential forces can be challenging, as different species have different population doubling times. Where he goes wrong is in claiming exponential growth results in population doublings in "...successive equal intervals of time." There is no example of any population of any species ever maintaining population doublings at a constant rate. It's impossible. In fact, all populations of all species experience variable rates of population growth which result in variable population doubling periods. However, regardless of whether or not the doubling periods are constant or variable, Hubbert would still be right in asserting that such growth cannot be sustained forever...such growth is impossible.

Hubbert uses the classic chessboard analogy to explain the power of exponential growth (see The Persian Chessboard from my article on Carl Sagan). He then extrapolates human population doublings from a single human pair (which he calls Adam and Eve), and asserts that the present world population (in 1976 this was...) would require 31 population doublings, with 46 population doublings being sufficient to increase population density to 1 person per square meter of the Earth's land surface. In another example, he combines the chessboard example with car production starting with just one car and explains that after 64 "population doublings" that the land surface of the Earth would be buried in cars up to 2,000 kilometres thick!

In fact, it is precisely because such population growth (exponential growth at a constant rate) is impossible that Hubbert claims (Hubbert, 1976):

"Therefore, any rapid rate of growth of such a component must be a transient phenomenon of temporary duration. The normal state of a a biologic population, when averaged over a few years, must be one of an extremely slow rate of change - a near steady state." (my bolding)

I agree with Hubbert that high rates of population growth are unsustainable, and that a "near steady state" must be the natural state of a population. However, it does not matter whether or not the population growth is constant or variable, and hence it does not matter whether the population doubling period is constant or varies. For example, the human global population doubled from 3 billion in 1960 to 6 billion in 1999 and growth rates varied year from year. Using the Rule of 70, if we take a growth rate of 1% a population will double every 70 years and if we take a growth rate of 2% a population will double every 35 years. Our global population grew at rates roughly between 1% and 2% from 1960 to 1999 and our population doubled in just 39 years. Hence, variable rate growth is just as powerful as constant rate growth.

I also agree with Hubbert on the seriousness of our human predicament, and the need to find a way forward with the least amount of pain and suffering (Hubbert, 1976):

"It appears that one of the foremost problems confronting humanity today is how to make the transition from the precarious state we are now in to this optimum state by a least catastrophic progression."

Yet the cultural challenge for humanity is enormous. In criticising our modern addiction to unsustainable economic exponential growth, Hubbert clearly assumes that human population growth for the period 1776 to 1976 was exponential (Hubbert, 1976):

"During the last two centuries we have known nothing but exponential growth and in parallel we have evolved what amounts to an exponential-growth culture, a culture so heavily dependent upon the continuance of exponential growth for its stability that it is incapable of reckoning with the problems of nongrowth."

As already explained in my criticism of Hawking, human population growth has never been exponential (at a constant rate). Hubbert previously drew attention to this 200 year exponential phase in human history in a paper called The Nature of Growth, (Hubbert, 1974):

"It is demonstrable that the exponential phase of the industrial growth which has dominated human activities during the last couple of centuries is drawing to a close." (my bolding)

So what does Hubbert mean by exponential?

Hubbert's definition of exponential is ambiguous. Does he mean growth at a constant rate (resulting in constant population doubling periods)? Or does he now just mean a "...rapid rate of growth"? The terminology of an exponential phase (two centuries long...maybe Hawking read Hubbert?) of population growth is misleading and confusing, especially when combined with the usual explanation that exponential growth requires constant rates of growth resulting in constant population doubling periods. For example, if the growth rate was constant at 0.1% - close to a steady state - this is still exponential growth at a constant rate. Hence, it is now both exponential and steady state! Furthermore, such growth is still unsustainable as it is growth at positive rates  - constant or varied - that is unsustainable.

Ten Thousand Years Of Exponential Growth?

Carl Sagan asserted that (Sagan, 1997):

p.18 "Exponentials are also the central idea behind the world population crisis. For most of the time humans have been on Earth the population was stable, with births and deaths almost perfectly in balance. This is called a "steady state". After the invention of agriculture ... the human population of this planet began increasing, entering an exponential phase which is very far from a steady state. Right now the doubling time of the world' population is about 40 years. Every forty years there will be twice as many of us. As the English clergyman Thomas Malthus pointed out in 1798, a population increasing exponentially - Malthus described it as a geometrical progression - will outstrip any conceivable increase in food supply. No Green Revolution, no hydroponics, no making the deserts bloom can beat an exponential population growth." (my bolding)

Sagan is making the same mistake as Hawking and Hubbert. They implicitly assume that some fundamental law of nature changed at some landmark time in human demographic history. A common misleading theme is the requirement for a constant population doubling period (Hawking and Sagan assume 40 years). This is already shown to be a fallacy.

Sagan's landmark is the agricultural revolution of 10,000 years ago, Hawking's and Hubbert's landmark is (effectively) the industrial revolution of 200 years ago. Nobody, least of all me, would deny that these were landmarks in human demographic history. Each marked a period when our global annual rate of population growth increased. 

Sagan also slightly misrepresents Malthus. As Malthus himself put it (Malthus, 1798):

"Population, when unchecked, increases in a geometric ratio. Subsistence increases only in an arithmetical ratio. A slight acquaintance with numbers will shew the immensity of the first power in comparison to the second."

As noted by Sagan, Malthus uses the older term - geometric - whereas these days we are used to the term exponential (even if it is commonly then incorrectly explained these days). It is also worth noting that the term arithmetic can be taken to mean linear.  

So, what Malthus means is that exponential growth is more powerful than linear growth. He's most definitely right about that. See Linear Growth Versus Exponential Growth for more. However, where I believe that Malthus is wrong is asserting that our food supply increases in a linear ratio. Confusing, isn't it?

Well no, not really. You see, if we assume that Malthus is right to believe that his Principle Of Population is a universal law of nature (for all populations of all species for all time), then suddenly it all becomes very clear. To make Malthus' assertion true, we must assume that all populations of all species are subject to variable rate compound interest ( a form of exponential growth) at all times. I call this the Couttsian Growth Model to differentiate it from the Malthusian Growth Model. We must assume that all populations are either growing exponentially, or shrinking exponentially (which is negative growth anyway). Most of the time, this can be simplified by saying that all populations are subject to population halving and population doubling with variable periods. See Malthus - An Exponentialist View for more.

How does this help? Does this mean that Sagan is wrong? Well, yes and no. Sagan was wrong to put words into Malthus' mouth. If Malthus had thought that food supply could increase exponentially then he would have been able to see how food supply could in theory keep pace with population. In fact, Malthus even provided two examples of the exponential growth of food supply - refer Malthus on Grain and Malthus on Sheep - but failed to see his own contradiction in then asserting that food supply is arithmetic (or linear) in nature.

Exponential growth can match exponential growth, though it is by no means guaranteed to to do - see Gigantic Inevitable Famine for more. This is, in fact, the deep secret of the "steady state" to which Sagan refers. The balance of nature, the state of dynamic equilibrium, is a never-ending struggle for existence between competing populations within and between species within Nature's dynamic and often deadly arena. This fine balancing act pits exponential forces against exponential forces for as long as any replicator exists.

Where Sagan and Malthus would rightly agree is that no population can sustain a positive rate of population growth indefinitely. Especially if that population is limited to the finite resources of the Earth. In time, all growing populations must face the limits to growth.

To read more on Sagan's otherwise excellent treatment of "exponentials" read my article Carl Sagan - The Secrets of the Universe.


As Hubbert said, humanity faces a serious predicament. We need to ween ourselves off unsustainable exponential growth and return to steady state economics and population growth. However, it will help if we agree on what is meant by exponential growth in the first place. This will then allow us to more clearly articulate what we mean by sustainable growth. Sustainable growth is an oxymoron - a contradiction in terms. Ultimately, any positive rates of growth are unsustainable (constant or variable) with higher positive rates of growth being more unsustainable than lower positive rates of growth. Like it or not, at best, steady state will require a mixture of low rates of positive growth combined with low rates of negative growth.

It is far simpler and less confusing to simply acknowledge that all populations of all species grow and shrink exponentially (at variable rates), all of the time. Or, if you prefer, you can claim that all populations of all species grow via variable rate compound interest all of the time. However, if we want to talk of a special phase in human history then simply point out that higher rates of growth will inevitably force us to face Malthusian limits to growth - and the fact that the Earth is finite - surprisingly soon. See Human Global Ecophagy for more.


Hawking, Stephen. The Universe In A Nutshell. Bantam Spectra. 2001

Hubbert, M. King. The Nature of Growth. 1974. Accessed from http://www.hubbertpeak.com/hubbert/ 5th July 2012.

Hubbert, M. King. Exponential Growth as a Transient Phenomenon in Human History. 1976. Accessed from http://www.hubbertpeak.com/hubbert/ 5th July 2012.

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

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: 06 January, 2013