Asimov - An Exponentialist View
Bartlett - An Exponentialist View
Darwin - An Exponentialist View
Dawkins - An Exponentialist View
Drexler - An Exponentialist View
Ehrlich - An Exponentialist View
Malthus - An Exponentialist View
Sagan - The Secrets of the Universe
Savage - An Exponentialist View
Turchin - An Exponentialist View
Wallace - An Exponentialist View
Witting - An Exponentialist View

Population Doubling Mechanism 

New Malthusian Scale 

External Links:
Paul R. Ehrlich - Bing Professor Of Population Studies, Stanford University

The Club Of Rome

Fair Warning?: The Club Of Rome Revisited by Keith Suter

Malthusian Growth Model - by Steve McKelvey.

The Doomslayer - Julian Simon interviewed by Ed Regis for Wired Magazine

"The World's Energy Future Belongs in Orbit" - by Dr. O'Neill. Space Studies Institute

The Case For Solar Power From Space - Arthur Smith, Space Daily

Under The Spell of Malthus - Ronald Bailey, Reason Magazine (2005)

The World At Six Billion - United Nations

Malthusian Catastrophe article on Wikipedia

Demographic Transition article on Wikipedia

Hunger Map - The United Nations World Food Program

Paul R. Ehrlich and the prophets of doom - An Exponentialist View


Ever since biologist Ehrlich wrote "The Population Bomb" in 1968 his name has become associated with the doomsayers and their pessimistic predictions of catastrophe. "The Population Bomb" was revised and reissued twice more (1971, 1978) , and then "The Population Explosion" was published in 1990.

In this article I will explore the explanation of exponential growth by Ehrlich and other people perceived as prophets of doom (Bartlett, Hardin, Malthus and Meadows). I think it is important to expose and dispel some myths and confusion regarding exponential growth which crop up in books about the ecological state of our planet.

Ehrlich's Malthusian heritage has been clearly articulated more than once by his many critics (for example Maddox [1972], Simon [1980], Bailey [1994] and Lomborg [2001]). The war of words continues. Ehrlich continues to hit back at the anti-environmentalist rhetoric of some of his critics (Betrayal of Science and Reason, 1998). I will examine some of the anti-Malthusian arguments and how anti-Malthusian argument are too naive in denying the Malthusian argument. I will argue that although Malthus' argument was flawed, it contains the key to understanding a universal law of nature that is not so far removed from Malthus' original argument.

I will argue that exponential growth does not require a constant growth rate. Instead, I will argue that sustained exponential growth at a constant rate is impossible. I will argue that populations grow through variable rate exponential growth.

Population Modelling - Positive Growth

This is what Ehrlich wrote in the endnotes for Chapter 1 of "The Population Explosion" (1990):

"Exponential growth occurs when the increase in population size in a given period is a constant percentage of the size at the beginning of the period. Thus a population growing at 2 percent annually or a bank account growing at 6 percent annually will be growing exponentially."

However, Ehrlich qualifies that although human populations (Ehrlich,1990, p.15) "...have often been described as growing exponentially..." they do not in fact grow exponentially because (Ehrlich, 1990, in the endnotes, p.265):

"True exponential growth is rarely seen in human populations today, since the percentage rate of growth has been changing."

Sadly, the view that exponential growth requires a constant rate is quite prevalent. This is sad because it undermines the underlying message that Ehrlich and others are trying to get across, that the Earth cannot sustain positive rates of human population growth (whether they are constant or vary). This is exactly the same issue that faces Professor Albert Bartlett (Arithmetic, Population, and Energy, 1998), who repeatedly warns of the dangers of exponential growth at a constant rate for human populations even though there is no evidence that human populations have ever sustained constant rates of exponential growth. For more see Bartlett - An Exponentialist View. Never mind human populations "rarely" growing exponentially "today" - if exponential growth is defined by a constant rate of growth (with compound interest) then I can say with certainty that no population has grown exponentially, is growing exponentially, or ever will grow exponentially. Because, by definition, if a population was growing exponentially (at a constant rate) then it must be doing so now. Yet there is no evidence of any human population (or any population of any species, for that matter) that has been sustaining the same (constant rate) of exponential growth for all of its history.

Here is another classic definition of exponential growth from the Club Of Rome's infamous report "The Limits To Growth" (Meadows et al, 1972):

"A quantity exhibits exponential growth when it increases by a constant percentage of the whole in a constant time period. A colony of yeast cells in which each cell divides into two cells every 10 minutes is growing exponentially."

The Club Of Rome explain how a 0.3% rate of growth in 1650 meant a population doubling time of 250 years for the then population of 0.5 billion people. In 1970 the situation was radically different, with 3.6 billion people growing at 2.1% which gives a doubling rate of 33 years. They conclude (Meadows et al, 1972):

"Thus, not only has the population been growing exponentially, but the rate of growth has also been growing."

So how could the authors of the Club Of Rome's report reconcile this statement with their previous assertion that exponential growth requires a constant rate of growth? Either their thinking is entirely illogical, or their explanation is clumsy and self-contradictory.

Another well-known prophet of doom was Garrett Hardin in which he naively repeats this key part of the Malthusian argument (Hardin, 1968):

"Population, as Malthus said, naturally tends to grow "geometrically," or, as we would now say, exponentially."

Hardin doesn't challenge this assertion, he just accepts it. Then he argues, in true Malthusian style (Hardin, 1968):

"No technical solution can rescue us from the misery of overpopulation."

Professor of population Cohen agrees with the definition of exponential growth requiring a constant rate (Cohen, 1995, p.82):

"...In an exponentially growing population, the relative growth rate r, or percentage of increase per year, is constant as time passes."

Cohen refers to the 1830 edition of  Malthus' work (which was actually only a summary called "A Summary View") in which Malthus uses the population growth of England's former American colonies as an example of "unfettered growth" in that the population was doubling every 25 years. In fact, Malthus made use of this figure in the 1st (1798) edition of his book too. Cohen then refers to Malthus' estimated population doubling time of 50 years for a number of European countries. In concluding his look at The exponential curve Cohen states (Cohen, 1995):

"Surprisingly, in spite of the abundant data to the contrary, many people believe that human population grows exponentially. It probably never has and probably never will."

Strangely, Cohen misses the point that Bartlett, Ehrlich, Hardin, Malthus and others are actually making. This is the fact that sustained exponential growth at a constant rate is actually impossible. It's not just unlikely, as Cohen suggests - it's impossible. As all the evidence indicates, Nature does not allow any population of any species to sustain a constant rate of exponential growth. Nor does Nature does allow any individual of any species to sustain a constant rate of exponential growth for its body. Logic must also dictate this fact. Nothing can grow at a constant rate of exponential growth because such growth would inevitably consume all available resources in a surprisingly short amount of time, even at modest rates of growth. Hence, when any such population hits the Limits to Growth it cannot continue to grow at a constant rate as the rate is forced to vary. In short, nature abhors exponential growth at a constant rate.

Cohen then examines a number of other population models (the logistic curve, the doomsday curve, and the sum-of-exponentials curve). He then uses a semi-log plot to examine world population history for the last two thousand years. He confidently concludes (Cohen, 1995):

"...Population growth was not exponential during these millennia, whatever Malthus and others may have thought."

Assuming positive rates of growth, I would be happy to take a bet with Ehrlich, Cohen and the Club Of Rome that variable rate compound interest results in a doubling of the original sum as surely as constant compound interest. Let's start by agreeing to use the Rule Of 70 (i.e. divide the growth rate into 70) and establishing some approximate population doubling times:

What Ehrlich, Cohen, the Club Of Rome, a host of other authors and a vast number of gullible readers may have failed to realise is that a population which experiences variable growth rates between 1% and 2% (inclusive) will still double somewhere between 35 and 70 years. This is simple logic, but a surprising number seem to be incapable of understanding and believing it. Take a look at the Total Midyear World Population for 1950 to 2050 AD from the U.S. Census bureau for an example. You can see a similar situation with the variable growth rates between 1960 and 1999, which resulted in our global population doubling in 39 years. This is variable rate compound interest in action (try it for yourself - the annual growth rates are provided). 

In fact, so long as the rate is positive, it's only a matter of time before the population doubles. I guarantee it. It is not simply a matter of convention or convenience to say that a population has doubled over a given time period. It is a provable mathematical fact. Variable rates of population growth result in population doubling. Population doubling results in exponential growth. Such a form of growth is just as powerful as exponential growth at a constant rate (see What Is Exponential? for more).

Malthus claimed that populations (and stressed human populations in particular) grow exponentially. He used population doublings (at a constant rate) to illustrate the power of such growth over arithmetic growth (also known as linear growth). Nobody disputes the power of exponential growth compared to linear growth (see Linear Growth Versus Exponential Growth for a comparison of these two growth models). However, the key point here is that variable rate compound interest (effectively variable rate exponential growth) is just as powerful constant rate exponential growth. The difference is that the evidence supports the fact that all populations of all species (including human populations) grow at variable rates of compound interest (aka variable rates of exponential growth).

Ironically for Cohen's case against Malthus, it was clear in A Summary View (1830) that Malthus understood this:

"It may be safely asserted, therefore, that population, when unchecked, increases in a geometrical progression of such nature as to double itself every twenty-five years. This statement, of course, refers to the general result, and not to each intermediate step of the progress. Practically, it would sometimes be slower, and sometimes faster."

And here is Malthus' argument for exponential population growth at variable rates:

"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."

In other words, for each period for which a growth rate is being calculated, Malthus is insisting that the projected population doubling period would vary. If we assume positive growth rates, and we accept that the population must eventually double, then logic would dictate that the variable growth rates experienced must influence the actual population doubling period. They do. I explore this topic further in The Scales Of 70 and the Scales of e, which describe the Couttsian Growth Model of variable rate compound interest which is, effectively, variable rate exponential growth.

For further evidence that Malthus clearly recognised that variable rates of population increase were commonplace, I refer the reader to Malthus' analysis of the "Proportions of Births to Deaths" of Prussia/Lithuania, Pomerania, Brandenburg, and Magdeburg in the early 18th century (Chapter VII, 1st edition, 1798). Malthus concludes that the average growth rates vary so much every 5-10 years that they are "...an inadequate criterion of the real average increase of population." Malthus concedes that the growth rate clearly shows the rate of increase during the 5-10 year period, but argues that:

"...we can by no means be thence infer what had been the increase for twenty years before, or what would be the increase twenty years hence."

I think the problem that modern demographers suffer from can be described as graphilism - they love graphs! Because they endlessly plot population growth on graphs they imagine that only constant compound interest rates result in exponential growth because only constant compound interest rates result in an exponential curve. Didn't it occur to anyone to question whether variable compound interest still doubles the original population? Modern demographers have lost the plot. What is important is whether the population doubles, not how pretty and neat your graph is! You can get exponential growth without an exponential curve!

Years   10 40 80 160 320 360 400 500 600 700
Variable Breeders (thousands) 1 2 4 8 16 32 64 128 256 512 1024
Years   70 140 210 280 350 420 490 560 630 700
Constant Breeders (thousands) 1 2 4 8 16 32 64 128 256 512 1024

Table A. Disproving demographer's assertions that exponential growth requires a constant rate of growth

Table A clearly shows that the Variable Breeders, despite their variable population doubling times, have doubled repeatedly to reach a population of 1024 thousand in 700 years. Meanwhile, our Constant Breeders have reached the same figure in the same time. Both are valid examples of exponential growth, but the Variable Breeder example is more likely in reality. 

Oddly, all demographers appreciate that our global population has in fact doubled historically. Most people would recognise this repeated doubling as an exponential series, and yet demographers regularly fail to explain that this exponential growth is caused by variable rate compound interest and not a constant rate of compound interest.

In describing a "slight slackening" in the global growth rate, Ehrlich (1990) states:

"The slowdown has only been from a peak annual growth rate of perhaps 2.1 percent in the 1960s to about 1.8 percent in 1990. To put this change in perspective, the population's doubling time has been extended from thirty-three years to thirty-nine. Indeed, the world population did double in the thirty-seven years from 1950 to 1987...."

Well, for the exponentialist the explanation for that doubling is variable compound interest. Repeated doubling means exponential growth, even if the doubling time is not constant.

Population Modelling - Negative Growth

Exponential growth can be negative, as well as positive. Getting back to the quote from the endnote from chapter 1 of The Population Explosion", Ehrlich (1990) continues:

"Exponential growth does not have to be fast; it can go on at very low rates or, if the rate is negative, can be exponential shrinkage."

Ehrlich clearly understands that exponential growth rates can be negative (though it's not clear here whether he imagined that only a constant rate would cause exponential shrinkage). 

Malthus' case for population shrinkage can be logically drawn from his original statement on population growth thus (Malthus, 1798):

"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."

Taking negative annual growth rates, a population which experiences variable growth rates between negative 1% and negative 2% (inclusive) will still halve somewhere between 35 and 70 years. In fact, so long as the rate is negative, it's only a matter of time before the population halves. Again, I guarantee it.

Years   10 40 80 160 320 360 400 500 600 700
Variable Losers (thousands) 1024 512 256 128 64 32 16 8 4 2 1
Years   70 140 210 280 350 420 490 560 630 700
Constant Losers (thousands) 1024 512 256 128 64 32 16 8 4 2 1

Table B. Negative growth at variable and constant rates for populations losing 
Malthus' Struggle For Existence - using Couttsian Growth Model

Table B shows two populations both losing the Malthusian struggle for existence in the same timeframe. This demonstrates exponential shrinkage, regardless of whether the negative growth rates are variable or constant.

Modelling Positive and Negative Growth - Egypt

Graphed, Egyptian demographic history looks like this:

Graph A. The chaotic history of Egypt's population

In "How Many People Can The Earth Support?" Cohen uses Egypt as an example of erratic population growth, demonstrating periods of negative and positive growth. Historical highlights (low points?) include the Persian Conquest (541 B.C.), Macedonian Conquest (332 B.C.), Roman Conquest (50 B.C.), pandemic begins (541), Arab Conquest (641), plague leaves (719), plague returns (1010), Black Death in Europe (1348), Turkish Conquest (1517), British occupations (1801, 1882), Egyptian independence (1956):

Years 575 560 550 540 490 525 332 60 75 ? ? 541
2 4 8 12 16 24 16 8 7 8 16 32
Variable Doubling or Halving Period   15 10   60     550   ? ?
Years 641 719 1000 1250 1517 1800 1810 1875 1930     1970
21 10 16 8 4 3 4 8 16     32
Variable Doubling or Halving Period     ? 250 267     65 55     40

Table C. Egypt experiences Malthus' Struggle For Existence (Years in red are BC)

Table C is an approximation of Cohen's graph for Egypt. I've taken the liberty of highlighting numbers in the exponential doubling series (1, 2, 4, 8, 16, 32 etc) which apply throughout Egypt's demographic history. Notice the peak of 24 million in 525 B.C. Where does this figure lie in the exponential doubling series? Well, this is double the 12 million in 540 B.C. which nicely demonstrates the fact that population doubling can apply to any sized population (not just those neatly on the exponential doubling series).

Notice how the Couttsian Growth Model (which I freely admit is a Malthusian inspired population model) holds throughout Egypt's demographic history, regardless of whether growth is positive or negative. Furthermore, the population model allows for variable rates of growth (and definitely does not require constant growth rates).

Effectively, for over 2,600 years, Egypt's population has moved up and down the exponential doubling series (never going below 1 million, or over 32 million). The cumulative effect of variable rates growth rates over the course of Egyptian history has resulted in variable population halving times (when the overall trend is for negative growth), or variable population doubling times (when the overall trend is for positive growth). 

For a growing population it is possible to prove mathematically that variable compound interest results in population doubling. Over time the population doubling period itself can vary but the result is still exponential growth.

For a shrinking population it is possible to prove mathematically that variable compound interest results in population halving. Over time the population halving period itself can vary but the result is still exponential shrinkage.

The Egyptian population has always been subject to the Couttsian Growth Model, and has always grown exponentially (via Couttsian Growth) or shrunk exponentially (via Couttsian Shrinkage). Given the ease with which the Couttsian Growth Model copes with Cohen's Egyptian example, it is clear that the same could be done for our global human population. This contradicts the assertion (Cohen, 1995):

"In any event, the available simple models do not describe human population history."

The deceptively simple Malthusian population model (constant rate exponential growth) which I have refined and called the Couttsian Growth Model (variable rate exponential growth) does describe human population history. To turn Mr Cohen's phrase on its head:

"Surprisingly, in spite of the abundant data to the contrary, many people believe that human population does not grow exponentially. Yet it always has and always will."

Even Zero Population Growth, that transitory moment when the birth rate (plus immigration) equals the death rate (plus emigration) for a given year, fits within the same Couttsian Growth Model. A zero percent growth rate applied to any figure results in no growth.

Overlapping Doubling Periods, Overlapping Halving Periods

There is one situation in which a model which uses population doubling and halving times might seem inappropriate. That is when a growth trend is not sustained for long enough to halve or double a population. However, Egypt provides examples which show that this is not as big a problem as it might first appear. In fact, in practical terms, this is not a problem at all in the Egyptian case.

Consider the 16 million in 490 B.C. They failed to double to 32 million and in fact only reached 24 million. Yet, as we saw, there were 12 million in 540 B.C. This population did double to 24 million in 525 B.C. - a population doubling time of just 15 years. 

Consider the 3 million in 1800, which is not half the 4 million in 1517 nor half the 4 million in 1810. Yet somewhere between 1250 and 1517 there were 6 million Egyptians, which consequently halved to 3 million by 1800. Somewhere between 1810 and 1875 there were again 6 million Egyptians, with the population rapidly doubling from 3 million in 1800 following the introduction of modern medicines to Egypt. 

Notice that population doubling periods overlap in Egyptian history:

This point is also illustrated by the following global population milestones are taken from the United Nations Population Fund:-

If the overall trend is for exponential shrinkage, then population halving periods can be demonstrated to overlap. 

As you can see, population doubling applies at every moment leading up to each peak throughout Egyptian history, and population halving applies at every moment leading up to each trough throughout Egyptian history. 

Why Use Doubling and Halving?

This is a good question, and highlights the fact that even the standard Malthusian Growth Model does allow us to use tripling and thirding, quadrupling and quartering, or even ten-fold increase and decimation (destruction of one in ten, once a form of punishment for Roman Legions which performed poorly in battle). Both constant and variable rates of growth can result in any exponential growth series (doubling, tripling, quadrupling etc) or shrinkage series (halving, quartering, decimation etc). 

In fact, an obvious demonstration of this principle is the fact that the quadrupling series is a sub-set of the doubling series:

Malthus was most definitely aware of the flexibility of the population growth model which he proposed. For example, in 1830 he wrote an example of sextupling for grain. Typically however, Malthus preferred to use population doubling. Given that any exponential series is valid, yet doubling one of the easiest to comprehend (together with the opposite case of halving), my own preference is also to use population doubling to demonstrate exponential growth and halving to demonstrate exponential shrinkage). 

Also see my article The Scales Of 70 which shows the usefulness of population doubling and halving, provides a clear and practical approximation of the nature of such growth, and the underlying universal law of nature underpinned by the transcendental number e. The full model - the Couttsian Growth Model - is best illustrated by the Scales of e.

Malthusian Catastrophe?

In "The Population Explosion", in the endnotes, Ehrlich provides a novel but powerful catastrophe scenario (Ehrlich, 1990):

"The potential for surprise in repeated doublings can be underlined with another example. Suppose you set up an aquarium with appropriate life-support systems to maintain 1,000 guppies. If that number is exceeded, crowding will make the fishes susceptible to "ich", a parasitic disease that will kill most of the guppies. You then begin the population with a pair of sex-crazed guppies. Suppose that the fishes reproduce fast enough to double their population size every month. For eight months everything is fine, as the population grows 2=>4=>8=>16=>32=>64=>128=>256=>512. Then within the ninth month the guppy population surges through the fatal 1,000 barrier, the aquarium becomes overcrowded, and most of the fishes perish. In fact, the last 100 guppies appear in less than two days - about 2 percent of the population's history."

Ehrlich assumes a constant rate of growth whereby the guppy population doubles every month. Of course, we can now see that the guppies would face the same catastrophe even with variable population doubling times (refer back to Table A). 

Ehrlich, like Malthus, has a reputation as a prophet of doom for humanity. For example, in 1971 Ehrlich was in a televised debate on Australian TV, the transcript for which was published by ABC-TV in Saving Our Small World (1974). In the chapter on Ehrlich he is reported to have made the following comment in his opening remarks:

"...it's quite clear that on a permanent basis we cannot support 3.7 billion people. In fact that's somewhere between three and seven times more people than the planet can support."

Robert Moore, the interviewer, later asked Dr Ehrlich the following question:

"To come back to the food problem for the moment, can you see any foreseeable technological advances that would enable the food balance to be brought into balance with the population?"

Ehrlich estimated that five to seven billion could live at subsistence levels, and noted the contribution to the Green Revolution made by Lester R. Brown of USAID and Nobel Prize winner Norman Borlaug. Ehrlich then states:

"...both say that if we're very fortunate, the Green Revolution will buy us perhaps ten to twenty years to solve the population explosion. If you know anything about demography, you know that, short of a rise in the death rate, there is no way whatever you can bring the population explosion to a halt in that time."

Just to be absolutely sure that we were all doomed, Robert Moore asked:

"There is no contemporary or foreseeable future technology that can cope with the present rate of population increase?"

Ehrlich's reply was quite definite, to say the least:

"None whatsoever. Absolutely hopeless."

Ehrlich was right to assert that the population explosion would not be stopped by 1994. In fact, today's most optimistic estimates give a date of between 2075 and 2100 (see Human Replicators - An Exponentialist View). Nevertheless, Ehrlich's catastrophic scenario hasn't happened. 

The Club Of Rome's pessimism in "The Limits To Growth", and Garrett's in "The Tragedy Of The Commons" were equally also unfounded. Somehow, the world's population continues to grow. Books such as L.T. Evans' "Feeding The Ten Billion" are not too despairing of our future. Lester R. Brown (now with Worldwatch) conscientiously contributes to annual reports such as "Vital Signs" and "State Of The World", plus books such as "Beyond Malthus" (which, disappointingly, does not mention Malthus). Worldwatch at least do their best to present the good news and the bad news, and not just the bad news.

However, as Malthus (1798) himself stated (see Malthus - An Exponentialist View for more):

"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."  

So it is that the name of Malthus is associated with pessimism, and prophecies of impending doom. Strictly speaking though, what Malthus was saying was that mankind has always been in this state, is in this state now, and always will be in this state. Apart from the "positive" checks" such as war, pestilence and famine Malthus advocated "moral restraint" to reduce growth and thus reduce unnecessary suffering, though he opposed contraception and abortion (which also act to reduce population growth).

Of course, Ehrlich's argument is essentially the same as Malthus' argument anyway. Ehrlich's key error (apart from his unrelenting pessimism) is his attempt to make a specific timed prediction rather than claim a universal law of nature that applies all the time to all populations of all species for all time as Malthus did. However, the human species cannot continue to grow exponentially (at variable positive rates of growth) on planet Earth. On that precise point I agree wholeheartedly with Malthus, Ehrlich, Bartlett and all other such prophets of doom.

A Simple And Attractive Theory

Lomborg (2001) quotes Ehrlich (1968) in "The Skeptical Environmentalist":

"The battle to feed humanity is over. In the course of the 1970s the world will experience starvation of tragic proportions - hundreds of millions of people will starve to death."

Lomborg then quotes Lester Brown (1965), founding president of the Worldwatch Institute:

"...the food problem emerging in the less-developed regions may be one of the most nearly insoluble problems facing man over the next few decades."

Lomborg (2001) concludes:

"They were both mistaken. Although there are now twice as many of us as there were in 1961, each one of us has more to eat, in both developed and developing countries. Fewer people are starving. Food is cheaper these days and food-wise the world is quite simply a better place for more people."

To the exponentialist, this would not have come as such a surprise. However, it is clear that Lomborg is not a Malthusian, nor an exponentialist (though he is an Associate Professor of Statistics). In the very next section entitled "Malthus and everlasting hunger", Lomborg (2001) then wrote of Malthus:

"Malthus' theory is so simple and attractive that many reputable scientists have fallen for it. But the evidence does not seem to support the theory. The population rarely grows exponentially, as we saw in the introductory section (Figure 11). Likewise, the quantity of food seldom grows linearly...."

Lomborg's Figure 11 is  from the U.N.'s medium variant forecast (see Human Replicators - An Exponentialist View for more). This shows the classic exponential curve from 1750 to 2050, which then flattens out giving the famous "S" curve.

It is easy to understand why Lomborg also interprets Malthus as saying that populations grow exponentially (at a constant rate). After all, Malthus used as his "general rule" a regular human population doubling time of 25 years (derived from population data then available from the USA's demographic history). This is the same mistake that Cohen makes. However, as demonstrated above, Malthus was well aware that populations grow at variable rates which result in variable doubling periods. The key point here is that variable rate exponential growth is just as powerful as constant rate exponential growth, and that the Earth cannot sustain positive rates of exponential growth whether those rates are constant or variable.

On Lomborg's other point, that food quantity seldom grows linearly, I make this same point myself in discussing Malthus on Plants And Animals, Malthus On Grain and Malthus On Sheep. Malthus hasn't helped his own argument that his Principle Of Population is a universal law of nature by then modelling linear growth for food quantity. This contradiction  has confused many people. After all, all food is derived from other species of replicator populations. However, rather than therefore assuming that Malthus is totally wrong I have simply dropped Malthus' linear model for growth of food quantity and instead I've applied Malthus' exponential model for population growth to the increase of food quantity (which, as I said, grow in populations anyway). I also allow variable rate exponential growth in lieu of Malthus' constant rate exponential growth (see What Is Exponential? for more). I find that this further simplification of Malthus' Principle Of Population is very attractive for its universal application to all replicator populations. This leads to the exponentialist definition of farming:

Activity which deliberately encourages and harnesses the exponential growth of replicators from one or more species for the supply of food.

Note that Limits to Growth, and the Malthusian Wall, must eventually apply to any replicator population which sustains positive population growth. Also, never forget the Struggle For Existence that exists between all replicator populations. Thus if one population is able to sustain positive population growth then it will be at the expense of one or more other replicator populations - see Darwin's Wedge for more.

Lomborg is widely reported as being inspired by an interview (issue 5.02) in Wired magazine with Doomslayer and Cornucopian Julian Simon, and here is what Simon had to say about Malthusian views on food and population.

Julian Simon - Doomslayer

Simon's war against "The Litany" (his term for what he regards as the false environmental doom saying of the likes of Ehrlich) began with an article in Science (Simon, 1980). The Malthusian Ehrlich then famously lost his bet with cornucopian Julian Simon over the rise or fall of certain commodity prices from 1980 and 1990, and even refused a subsequent bet with Simon. However, my focus here is Simon's views on population growth and the growth of food supply. Central to Simon's cornucopian views are his criticisms of Malthus' mathematical Principle of Population that population grows geometrically (exponentially) and food grows arithmetically (linearly) [Regis, 1997]:

"Population has never increased geometrically" says Simon. "It increases at all kinds of rates historically, but however it increases, food increases at least as fast, if not faster. In other words, whatever the rate of population growth is, the food supply increases at an even faster rate."

I almost find myself completely agreeing with Simon, but not quite. Simon is absolutely correct to assert that population has never increased exponentially (at a constant rate), but grows at variable rates. One point that he misses here is that variable rate exponential growth is just as powerful as constant rate exponential growth (see What Is Exponential? for more). Hence, Malthusian fears of limits to growth are still valid. Simon is close to the truth in his observation that the growth of food supply matches population, but not quite. I think he overstates the case. I believe Malthus meant  - and is correct in asserting - that localised (not global) famine has been, is now, and will always be a "periodic misery" for all species, including humanity (along with other checks such as war for humans, and pestilence for all species). However Simon's observation could be reasonably taken to mean that, historically, overall world food supply has kept pace with world population growth for humans. This is true, otherwise our global population could not have grown from around 1 billion in 1798 when Malthus penned his essay to close to 6 billion at the time of Simon's death in 1998. But just because world food supply has kept pace with population in this time does not mean that world food supply inevitably will keep pace with, or exceed, the growth of population.

Simon's interviewer, Ed Regis (possibly paraphrasing Simon), notes that Simon's observation regarding the growth of food supply shouldn't come as much of a surprise as (Regis, 1997):

"Plants and animals used for food constitute 'populations' just as humans beings do, and so they, too, ought to increase not arithmetically, as Malthus claimed, but geometrically. The food supply, in other words, ought to keep pace with human population growth, thereby leaving us well-fed, happy, and snug in our beds."

Again I almost completely agree, but not quite. Yes, plants and animals used for food do grow in populations. Hence, by Malthus' logic, the same geometrical (exponential) growth model should apply to food. I make this same point repeatedly in my Exponentialist site, and it amazes me that Malthus missed this logical fallacy in his own argument. However, if Simon can dismiss exponential growth as a valid model for population growth for humans then surely - logically - it is an equally invalid model for the population growth of animals and plants? After all, there is no evidence that food supply grows at a constant rate of exponential growth (geometrically) either. Simon can't have it both ways. So how is it possible that the growth of food supply can - but not necessarily must - keep pace with or even exceed human population growth?

The answer is that all populations of all species obey the same law of nature for all time - a concept first proposed by Malthus, albeit in a flawed form, in 1798. What law of nature is this? It is that populations grow via variable rate exponential growth (or some might prefer variable rate compound interest). See my Scales of 70 and Scales of e for how this works in practice. Thus, the exponential power of each  discrete (e.g. national) human population is pitted against the exponential power of each and every population of animals and plants that make up our food supply.  Malthus' lurking threat of "inevitable famine" will reduce any population that is too successful for too long, due to limits to growth. It may also strike simply due to a natural balancing of exponential forces as we see in predator-prey populations, or herbivore-plant populations. Or it may strike human populations through (for example) crop failure, natural disasters, or pestilences (which are microbial populations also growing at variable rates of exponential growth). Each of these populations (humans, animals, plants and microbes) is subject to Malthus' population checks (with war and moral restraint being applicable mainly just to humans).

For more on Julian Simon's views on Malthus' Principle of Population, and why I think the Malthusians and Cornucopians are both wrong (and both partially right) see Albert Bartlett - An Exponentialist View.


Ehrlich (1990) seems to understand perfectly well the connection between population and evolution:

"All of us naturally lean toward the taboo against dealing with population growth. The roots of our aversion to limiting the size of the human population are as deep and pervasive as the roots of human sexual behaviour. Through billions of years of evolution, out reproducing other members of your population was the name of the game. It is the very basis of natural selection, the driving force of the evolutionary process. Nonetheless, the taboo must be uprooted and discarded."

Well, I'd agree with the part about uprooting the taboo against dealing with population growth, but perhaps not in the way that Ehrlich would like. 

Zero Population Growth (replication rate equals death rate) is the evolutionary equivalent of treading water. Reaching land would allow population growth to continue. If you stop treading water (leading to Negative Population Growth), you start to drown. This view is neatly summed up as follows (Whitfield, 1993, p. 182):

"The extinction of a species does not usually involve the sudden death of all its individual members. Rather, it is a function of the dynamics between rates of birth and death. Species will persist when their overall birth rate equals or exceeds their death rate. But if the latter exceeds the birth rate for a long enough period, replacement of one generation by the next ceases to exist. If no new factor intervenes then the species will go extinct."

I agree with Ehrlich that, for the moment, we should limit the size of our global population. Why? Because I am saddened by the decreasing biodiversity on Earth caused by human overpopulation, and the loss of species in mere decades after evolution over millions of years.

What I disagree with is Ehrlich's apparent conclusion that we can together fight evolution. On Earth, if you check the figures, you will find that nations do not share the same growth rates. This is how it has always been, and this is what the predictions indicate for the future (see Human Replicators - An Exponentialist View for more). Hence, through differential replication, some populations are destined to fade into history and some will sustain growth for long enough to surpass them in number. Therefore, it is a fallacy to imply that limiting population will alone stop the process of differential replication amongst human populations. At the same time, I hasten to add that I do not favour the differential reproduction of any human population over any other human population.


Another factor which would prove Ehrlich wrong is the colonisation of space. Put simply, the exponentialist view is that evolution will inevitably favour any population which can adapt to live and grow in space. By grow I mean be capable of sustaining population growth. Such a population, by definition, will be the fittest in evolutionary terms and will face no real limits to growth for the foreseeable future. Yes there will be practical limits to growth faced with each moon, each planet, each solar system. As each limit to growth is faced, evolution will again favour growth over stagnation or decline. Each limit to growth is thus overcome, and so life will continue to grow exponentially for the foreseeable future.

This is not to say that the colonisation of space is inevitable (though personally I am hopeful), but merely that evolution will favour any population which breaks free of Earth's limit to growth and is able to sustain growth in space. Critical to their success will be the ability to identify the available resources in space, and to then exploit them to the full. This will require cultural, technical and biological adaptations never seen in the history of the Earth. Once life breaks free from Earth, Earth will become an evolutionary sideshow (both in terms of numbers, and diversity).

It is not so much a case of the meek inheriting the Earth, but more a case of a tiny minority of humans from Earth shaping the destiny of future generations in space. It might be as few as a thousand, or perhaps several millions over the coming centuries. In the end, with Earth still struggling to contain its population, evolution will once again demonstrate that the future belongs to the few ... and their many descendants.

Addendum - John Maddox and The Doomsday Syndrome

I recently (May 2003) came across this 1972 attack on pessimism in a second-hand book store. I haven't read it all yet, though I immediately read through Chapter 2, The Numbers Game. This attacks (as do I) the arguments of the likes of Paul Ehrlich  - the Doomsayers, and makes no bones about the central tenet of their dogma (Maddox, 1972):

"...it is no surprise that the growth of population is central to the doomsday movement. The most common assertion is that the population of the world is growing more quickly than the supply of food, with the result that famine will occur."

Maddox traces this argument back to Malthus' eloquent and apocalyptic essay, and notes Malthus' argument that starvation, disease, war and "moral corruption" would each serve to check population growth. Perhaps because he's read too much Ehrlich, and not enough Malthus, Maddox is a little too quick to question the overriding role of famine as stated by Malthus and Ehrlich in their own separate ways. Although Maddox does acknowledge a role for famine, much is made of the absence of an Ehrlich-like famine-related catastrophe, as if this proves Malthus wrong. This ignores a long history of famine in nations such as China (Peterson, 1979):

"Between 108B.C. and A.D. 1911, China withstood 1,823 famines, or nearly one per year during those two millennia. Most were over only a portion of the country; the worst were nation-wide. But throughout China, every district experienced a famine at least several times during each normal lifetime...

Millions upon millions died in these ongoing - and largely unpublicised - catastrophes. And during that time China's population has, despite faltering in it's Malthusian struggle for existence, continued to grow subject to Malthus' Principle Of Population. China is not an isolated case. 

In India, Peterson (1979 ) notes (quoting Ambirijan, 1976):

"It was hardly necessary in nineteenth century India, as another example, for 'Malthusians among the Indian civil servants' to conclude that 'if there was not sufficient land to sustain the population, the surplus should be removed either by emigration or by death.' On the contrary, the British brought the endemic famines under control partly by improving irrigation systems, mainly by building a rail net that provided the means of transporting food to areas where shortages started."

This is basically what Malthus would have predicted, but not what Ehrlich would have predicted. Famine is - or, in some cases, was - a key "endemic" factor, and has and will regularly sweep away those that don't die due to war and disease and other vices of mankind. But when the means of sustenance increases, so too does the population. The key failing of Malthus and Ehrlich was that they couldn't conceive how such an increase could be achieved (or even that better food distribution alleviates famine as an endemic factor even without an increase in food!). In short, science and technology made this sustained increase possible (though they cannot sustain such population growth forever if we stay on Earth, as Maddox himself notes). 

Malthus - False Prophet of Doom?

Also, Maddox makes a great fuss of Malthus' later inclusion of "moral restraint" from the second edition of his essay onwards, claiming this was the clincher in reducing Malthus to a false prophet of doom (Maddox, 1972):

"The change of ground, of course, is crucial. Malthus began by ignoring the evidence even then available that human populations can regulate their fertility without the help of external catastrophe, with the result that he became a prophet - a false prophet as it turned out - of doom. He finished on firmer ground, but with an argument of much less awesome significance. Who needs to be alarmed if disaster can be avoided by fertility restraint of a kind even then widespread?"

Apart from attacking Malthus for modifying his initial argument (which I think speaks volumes for Malthus' maturity as a scientist), what Maddox has failed to acknowledge throughout this chapter is Malthus' 1798 discovery of an approximate Law Of Nature, namely the Exponential Law, applicable to all populations of all species. This is where I disagree with Maddox. Malthus was not a false prophet of doom as he did not predict the catastrophes that Maddox and Ehrlich discuss. Rather, he predicted that humanity (and all species) would constantly press upon their means of sustenance, and increase in numbers should those means increase. However, I agree with Maddox that Ehrlich is a false prophet of doom. I also think Maddox makes a gross miscalculation in thinking Malthus - or even Ehrlich - should not be alarmed, simply because moral restraint has the potential to limit growth. In Malthus' day there were roughly 1 billion people worldwide, and in Ehrlich's day roughly 3 billion. Today there are over 6 billion people worldwide. This was an alarming trend, and clearly "fertility restraint" (even if we agree it was commonplace back then) wasn't much practical use in stemming that trend. Thus, I also disagree with Maddox that this late introduction of "moral restraint" is "of course, is crucial."

Even as recently as 15th May 2008, The Economist published an article entitled Malthus, the false prophet (6th May, 2008) claiming that the "pessimistic parson" is "wrong as ever" calling Malthus' work "crude demography." The claim is that the industrial revolution invalidated the "Malthusian heresy" (that populations growth rates would rise in times of plenty) as industrialised nations underwent a demographic transition and growth rates fell. The Economist article claims that there is "no limit to human ingenuity" in overcoming any future limits to growth.

As noted in the article, Malthusian fears peaked along with the global growth rate as recently as the 1970s though the growth rate has since fallen to 1.2 percent. The United Nations Population Fund claims (though with little certainty) that our global population will "nearly stabilise" at just over 10 billion soon after 2200 (The World At Six Billion, United Nations), whereas the US Census Bureau (Total Midyear World Population for 1950 to 2050 AD) predicts a global population at over 9.5 billion as early as 2050. Time will tell. In the meantime I note that the Telegraph (Malthus, the false prophet) claims that in England's population will rise to 68 million by 2056 to be the "the most crowded in Europe". So much for the demographic transition in England, Malthus' home country. Another nation that received Malthus' attention in terms of its population was the USA. This is another of those nations (industrialised after Malthus wrote his essay) that presumably underwent the much lauded demographic transition, and yet the population of the USA has remained positive since before the time of Malthus and is predicted to remain positive longer than most nations on Earth (Population Set To Decline - Nature Magazine online, 2001).

A quick glance at The United Nations World Food Program Hunger Map shows that ongoing Malthusian concerns of periodic famine around the world continue to be valid. This is just as Malthus (1798) predicated:

"...this constantly subsisting cause of periodical misery has existed ever since we have had any histories of mankind, does exist at present, and will for ever continue to exist, unless some decided change takes place in the physical constitution of our nature." 

The Economist continues the attack in the same manner in The Malthus Blues (9th June 2008). Oddly, given the Economists' anti-Malthusian sentiments,  the article notes the simple and appealing Malthusian argument:

"It is an inescapable truth of truth of mathematics that that any geometric progression will overtake any arithmetic one given enough time."

Yes, this is an inescapable mathematical truth, but so what? The key is to ask whether was Malthus right to claim that population grows geometrically and food grows arithmetically? The article doesn't even challenge this classic Malthusian argument. Why note this inescapable mathematical truth? Simple - it provides ammunition against all arguments pertaining to overpopulation. It's almost a form of straw man fallacy, creating a false position for your adversary's argument in order to more easily destroy it. In this case Malthus' argument is repeated fairly faithfully, and it is hard to read arguments for sustainable ecology (e.g. Bartlett, Ehrlich, Hardin) without tripping over Malthus' argument. But the Economist notes Ehrlich's failed predictions of global famine as an embarrassment for the ecologically concerned. So we have a flawed Malthusian argument regularly presented by its advocates and its enemies as if it were an inescapable mathematical truth. This doesn't make sense, especially if  - like the Economist - you clearly don't buy the Malthusian argument. The optimistic Economist article is anti-Malthusian, and clearly doesn't believe that we need suffer "Malthusian Blues." So the Economist repeats the simple and flawed Malthusian argument and elevates it to the status of an inescapable mathematical truth simply to make it easier to claim that everything is ok now and so we can have "cheering thoughts about population." This argument is then supported by United Nations predictions of falling growth rates and a stable global population of around 9 billion in the last twenty-five years of the 21st century (different again to the predictions on global population stabilisation noted earlier). We don't need to worry about overpopulation and that foundation of ecological science "carrying capacity" any more, it is hoped. Just worry slightly about population shrinkage, and forget overpopulation - it was just a "uniquely 20th century phenomenon."

I disagree, as it can clearly be demonstrated that variable rates of population growth are just as powerful as constant rates of population growth (see What Is Exponential? for more), and hence just as unsustainable. Overpopulation is not just a uniquely 20th century phenomenon, it is a universal phenomenon curtailed through various checks (and voluntary restraints) on population as Malthus claimed.  The Couttsian Growth Model, best illustrated by the Scales of e, will always apply. Whatever we call it, this same universal law of population growth has always applied to all populations of all species, does now apply to all populations of all species, and will always apply to all populations of all species. Limits to growth and carrying capacity still apply - the question is for how long will our global growth rate "stabilise?"

The Numbers Game

Maddox in fact acknowledges that (at the time of writing) the world population has been growing faster than ever before, and cannot continue at its present rate. Few  - including Malthus and Ehrlich - would disagree. However, he continues (Maddox, 1972):

"The most gloomy prophecies are, however, unwarrantable. Too often these arguments rely on arithmetic that is misleading in its simplicity...

...One of the saddest features of the present alarm about he growth of population is that so little has been learned from Malthus' mistakes. Simple arithmetic is given more respect than it deserves. As Dr Ehrlich explains, ...'no matter how you slice it, population is a numbers game."

...The trouble with the numbers game is that no amount of arithmetic accuracy can make up for faulty assumptions."

As already acknowledged earlier in this article, I believe (as others have stated) that food does not grow linearly but follows the same exponential growth model as do all populations of all species. So, I agree with Maddox about reading too much into "arithmetic that is misleading in its simplicity..."

The problem is that everyone also seems to miss Malthus' transition - between 1798 and 1830 - from the constant rate exponential growth model to the variable rate exponential growth model. This is a problem because the constant rate exponential growth model (sadly now known as the Malthusian Growth Model) does come with its fair share of "faulty assumptions", as demonstrated by Coen and Lomborg above, and is thus used to attack Malthus. In short the Malthusian Growth Model, as it is now called, assumes no limit to growth. It also assumes a constant rate of exponential growth, which never exists in Nature. Hence the immediate - but unfortunate - jump by many mathematicians, scientists and demographers to the logistic growth model (another failed growth model with faulty assumptions - see Logistic Growth versus Exponential Growth for more).

Maddox explains how, in The Population Bomb, Ehrlich projects constant rate exponential growth - doubling every 37 years - for the next 900 years. This reductio ad absurdum argument is used by Ehrlich to demonstrate that such growth cannot happen, otherwise there would be 100 people for every square yard of ground. Hence, Ehrlich repeats the Malthusian mantra that famine must thus ensue. So what does Maddox think of all this? Not much (Maddox, 1972):

"The real absurdity, however, is not the estimate of how many people there would be in 900 years but that demography can ever be as simple as this. The rate of increase is the balance between the birth rate and the death rate, both of which are continually changing, differently in different places....

... If demography is a numbers game, it is much more intricate than Dr Ehrlich implies."

Here Maddox is right, though conceptually all it takes it a single word change from constant rate exponential growth to variable rate exponential growth (which I now call Couttsian Growth). The Couttsian Growth Model comes with fewer assumptions (though it does have at least one), and can be applied to all populations of all species at all times. This model - which deserves to be credited to Malthus - does not assume a limit to growth but neither does it assume the absence of a limit to growth. Above all, just as Maddox would no doubt prefer, it does assume that the rate of increase is "continually changing, differently in different places...."

Years later Maddox again used the chapter heading "The Numbers Game" (in "What Remains To Be Discovered"), this time in a discussion of the contribution of mathematics to scientific discovery and what we might expect over the horizon. Here is Maddox on mathematics and reality (Maddox, 1998):

"It is an illusion that mathematics necessarily reflects the real world. Rather, mathematics is a means by which conclusions are drawn from stated assumptions, including the rules of inference and axioms (defined in Euclid as "that which is self evident")."

Please read my articles the Scales Of 70 and the Scales of e, which I hope you will agree demonstrates that it is axiomatic that populations grow and shrink according to variable rates of exponential growth (the Couttsian Growth Model). And I recommend reading Malthus' essay with this model in mind, in place of both the Malthusian Growth Model for populations and the linear growth model for food. 

Sadly, despite a chapter entitled "Avoidance Of Calamity", there was no attempt by Maddox to revisit his arguments from the earlier book, and no mention anywhere of any "Problems Unsolved" pertaining to population modelling or overpopulation. Indeed, population barely got a mention...problem solved?

I don't think so - see  BNBG - 6 Billion (The Cassandra Prediction - Exploding The ZPG Myth).


Australian Broadcasting Commission, Saving Our Small World - Monday Conference on population, ecology and resources. 1974. Permission to reproduce text from this book was granted by the Australian Broadcasting Corporation.

Bailey, Ronald, Ecoscam: The False Prophets of Ecological Apocalypse.1994

Bartlett, Albert, Arithmetic, Population, and Energy. 1998.

Coen, Joel E., How Many People Can The Earth Support? 1995.

The Economist, Malthus, the false prophet. Economist.com, 15th May 2008

The Economist, The Malthus Blues. Economist.com. 9th June 2008

Ehrlich, Paul R, The Population Bomb.1968.

Ehrlich, Paul R.&  Anne H, The Population Explosion. Touchstone, Simon & Schuster. 1990.

Ehrlich, Paul R., Betrayal of Science and Reason: How Anti-Environmental Rhetoric Threatens Our Future. 1998

Hardin, Garrett, The Tragedy Of The Commons - Science magazine (162:1243-1248). 1968 (accessed from Dieoff.org on 16th September 2008)

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Planetary Overload Global Environmental Change and the Health of the Human Species A.J. McMichael (1993)

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Feeding the Ten Billion Plants and population growth - L. T. Evans (1998)

The Future Eaters - Tim Flannery

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Naked Ape To Superspecies A Personal Perspective on Humanity and the Global Ecocrisis - David Suzuki and Holly Dressel

Time To Change - David Suzuki

Respect For Earth Sustainable Development - the contributors and the BBC (2000)

Believing Cassandra An Optimist Looks at a Pessimist's World - Alan Atkisson (1999)


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