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Whilst evolutionists generally accept the Malthusian (constant rate exponential) nature of population growth, others such as Dawkins (1996) sometimes point out that an exponential growth model (for example, when applied to local doubling of cell populations) is naive. However, even Dawkins (1996) does not always qualify his (typically Malthusian) explanation of exponential growth:

"It is the nature of a replicator that it generates a population of copies of itself, and that means a population of entities that also undergo duplication. Hence the population will tend to grow exponentially until checked by competition for resources or raw materials. ...Briefly, the population doubles at regular intervals, rather than adding a constant number at regular intervals. This means that there will soon be a very large population of replicators and hence competition between them."

Demographer Joel E. Cohen (1995) is highly critical of such views, arguing that:

"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."

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 dismisses popular opinion:

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

Cohen is equally critical of the Logistic Growth Model, too, pointing out the frequency of failed predictions by Verhulst and others. He concludes:

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

Cohen presents Egypt's demographic history as a classic example of intractible population modelling complextiy.

So does population grow exponentially, or not? Does a simple but universal physical law of population growth exist? Can we simply model Egypt's demographic history?

Re-examining Malthus[]

Malthus typically considered the growth rate (dubbed by Fisher as the Malthusian parameter), and from that deduced the population doubling time. From A Summary View (1830):

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.


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.

There is more to Malthus that he has been given credit for.

The Scales of e[]

Taken together, as demonstrated by my Scales of e, these statements indicate that variable positive rates of population growth result in variable population doubling periods. Conversely, it has been shown that variable negative rates of population growth result in variable population halving periods. Yet, such growth is more akin to Malthus' geometric (or exponential, constant rate) model of population growth rather than any arithmetic (or linear) model.

The Scales of e takes the concept of the Rule of 70, (or the Rule of 72), and expands it to cater for any mixture of positive and negative rates of variable compound interest. Variable rates of compound interest result in variable population doubling and population halving periods. The timeframes involved are entirely comparable to those for fixed rate compound interest.

This explains the demographic history of Egypt, and does so simply.

To avoid confusion with the existing Malthusian growth model, I call this revised Malthusian model the Couttsian Growth Model. Linguistically, all it takes is one word to change fixed rate compound interest (Malthusian Growth Model) to variable rate compound interest (Couttsian Growth Model).

Two Centuries of Exponential Growth (a fallacy)[]

Conceptually, it has taken over two centuries to make this simple leap.

No less a person than Stephen Hawking (2001) is one of many luminaries who get confused over the nature of human population growth:

"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..."

So the population grows exponentially, and the rate is currently 1.9%. But it wasn't a constant rate of 1.9% for those two centuries! What sort of growth model applies at other times? What physical law of nature determines which model applies, and when?

Paul Ehrlich (1990) is another one that promotes the concept of the exponential nature of human population growth, and notes:

"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."

Yet he also 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"

The self-styled Club Of Rome (1972) wrote:

"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."

Discussing the slowdown in world population growth rates, and with no hint of self-contradiction, they concluded:

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

Carl Sagan (1998) puts the exponential phase of human population growth back to the time of the agricultural revolution:

"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."

Again, the idea that there can be a "phase" of exponential growth implies a separate growth model (and thus a separate physical law of nature) applied during our hunter-gatherer days. Furthermore, Sagan needlessly reinforces the (Malthusian) delusion that a separate growth model applies to food which, after all, grows in populations too. There is only one universal growth model needed - the Couttsian Growth Model.

Sagan rather poigniantly states:

"If you understand exponentials, the key to many of the secrets of the universe is in your hand.""

Modern day Malthusian, Albert Bartlett has lectured on Arithmetic, Population, and Energy over 1,500 times. He too uses the classic (constant rate) exponential growth model for human populations, thus unfortunately allowing his critics to easily dismiss his prophecy of doom as human populations don't grow exponentially. Ironically, Bartlett has stated:

"The greatest shortcoming of the human race is our inability to understand the exponential function."

Even the US Census Bureau, who acknowledge that rates of population growth vary from year to year, do not see the simple population growth model of variable rate compound interest. If they had done so, they would have detected the errors that I found in their use of the exponential method.

Universal Physical Law[]

Today the Malthusian growth model bears Malthus' name, and is nonetheless widely recognised as an approximate physical law of population growth.

If the Malthusian Growth Model is an approximate physical law, then the Couttsian Growth Model is an actual physical law that applies to all populations of all species for all time. Just four words - variable rate compound interest.

Note that variable rate compound interest is simply consecutive periods of fixed rate compound interest (constant rate exponential growth), at variable rates. This makes sense, for all examples of actual exponential growth are at best temporary periods of growth at a constant rate. See Biology examples in Wikiopedia article Exponential Growth.

Fixed rate compound interest for consecutive years can be regarded as nothing more than a special (and artificial) case of variable rate compound interest. Indefinitely sustained exponential growth at a constant rate is logically and practically impossible, due to Malthusian limits to growth.

A classic example of the Couttsian Growth Model in action is the population doubling of the human world population from 3 billion in 1960 to 6 billion in 1999. Each year of growth represents classical (averaged) exponential growth at a constant rate for all 12 months. Yet each year, the global growth rate varies. Consectuve periods of exponential growth, yet the growth rate varies every year. The explanation is simple - variable rate compound interest.

Everyone agrees that fixed rate compound interest is equivalent to constant rate exponential growth (even for one year's growth). Yet popular perception (confused by definitons of exponential) is blind to the fact that variable rate compound interest is equivalent to variable rate exponential growth, because variable rate exponential growth is not supposed to exist!. But it must be so, because variable rate compound interest is just consecutive periods of fixed rate compound interest (at different rates).

Thus if exponential means "indefinitely sustained constant rate of exponential growth" then it never occurs for real populations in Nature. There is not one single documented example. If exponential means "temporary periods of constant rate exponential growth" then this is just another way of saying that you have variable rate exponential growth (i.e. variable rate compound interest). Every example of "exponential growth" turns out to be temporary, and subsequent growth must either be at a zero rate (and equally impossible to be maintained indefinitely) or a negative rate (resulting in extinction within a finite timeframe if maintained).

The logistic growth model may appear to offer a satisfying "get out of jail" clause, but appearances in this case are deceptive. The logistic growth model is nothing more than a myth.

Malthusian population thinking is the key to understanding a simple but universal physical law of population growth. According to Wikipedia, such a simple law of population growth is not supposed to exist (from the Malthus article): is widely acknowledged that population growth is almost never exponential, but instead influenced by so many factors that no simple mathematical model can describe it.


Malthus' Principle of Population is finally brought up to date. The arithmetic (or linear) model of the growth of food can be dropped altogether. To suggest a universal physical law that applies to all species, and then propose that food (which grows in populations!) grows arithmetically is logically contradictory anyway. The exponential nature of population growth remains, but is now based on variable rate compound interest rather than fixed rate rate compound interest.

Regarding the concept of Malthusian catastrophe, Malthus himself noted:

"...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.

Malthus was proposing that his physical law - the Principle of Population - has always applied, applies now, and always will apply.

Most interpretations of Malthus' pessimistic prediction emphasise Malthus' assertion that food supply cannot keep pace with population, and famine will result. It is estimated that the Chinese have experienced at least 1,800 famines. The Indians have not fared much better. Famine has always definitely been a factor. Yet human history is also rife with infanticide, murder, droughts, floods, fires, epidemics, pandemics, and warfare. This is what Malthus explained:

"The power of population is so superior to the power of the earth to produce subsistence for man, that premature death must in some shape or other visit the human race. The vices of mankind are active and able ministers of depopulation. They are the precursors in the great army of destruction; and often finish the dreadful work themselves. But should they fail in this war of extermination, sickly seasons, epidemics, pestilence, and plague, advance in terrific array, and sweep off their thousands and tens of thousands. Should success be still incomplete, gigantic inevitable famine stalks in the rear, and with one mighty blow levels the population with the food of the world."

Couttsian growth applies to our livestock and crops, and well as to human populations. The periodical misery of the human species of which Malthus spoke is caused by occasional but inevitable localised imbalances of the complex interplay of powerful exponential forces of Couttsian population growth. Murders, infanticide, wars, epidemics, droughts, floods and famines are with us still. So too are Malthusian "vices" such contraception and homosexuality, and Malthusian moral restraint.

The concept of Malthusian catastrophe needs to be refined. The failed Malthusian Growth Model appears inevitably to compel populations towards distant limits to growth. Limits to growth still apply, but populations are not compelled towards them by the Couttsian Growth Model unless they sustain variable and positive rates of population growth. However, the Couttsian Growth Model does not inevitably compel populations to sustain variable (or constant) positive rates of population growth.

Without the naive feedback mechanism of a limit to growth built into the failed Logistic Growth Model, the Couttsian Growth Model allows for any and all processes to affect the differential rates of reproduction of populations of any and all species. These processes are natural selection, artificial selection, and Malthusian Selection.

The Couttsian Growth Model is proposed as a universal physical law. Thus, all populations of all species are always subject to Couttsian Growth, or Couttsian Shrinkage, as per the model.

Although the focus of this article is populations, the Couttsian Growth Model works for any application of variable rate compound interest, or indeed for any application of fixed rate compound interest. The Couttsian Growth Model would certainly apply to the world of finance (eg. loans, investments, stocks and shares), and may also apply to the emerging field of memetics.

Einstein referred to compound interest as the most powerful force in the universe. Sure it exists in the world of finance. However, most significantly, compound interest is a force of life and its full potential is latent in all populations of all species.

See Also[]

Malthusian Selection


  • Club of Rome (1972), The Limits to Growth. Earth Island. ISBN 0-85644-008-6
  • Joel E Cohen (1995), How Many People Can The Earth Support?. Norton. ISBN 0-393-31495-2
  • Richard Dawkins (1996), Climbing Mount Improbable. Penguin Group. ISBN 0-14-026302-0
  • Paul R. Ehrlich (1990), The Population Explosion. Touchstone. ISBN 0-671-73294-3
  • Stephen Hawking (2001), The Universe in a Nutshell. Bantam Press. ISBN 9-78593-048153
  • Carl Sagan (1998), Billions and Billions. Headline. ISBN 0-7472-5792-2
  • Wikipedia Malthus article

External Links[]

Wikipedia disambiguation page for exponential - see pages for exponential growth and exponential decay Wikipedia article Interest - includes compound interest