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

External Links:

The Galilean Turn in Population Ecology - Mark Colyvan and Lev. R. Ginzburg (2001)

Laws of nature and laws of ecology - Mark Colyvan and Lev. R. Ginzburg (2001)

Ecological Kinds and Ecological Laws - Gregory M. Mikkelson (2005)

Does Population Ecology Have Laws? - Peter Turchin

Scientific (physical) Law - Wikipedia

Scientific Method - Wikipedia

Occam's Razor - Wikipedia

The Test Case As A Scientific Experiment - David Coutts (2005) - Stickyminds

Peter Turchin - An Exponentialist View

"In science there is only physics; all the rest is stamp collecting" Ernest Rutherford

"If you can't explain it simply, you don't understand it well enough" Albert Einstein


In this article I will examine current scientific and philosophical thinking regarding the fields of population dynamics and population ecology. What is population dynamics? 

"Population dynamics is the study of how and why population numbers change in time and space." Turchin (2003)

Much of the current thinking in the field appears to be geared towards convincing the scientific community that population ecology is a mature science and has scientific laws comparable to those of physics. Peter Turchin is one of those leading the charge, and so this Exponentialist article will examine Turchin's arguments.

I will start by examining what scientists believe it takes for a scientific theory to become scientific law. I will then examine Turchin's treatment of simple population dynamics (for single species populations, and exponential growth).

 I will argue that population dynamics does have at least one simple universal scientific law, but one that is unrecognised by those immersed in the obvious complexity of their field. To do that I will show that the traditional interpretation of the Exponential Law introduced by Malthus in 1798 is false, and I will provide a simpler and more useful interpretation. In fact, I will argue that scientific understanding of the nature of exponential growth is itself flawed.

Laws of Science

There appears to be a consensus amongst writers in the field of population dynamics (or population ecology) that the Malthusian Exponential Law is a prime candidate for the founding principal of the field (see references, some of which are also available online in the external links). Called a "law" by some, this is the law that population increase proceeds at an exponential rate (at a constant rate) providing the environment for that population is constant. Exponential growth is described as the "default" state.

The argument that it is a scientific law goes something like this. Newton's First Law (the law of inertia) states that an object moving in a straight line will continue to do so unless acted upon by an external force.

In Malthus' final treatise on his principle of population (A Summary View, 1830) he stated:

"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."

The modern interpretation of geometrical progression is that of exponential growth (at a constant rate). The doubling period of 25 years was based on contemporary observations regarding the doubling of the population of the former colonies in North America (now the USA). 

As Turchin concludes, both scientific statements (by Newton and Malthus) can only be derived via "speculative thinking". Both statements have the form "X will happen in the absence of external influences". Turchin argues that the two laws are completely analogous, but then conveniently demurs regarding a hard definition of the word "law".

So it is a law? From my reading of the prevailing literature, it seems to me that those in the field of population dynamics (or population ecology) feel that they are still playing second fiddle to physics. Borrowing from the words of Ernest Rutherford, they feel that they are regarded as mere "stamp collectors". 

James Trefil, a physicist, is the author of Cassell's Laws of Nature. In his introduction, he freely admits (Trefil, 2002):

"Scientists are remarkably sloppy about their use of the word 'law.'

For example, he mentions that one of the most verified "laws" of the life sciences is the "theory" of evolution!

Malthus is not even credited with the Exponential Law in Trefil's entry for Exponential Growth, and Trefil doesn't regard Exponential Growth as a realistic representation of real population growth. Trefil regards the logistic growth model as (Trefil, 2002):

"...a better representation of the growth of real populations than the simple exponential".

Trefil only implicitly credits Malthus with the law of exponential growth (see Green Revolution in Trefil's book) and by dating the law of exponential growth to 1798 (when Malthus published his essay on population).

In essence, Trefil (2002) is saying that the Exponential law is at best only an approximation as a law (and better approximations - such as the logistic growth model - are available). Yet Trefil (2002) sees Newton's First Law as a "truly revolutionary concept", and Newton's 3 laws of motion together as "...either the end or the beginning of the end. They mark a crucial turning point in the history of science...".

The contrast between Trefil's views on Newton and Malthus could not be more stark. If Trefil is anything to go by then, it appears that physicists do not see the two laws as equals in any way. 

The contrast in these two scientists' views on the nature of science is also significant. Turchin (2003) rejects Popperian falsification as "...the way of doing science." Trefil (2002) argues explicitly in favour Popperian falsification "...scientific ideas must be falsifiable."

Working in the field of software testing, I tend to favour Popperian falsification as the way to do science. I draw parallels between the scientific method and the practice of software testing in my online article The Test Case As A Scientific Experiment (2005). As a software test manager, challenging expert opinion and testing assertions by experts comes naturally. As an example of my own use of the scientific method relating to population modelling, read my article US Census Bureau - Incorrect Use of The Exponential Method

I believe that the field of population dynamics (and population ecology) would benefit from embracing the value of the Popperian falsification aspect of the scientific method.

Simple Population Dynamics

Favouring the Malthusian approach, let's stick to a well-documented human population to examine the simple population dynamics of a single species population.

The question is, for the period 1800 to 1999, is this population growing exponentially as per the Malthusian Exponential Law? In the absence of an exponential curve most experts would say no. See my articles Paul R. Ehrlich - An Exponentialist View for some examples from the "no" camp, and Albert Bartlett - An Exponentialist View and Two Centuries of Exponential Growth for some experts who explain human population increase in terms of exponential growth (at a constant rate). Read What Is Exponential? for my own opinion on what constitutes exponential growth.

In essence, what I see is variable rate compound interest resulting in variable population doubling periods. In early 2003 I put the following classic Exponentialist thought experiment to Turchin (Coutts, 2003):

"Using the Rule Of 70, a population which grows at 1% per annum doubles in 70 years. A population which grows at 2 % doubles in 35 years. Both are considered fine examples of exponential growth (each at a constant rate of  growth, producing a lovely exponential curve). The question is, if a population grows at variable rates, but always between 1 and 2 % (and thus is guaranteed to double in 35 to 70 years) - is this exponential growth, or based on some exponential function?"

The response that I received from Turchin was that exponential growth with a variable rate has been extensively studied by theoretical population ecologists, and that almost all ecological models have the form of an exponential equation with a variable rate. Turchin referred to his coverage of a stochastically varying rate (Complex Population Dynamics, p.58-59) and specifically to stochastic exponential growth (P.146 and p.404 - population dynamics where the growth rate is independent of population density). Again, Turchin emphasised that this is a topic well explored by theoretical population ecologists. It's all been done before, it seems, and there is nothing new in what I was proposing.

Yet I am at a loss to understand which of the dozens of growth models mentioned by Turchin throughout his book - if any - can explain perhaps the best documented single species population growth of all time, as depicted by the graph above. After all, if Malthus (who focussed his attentions on human populations) is the founding father of population ecology then shouldn't Turchin's comprehensive study of population dynamics be able to explain human population growth?

Yet I can explain such variable population doubling periods, and I can do so quite simply. As a rule of thumb, and a subtle variation of the Rule of 70, all you do is add up your variable positive growth rates on a year by year basis. When you hit 70, your starting population will have doubled. The same mixed bag of growth rates will double any sized starting population in the exact same number of elapsed years. A similar principle can explain population halving. 

It is thus no surprise to me that out our human population doubled from 3 billion in 1960 to 6 billion in 1999. The smallest growth rate for this 39 year doubling time was 1.26% and the largest was 2.19%. The Rule of 70 shows that the doubling time for a rate of 1.25% is 55.55 years, and for 2.19% it is 33.34 years. The 39 year doubling time falls inevitably within the 33.34 year doubling time and the 55.55 year doubling time. 

See my Scales of 70 for a fuller explanation, and my Scales of e for a 100% accurate model of single species population growth that works with the minimum of assumptions for all populations of all species, all of the time! This allows for both positive and negative variable rates of growth in any combination. The emphasis on my approach is to concentrate on how populations grow and shrink, not why they grow and shrink. 

Unlike the traditional interpretation of the Malthusian Exponential Law favoured by Turchin, it is unnecessary to draw tenuous conclusions from a superficial similarity between Newton's First Law and Malthus' Principle of Population. The fact is that there is no evidence of any population of any species ever sustaining exponential growth at a constant rate, providing the environment is constant. Forget it, it doesn't happen. For one thing, environments do not remain constant. For another, a growing population must hit a limit to growth and thus cease growing. Sustained exponential growth at a single constant rate is not the "default" state of any population of any species. As the basis for a field of science, or as an explanation of real-world population growth, it is a poor choice. 

For human populations the annual growth rate (which is rarely the same year to year) represents 365 days, 52 weeks, or 12 months of exponential growth at a constant rate, for a limited period. For an annual growth rate of 1%, the daily rate is 1.011/365 percent for a limited period of 365 days. Graph any growth rate for 365, 52 or 12 growth periods and you will get an exponential curve.  Each year the rate can vary, without having to assume the presence or absence of any influences on the growth rate (exogenous, endogenous, or otherwise). 

The Exponentialist view, therefore, is that populations grow via consecutive periods of exponential growth and that the growth rate can vary each period. Essentially, this is variable rate exponential growth. No matter how much the  rate varies from period to period, the population is growing exponentially in each period. Another name for it is variable rate compound interest. Many a mortgage is governed by a variable rate compound interest model.

There is no need to assume a "default" exponential state of growth, and some unspecified non-default non-exponential state of growth. There is only ever one state - exponential growth at variable rates.

The Scales of 70 and the Scales of e represent something that is not supposed to exist - namely a simple population growth model that is a true law of population ecology. The model works whether the environment is constant or not. Despite Turchin's claims that it has all been thought of before, I appear to be the first person to have revealed this scientific law. 

My explanation fulfils Einstein's requirement to be able to explain something simply, fulfils the parsimonious requirement of Ozzam's Razor, and has the added beauty of being consistent with the original Malthusian approach of using population doublings (and halvings).

A Mature Science 

The fields of population dynamics and population ecology do have a simple but universal law. It is one not recognised by the broader scientific community, nor by the experts in those fields of research. It was sitting in plain view on the surface of a deeply complex subject. Ever since Malthus first hinted at its discovery, its simplicity was lost in the background noise of that complexity. 

It was necessary to reject the assumption of a default exponential state (at a sustained constant rate). Instead, all populations of all species grow via variable rate compound interest, all of the time. 

Yet everyone, it seems, "knows" that populations grow via variable rates. Many "know" that this involves exponential growth, or compound interest, or both (being two sides of the one coin). Yet nobody has ever used simple logic to extend the Rule of 70 to variable growth rate, or even negative growth rates. Approximate as such an approach is, the Scales of 70 is better than any previous model for single species population growth on offer. 

The Scales of e then provides the coup de grace, a universal law of nature that accurately explains the population growth of all populations of all species for all time. The Exponentialist preference is to use the Scales of e in terms of population doubling and population halving (just like the Scales of 70), but it can be used for any growth factor.


Unlike Turchin, I am confident that Popperian falsification and the scientific method are the right approach to science. I am also confident that the Scales of 70, and the Scales of e, can be used to explain the how in Turchin's definition of population dynamics. 


Berryman, Alan, On principles, laws and theory of population ecology. Washington State University, 2003

Colyvan, Mark  and Ginzburg, Lev. R., Laws of nature and laws of ecology. 2001

Colyvan, Mark  and Ginzburg, Lev. R., The Galilean turn in population ecology. 2003

Haemig, Dr. Paul D., Laws Of Population Ecology. 2005

Trefil, James, Cassell's Laws Of Nature. 2002

Turchin, Peter, Does population ecology have general laws? 2001

Turchin, Peter. Complex Population Dynamics. Princeton University Press. 2003


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Copyright 2006 David A. Coutts
Last modified: 02 July, 2012