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:
A General Theory Of Evolution - Lars Witting

Some Related Wikipedia Links:

Evolution of Complexity
Population Dynamics
Great Chain of Being
Occam's Razor


 Lars Witting - An Exponentialist View

"What exponential growth can bring..." Lars Witting

(dedication in my copy of A General Theory of Evolution)


The object of this article is not to assess the validity of Witting's hypothesis (which he calls Malthusian Relativity). Instead, my intention is to examine Witting's views on Malthusian concepts such as exponential growth in relation to populations of replicators, and Malthus' influence on evolutionary theory and population dynamics. Nonetheless, a brief summary of Malthusian Relativity is included.

Witting is perhaps the least "famous" of the Famous Exponentialists in this site. However, if he proves successful in persuading the scientific community of his views then his fame would be assured for it is rare to overturn scientific consensus, especially where the consensus is so strong as in the case of evolutionary theory. Witting's approach is firmly scientific and thus falsifiable, and I wish him every success.

Regardless of the result of Witting's efforts in this regard, I believe that he has an admirable grasp of Malthusian concepts and influence and that is why is is included in my list of Exponentialists.

Malthusian Relativity

The main focus of Witting is on life history evolution by means of natural selection by density dependant competitive interactions. He refers to his theory as Malthusian Relativity which, in contrast to classical evolutionary theory, suggests that (Witting, 1997, p. xiii):

""...self-replicating molecules automatically evolve toward the complex organisms on Earth."

In other words, Witting argues that evolution is deterministic (directional). It is not an easy thing to overturn any aspect of "established" evolutionary theory as Witting himself experienced when unable to find a prestigious publisher for his ideas (Witting, 1997, p. xiv):

"For those of my readers who wonder why my study is published in this non-prestigious way there is only to say that the resistance against my theory was too great among the scientists who control the established scientific literature."

This is a stark reminder of the very human nature of scientific progress. It is not enough to come up with a testable hypothesis. In Witting's case no peer reviewer could dispute Witting's conclusions, whilst at the same time they persuaded editors of various scientific journals and publishers not to publish his work with the exception of a single paper on body mass allometries (Witting, 1997, p. xiv). Witting's ideas were deemed too innovative [Today, more than ten years later, most of the theory has been published in peer-reviewed journals, a review is given by Witting, 2008].

The success of any new scientific idea sometimes may require a lifetime of persistence and persuasion. Witting may like to take heart from the thoughts of a man that we both admire, Thomas Robert Malthus (1798):

"The finest minds seem to be formed rather by efforts at original thinking, by endeavours to form new combinations, and to discover new truths, than by passively receiving the impressions of other men's ideas."

Following the principle proposed by William Occam known as Occam's Razor, Witting's argument is elegantly parsimonious. He has reduced his assumptions to just one (Witting, 1997, p. 4):

"...self-replication is the origin from which all living organism have evolved."

Witting's assumption echoes the law of biology proposed by evolutionist Richard Dawkins who wondered whether the field of biology could boast any universal laws comparable to the laws of physics (Dawkins, 1976):

"This is the law that all life evolves by the differential survival of replicating entities."

See Richard Dawkins - An Exponentialist View for more.

A similar sentiment is also expressed by molecular nanotechnologist K. Eric Drexler (Drexler, 1990):

"Evolution proceeds by the variation and selection of replicators."

See K. Eric Drexler - An Exponentialist View for more.

I agree with this emphasis on self-replication as fundamental to life and fundamental to evolution. These are all statements that fit nicely with the Exponentialist view of evolution. However, I also believe that Natural Selection is only one of the forces that influence the Malthusian Parameter, and that the overriding principle of differential replication is insufficiently explained. See my articles Evolution - An Exponentialist View and Differential Reproduction - An Exponentialist View for more. 

In contrast to Witting's view that evolutional is directional, Darwinists and modern evolutionists reject the notion of a "ladder of life" (for example Aristotle's teleological scala naturae, also present in Lamarckism). Hence, the current theory of evolution does not explain the occurrence of large animals nor intelligence.

Witting's theory is aimed at (Witting, 1997, p. 8):

"...identifying a unifying force of selection in order to explain the general structuring of organic matter."

As the sub-title of his book suggests, Witting believes that the unifying force of selection is due to "...density dependant competitive interactions that exist among the individuals within populations." These interactions induce relativity in the values of the Malthusian Parameters among the individuals of those populations. Hence the term Malthusian Relativity. The Malthusian Parameter is defined as the fitness of the genome in relation to the environment (Fisher, 1930).

The restricted form of Malthusian Relativity is based entirely on the selective force of density dependant competitive interactions, whereas the general form (referred to in the title of Witting's book) aims to combine the restricted form with classical evolutionary theory (Witting, 1997, p. 9).

Witting explains that his theory also merges with the classical theory of population dynamics only when the genome is identical , as this theory (which arose from Malthus in 1798) is based on a lack of evolutionary change in the population growth rates (Witting, 1997, p. 15). For more on population dynamics, refer Peter Turchin - An Exponentialist View.

Malthusian Increase

Witting acknowledges that exponential growth, known as Malthusian increase or the Malthusian Law, is the "...cornerstone of theoretical ecology", laid down by Malthus in 1798 (Witting, 1997, p. 19). What is significant is that Witting also acknowledges  that (Witting, 1997, p. 19):

"...it is essential in the way it recognises that self-replication is a fundamental character of all organic beings."

This is entirely in keeping with the exponentialist definition of life:

A living entity is the result of a replication event by one or more replicators, and is encoded with the instructions of its own assembly.

Given that not all living entities are able to individually or jointly replicate, the ability to metabolise is also central to the definition of a living entity:

A living entity can either metabolise or replicate, or it can do both.

Thus, sterility or infertility do not prevent a creature from being regarded as living. Nonetheless every creature that has ever lived, including every one alive today, is the result of multiple replication events that lead all the way back to the very first replicators.

Witting notes that it is Fisher's idea of natural selection and not Darwin's which underpins most of the predictions of classical theory of evolution (Witting, 1997, p.19):

"Where Darwin defined natural selection from the competitive interactions that arise from Malthusian increase, Fisher disregarded these interactions, and defined natural selection from Malthusian growth in itself."

The theoretical analysis of Witting's book focuses on the questions that arise from the differences between these two definitions, and so Witting poses the question of who is right - Darwin or Fisher - and he concludes that (Witting, 1997, p.20):

"...Darwin was right in the sense that selection by density dependent competitive interactions is essential for our understanding of evolution by natural selection."

Or, in Exponentialist terms, we should consider the effect of the Malthusian Parameter as a force for selection at the level of the population via differential replication. However, the difference between the Exponentialist view and Witting's view is that Witting is considering the impact of Malthusian Relativity on Natural Selection, whereas I am considering everything from a purely population perspective.

Witting argues that the reason that Fisher failed to see Darwin's insight is because his focus was on genomic processes, whereas Darwin's focus was the "struggle for existence" (Witting, 1997, p.20) which is - as Darwin acknowledged - a Malthusian concept (Darwin, 1859, p.8):

"In the next chapter the Struggle for Existence amongst all organic beings throughout the world, which inevitably follows from their high geometrical powers of increase, will be treated of. This is the doctrine of Malthus, applied to the whole animal and vegetable kingdoms."

The Malthusian Principle

In examining Malthusian increase, Witting differentiates between continuous exponential increase (where the Malthusian Parameter is constant) and "stepped" geometric increase (with discrete values of the Malthusian Parameter per generation). 

Continuous form of Malthusian Law, or law of exponential increase (Witting, 1997, p.20):

Nt = N0ert


t = time period
r = Malthusian Parameter
N0 = initial abundance
Nt = abundance after t

Discrete form of Malthusian Law, or law of geometrical increase (Witting, 1997, p.21)

Nt = N0λt


λ = er (discrete Malthusian Parameter)
t = time in generations
r = Malthusian Parameter
N0 = initial abundance
Nt = abundance after t

When graphed the former gives the classical exponential curve, whereas the latter gives a series of discrete steps that roughly match an exponential curve (Witting, 1997, p.21).

Witting specifically notes that exponential increase is unrealistic as density dependant ecological forces cause population growth to level off as they approach the limits to growth. Witting refers to the limits to growth as the Malthusian Principle (Witting, 1997, p. 24) based on Nicholson (1933) and Smith (1935) - see Witting's references below. More generally, Witting notes that the Malthusian Law (both forms) is unrealistic (Witting, 1997, p.27), which historically led to the preferred continuous logistic growth equation of Verhulst (1838) - which was also inspired by Malthus - and later discrete logistic growth equations.

Although I agree that the Malthusian law is unrealistic, the Exponentialist approach is to reject the use of generations as naive (see Richard Dawkins - An Exponentialist View for more) and to re-examine our understanding of exponential growth altogether - see What Is Exponential?. The Exponentialist view is that the Logistic Growth Model is no more effective than the Malthusian Growth Model - see Logistic Growth Versus Exponential Growth for more.

Instead I start by using the simple yet powerful Malthusian concept of population doubling - to read more on population doubling see The Mechanism of Population Doubling. But population doubling alone is not enough. In his definition of the Malthusian Parameter, Fisher criticised Malthus for failing to adequately consider negative growth in his theory of population growth (Fisher, 1930):

"It evidently supplies in its negative values an equally good measure of population decrease, and so covers cases to which, in respect to mankind, Malthus paid too little attention."

I agree with Fisher's criticism, which I believe I have addressed by coupling the concept of population doubling with the opposite concept of population halving. As Fisher states, negative growth is equally powerful.

To get a rough idea how these two opposite concepts come together see The Scales of 70. This is an extension of the heuristic tool the Rule Of 70, which is normally only ever used to explain positive growth at a constant rate (the continuous form of the Malthusian Law). Note that the Rule of 70 is an approximation only, as is the Scales of 70. I extend the normal use of the Rule of 70 by proving that variable rates of positive growth also conform to the Rule of 70. Hence, population doublings occur regardless or whether or not the rate of growth is constant, or variable, and they always occur when the total of the growth rates equals 70. I then show how the Rule of 70 can be used for negative growth at variable or constant rates. It is worth noting at this point that there is a small imbalance between the positive and negative values of any growth rate (e.g. for a growth rate of 1%, 1.01 does not equal 0.99).

Imagine then a set of weighing  scales that "weigh" growth rates. You may add any growth rates that you like, constant or variable, positives and negatives, in any order you like. However, the positives go on one side and the negatives go on the other side. The Scales Of 70 predicts that, when the positive side "outweighs" the negative side by 70 then you have a population doubling and when the negative side "outweighs" the positive side by 70 then you have a population halving. It's that simple.

I then extend the Scales Of 70 even further into an accurate model of population growth called The Scales of e, which is based entirely on the use of natural logarithms (the "e" refers to the base of the natural logarithms). Hence I have replaced the concept of generations with the Malthusian concept of population doublings. I have replaced the Malthusian concept of exponential growth at a constant rate with variable rate growth. I have also completed the picture by accommodating negative growth rates into the model.

Essentially, I argue that populations grow and shrink via variable rate compound interest. Note that the Scales of e does not need to assume limits to growth, because it includes negative growth alongside positive growth. Limits to growth definitely do exist (see my article Human Global Ecophagy for an example), but they do not need to exist in our mathematical models in order to explain how populations actually grow and shrink. I believe that the Scales of e holds true for any population of any species for all time. I think it manages to do this by making zero assumptions, and just modelling what happens as the Malthusian Parameter varies. What it lacks in formal mathematical construction it makes up for in explanatory power.

Of course, the very thing that is missing from the Scales of e is the very thing that Witting examines in his argument for deterministic evolution - the density dependant competitive interactions of discrete populations. Such work is beyond my capabilities, but perhaps Witting may be sufficiently intrigued?


I have only briefly outlined Witting's concept of Malthusian Relativity. It is certainly an exciting concept, that evolution is inevitably directional. Reading between the lines, Witting's theoretical analysis is clearly thorough, logical and well argued.  However, I am not qualified as a mathematician nor as a scientist and I must leave it to the scientific community to consider its validity. I hope in time they will be less cautious of Witting's innovative ideas and give him a fair hearing. The interested reader can follow the link to an online copy of Witting's work via the external links listed above.

Unlike Witting, I am not attempting to explain the incredible complexity of population interactions. Rather, like Malthus, the Exponentialist approach is to assume that species do not change and instead to focus on the differential replication of discrete populations within limits to growth. This is not because I do not believe in evolution and natural selection - unlike Malthus, I do believe in evolution and natural selection. Taking a purely population view of evolution is a useful exercise in its own right, revealing the broader principle of differential replication in a true light.

I believe that overpopulation is of overriding concern for the future of humanity and for the future of life as a whole on Earth. I believe that Malthus' infamous argument in this regard - when it is not misrepresented - is frequently ignored or dismissed chiefly because his argument was flawed (see the summary on the Exponentialist Homepage). I therefore find myself drawn to any work which deals with Malthusian concepts, in the hope of re-examining Malthus and correcting his arguments. Re-examining Malthus is a constantly rewarding experience, and examining Malthus through Witting's work has been no less rewarding than usual, especially given the exciting new addition of the term Malthusian Relativity to the Malthusian lexicon.

As an amateur mathematician and scientist, also with something original to say like Witting, I am even further on the fringes of science than Witting. Witting, a practising professional scientist, could so easily have dismissed my Exponentialist attempts to correct Malthusian population theory. Instead he has been open-minded, generous and friendly. I thank him for that.


Darwin, Charles. Origin Of Species The Illustrated Edition*. Sterling Publishing Co. 1859, 2008*.

Dawkins, Richard. The Selfish Gene. Oxford University Press. 1976, 1989

Fisher R. A.. The Genetical Theory of Natural Selection - A Complete Variorum Edition* - Oxford University Press. 1930, 1958, 1999, 2003*.

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

Nicholson, A.J. The Balance of Animal Populations. J. Anim. Ecol. 2:132-178. 1933.

Smith, H.S. The role of biotic factors in the determination of population densities. J. Eco. Entom. 28:873-898. 1935.

Witting, Lars, A General Theory of Evolution By Means of Selection by Density Dependant Competitive Interactions. Peregrine. 1997

Witting, Lars. Inevitable evolution: back to The Origin and beyond the 20th Century paradigm of contingent evolution by historical natural selection. Biol. Rev. 83:259-294. 2008.

Back to Top

Send email to exponentialist@optusnet.com.au with questions or comments about this web site.
Copyright 2009 David A. Coutts
Last modified: 04 February, 2010