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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 

Generations and Population Doublings

External Links:
The World Of Richard Dawkins

Viruses Of The Mind - Richard Dawkins

The infidels.org Richard Dawkins page

Oxford University Press

Natural Selection and Differential Reproduction
- from Replicators: Evolutionary Powerhouses

Richard Dawkins - An Exponentialist View

Introduction

Dawkins is a rare case. On the one hand he never mentions Malthus once, in any of his books1. On the other hand, Dawkins is one of the best exponents of the mathematical Principle Of Population explained by Malthus between 1798 and 1830. For those of you with copies of Dawkins' work, the key chapters are Family Planning (Chapter 7) from The Selfish Gene, The Replication Bomb (Chapter 5) from A River Out Of Eden, and The Robot Repeater (Chapter 9) from Climbing Mount Improbable. In this article I also make use of a couple of quotations from The Extended Phenotype.

Malthus' Principle Of Population is generally stated in terms of an exponentially growing population outstripping its food supply. Overly successful populations are ultimately kept in check by the Malthusian limits to growth. All populations are driven, through the exponential growth of their populations, to a struggle for existence from which only the "fit" will emerge. Population growth is constantly checked by the Four Riders Of The Apocalypse, War, Famine, Pestilence and Death. In Malthusian terms these are all referred to these as "positive checks" on population. Malthus, in the second edition of his essay, added "moral restraint" as a more humane alternative. See Malthus - An Exponentialist View for more.

Replicators and Vehicle For Replicators

Dawkins differentiates between replicators and vehicles for replicators such as mice, humans, elephants or trees (Dawkins, 1982, my bolding):

"Genes are replicators; organisms and groups of organisms are best not regarded as replicators; they are vehicles in which replicators travel about. Replicator selection is the process by which some replicators survive at the expense of other replicators. Vehicle selection is the process by which some vehicles are more successful than other vehicles in ensuring the survival of their replicators."

I would like to make it clear to the reader that from an exponentialist view - with its focus on populations - there is no need to distinguish between replicators and vehicles for replicators. The same Couttsian Growth Model applies to both, and so I use the term "replicator" as short-hand for both. Perhaps the exponentialist replicator might be thought of as a "populator", though I personally find the term "replicator" works perfectly well. Also, from an exponentialist perspective, there is no need to differentiate between replicator selection or vehicle selection. Dawkins' replicator genes and their "vehicles" all exist in populations, and so I prefer to use terms for selection at the level of populations. The terms I use are differential replication and Malthusian Selection, but more on this later.

I should also explain that I have no problem with Dawkins' distinction where it applies to Natural Selection (plus Artificial Selection and even Unnatural Selection), for Dawkins is concerned with the genome and the origin of species. However, if we take an exponentialist view then the focus is on the "origin of populations" and not the origin of species.  

Dawkins does provide a broader definition of a replicator (Dawkins, 1982):

"I define a replicator as anything in the universe of which copies are made."

Today this broader definition has come to replace Dawkins' earlier, more restrictive use of the term (Drexler, 1990; Witting, 1997; Freitas and Merkle, 2004). From an exponentialist point of view, it is enough to be able to state that a population of replicators will give rise to more copies of those replicators. Thus Dawkins' broader definition is much closer to the exponentialist definition of a replicator:

Replicator - anything in the universe that is individually or jointly capable of causing a replication event.

For more on replicators see Replicators - An Exponentialist View.

Reproducers?

Darwin's Dangerous Idea by Daniel C. Dennett (1995) is described by Dawkins as "A surpassingly brilliant book". In it, Dennett declares Malthus' essay on population to be "chief among" the earlier ideas which led Darwin to his theory of natural selection. Dennett re-states Malthus' case that exponential growth for any population must inevitably lead to a crunch time when such a population must exceed the available resources (Dennett, 1995):

"It was Malthus who pointed out the mathematical inevitability of such a crunch in any population of long-term reproducers - people, animals, plants (or, for that matter, Martian clone-machines, not that such fanciful possibilities were discussed by Malthus."

Showing a clear understanding of Malthus (see The First Exponentialist), Dennett concludes (Dennett, 1995):

"So the normal state of affairs for any sort of reproducers is one in which more offspring are produced in any one generation than will in turn reproduce in the next. In other words, it is always crunch time."

It is Dennett's use of the term "reproducers" which interests me here. The problem is that viruses do not reproduce, and yet the same population growth model applies to viral populations - see Population Growth Models  for more. For more on viral populations see Viral Replication - An Exponentialist View . Also, a hypothetical Martian clone-machine (or any von Neumann machine) would be better described as a replicator than a reproducer. Thus I believe it would be better to adopt a Dawkinsian terminology, and use the term "replicator" rather than "reproducer", and "replication rate" in lieu of "birth rate".

The Selfish Gene. Chapter 7 - Family Planning. (Dawkins - 1976, 1989)

The First Replicators

In discussing the first replicators Dawkins explains that the fidelity of their replication is less than perfect (Dawkins, 1976, p.16):

"So we seem to arrive at a large population of identical replicas. But now we must mention an important property of any copying process: it is not perfect."  

He then briefly uses the analogy of text copying, which has a long history of making bad copies, and compares that with the high fidelity of the DNA replication process (Dawkins, 1976, p.16):

"We do not know how accurately the original replicator molecules made their copies. Their modern descendants, the DNA molecules, are astonishingly faithful compared with the most high-fidelity human copying process, but even they occasionally make mistakes, and it is these mistakes that make evolution possible."

Hence, evolution depends upon less than perfect replication events.

The Unit Of Selection

In my readings relating to evolution, I keep coming across the debate regarding the "level" (species, group, individual, genetic) at which evolution works. Some people think they are exclusive, some think they're not. Of those that think they're not exclusive, they then argue over whether they are concurrent or hierarchical etc. Dawkins however is quite definite that the unit of selection in evolution is the gene (Dawkins, 1976, p.11):

"...the fundamental unit of selection, and therefore of self-interest, is not the species, nor the group, nor even, strictly, the individual. It is the gene, the unit of heredity."

Dawkins thus refers to all animals, plants and bacteria as "survival machines" for genes. Taking a reductionist view of evolution, and Natural Selection in particular, I believe Dawkins is right. However, from a Malthusian (population) perspective, I believe it is valid to say that all animals, plants, fungi, bacteria and viruses are replicators. So are DNA molecules, cells and the much-anticipated replicators from the world of molecular nanotechnology. Even artificial life-forms (A-life) are replicators.

From an exponentialist perspective, which might be best described as a Malthusian Darwinist view. Bearing in mind that the exponentialist view in not concerned with the origin of species (how species evolve through Natural Selection), the exponentialist view instead focuses on the principle of differential replication as applied to all populations of replicators.

Overpopulation

Without making what should be an obvious reference to Malthus, Dawkins nonetheless provides an excellent Malthusian explanation of population growth. He even goes through the same dismissal of the factors of immigration and emigration, using the Earth's human population (just as Malthus did - see Population Doubling - The Earth) as an example of a population for which we only need consider the birth rate and death rate (Dawkins, 1976, p.110):

"Mankind is having too many children. Population size depends upon four things: births, deaths, immigrations and emigrations. Taking the world population as a whole, immigrations and emigrations do not occur, and we are left with births and deaths. So long as the average number of children per couple is larger than two surviving to reproduce, the numbers of babies born will tend to increase over the years at an ever-accelerating rate. In each generation the population, instead of going up by a fixed amount, increases by something more like a fixed proportion of the size that it has already reached. Since this size is itself getting bigger, the size of the increment gets bigger. If this kind of growth was allowed to go on unchecked, a population would reach astronomical proportions surprisingly quickly. 

It's a pity Dawkins didn't extend his explanation to show how replication rates and death rates lead to exponential growth, preferring as he did to use the standard replacement rate argument of modern demographers. See Human Replicators - An Exponentialist View for more. 

Dawkins then continues by correctly pointing out, in a similar vein to Malthus' "moral restraint", that delayed reproduction over the generations would be as effective as the "Stop at Two" slogan of today's Zero Population Growth protagonists (Dawkins, 1976, pp.110-111):

Incidentally, a thing that is sometimes not realised even by people who worry about population problems is that population growth depends upon when people have children, as well as on how many they have. Since populations tend to increase by a certain proportion per generation, it follows that if you space the generations out more, the population will grow at a slower rate per year. Banners that read 'Stop at Two' could equally well be changed to 'Start at Thirty'! But, in any case, accelerating population growth spells serious trouble."

Dawkins proceeds to give an example using the population of Latin America which does get somewhat fanciful (Dawkins, 1976, p.111):

"For instance, the present population of Latin America is around 300 million, and already many of them are under-nourished. But if the population continued to increase at the present rate, it would take less than 500 years to reach the point where the people, packed in a standing position, formed a solid carpet over the whole area of the continent. This is so, even if we assume them to be very skinny - a not very unrealistic assumption. In 1,000 years from now they would be standing on each other's shoulders more than a million deep. By 2,000 years, the mountain of people, travelling outwards at the speed of light, would have reached the edge of the known universe."

Pops (individuals) 1 2 4 8 16 32 64 128 256 512 1024
A-pops 1 2 4 8 16 32 64 128 256   512 1024
B-pops 1 2 4 8 16 32 64 128 256 286 512 1024
C-pops 1 2 4 8 16 32 64 128 256   512 1024
D-pops 1 2 4 8 16 32 64 128 256   512 1024
E-pops 1 2 4 8 16 32 64 128 256   512 1024
F-pops 1 2 4 8 16 32 64 128 256   512 1024
G-pops 1 2 4 8 16 32 64 128 256 286 512 1024

Table A. New Malthusian Scale Fanciful projected human population doubling (every 40 years), Latin America

Note:  1A-pop = 1024 pops, 1B-pop = 1024 A-pops, 1 C-pop = 1024 B-pops etc.

Let's assume a constant population doubling time of 40 years (an annual growth of roughly 1.75%). Using my New Malthusian Scale you can see that each row corresponds to 10 population doublings. Hence, each row represents 400 years of population doubling. Therefore, between 12 doublings and 13 doublings later (480 and 520 years respectively) the 286 C-pops (300 million people) would have doubled to a population between 1 D-pop and 2 D-pops. After 2000 years of doubling, the 286 B-Pops would have doubled to 286 G-Pops (337,649,203,525,179,632,779,264 people). 

Dawkins' example of runaway population growth and mountains of flesh is an example of ecophagy by a human population. For an exponentialist example see Human Global Ecophagy.

Here Dawkins appears to have the advantage of hindsight over Malthus, and can see that medical advances can fuel the exponential growth of human populations just as much as an increased food supply (Dawkins, 1976, p.110, my bolding):

It will not have escaped you that this is a hypothetical calculation! It will not really happen like that for some very good practical reasons. The names of some of these reasons are famine, plague, and war; or, if we are lucky, birth control. It is no use appealing to advances in agricultural science - 'green revolutions' and the like. Increases in food production may temporarily alleviate the problem, but it is mathematically certain that they cannot be a long-term solution; indeed, like the medical advances that have precipitated the crisis, they may well make the problem worse, by speeding up the rate of the population expansion. It is a simple logical truth that, short of mass emigration into space, with rockets taking off at the rate of several million per second, uncontrolled birth-rates are bound to lead to horribly increased death-rates. It is hard to believe that this simple truth is not understood by those leaders who forbid their followers to use effective contraceptive methods. They express a preference for 'natural' methods of population limitation, and a natural method is exactly what they are going to get. It is called starvation.

But of course the unease that such long-term calculations arouse is based on concern for the future welfare of our species as a whole. Humans (some of them) have the conscious foresight to see ahead to the disastrous consequences of over-population."

It was Malthus who, writing between 1798 and 1830, provided the mathematical certainty to which Dawkins refers. Dawkins is quite right to reject the mass emigration into space by rocket as a means of avoiding the inevitable (though the possibility of one or more space elevators may alleviate the problem slightly). He is also correct to criticise those opposed to contraception - here Dawkins could have included Malthus himself. However, Dawkins and Malthus are in full agreement that famine is likely to be the final executioner. As Malthus wrote in the 1st edition of An Essay On The Principle Of Population (Malthus, 1798):

"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."

[2009 - See Footnote - this is the exactly the same quote that Dawkins now uses in his new book!]

John Maynard Smith (an aside)

Not all evolutionists agree with the view that famine is the final arbiter in the struggle for existence (Smith,1958, p.43):

"Darwin was consciously influenced by the ideas expressed by Malthus in his Essay on Population....he [Malthus] argued that the human population is capable of increasing indefinitely in a geometric progression, and must therefore be held in check by the limited quantity of food available, and so by starvation. The argument is in part fallacious, since there is no evidence that the main factor limiting the human population is the shortage of food. However, the observation that animal and plant species, including the human species, are capable of indefinite increase in numbers in optimal conditions, is correct, and plays an important part in the theory of natural selection."

I have a few issues with this. I have already quoted enough of Malthus to show that Malthus was well aware of  other "ministers of depopulation" than famine. Dawkins' position is closer to the Malthusian "gigantic inevitable famine" which  "stalks in the rear". So Smith is only able to call Malthus' position "in part fallacious" by misrepresenting it in the first place. 

As Malthus, writing between 1798 and 1830, would be unaware of the later theory of Natural Selection, it is reasonable he does not mention it. However, it is worth noting that Natural Selection results in differential replication, but is not the sole cause of it. For the moment, like Malthus, focus on human populations. Ask yourself why the Australian population is so puny compared to the American population, and yet the land size is comparable in size. Environment (less water). Ask yourself why the African population (despite AIDS) is growing as 2 to 3 times the Chinese population. Behaviour (modern medicine, poverty, politics etc). Many evolutionists refer to stochastic processes to describe those other factors, but I prefer the term Malthusian Selection in honour of the man who provided the approximate Exponential Law of Population, and - from the second edition of his essay onwards - an in depth demographic analysis of the factors affecting national populations over time. It is not for nothing that demographers refer to Malthus as the "Founder Of Modern Demography" (William Peterson, 1979, 1999). 

It was Malthus who provided what is now known as the Malthusian Growth Model, and he that first explored what I call differential replication (you may know it as differential reproduction) within a mathematical population model.  When it comes to differential replication, it is Darwin who provided a correct observation on a theory first explored by Malthus. 

Lastly, although Smith's comment on "indefinite increase in optimal conditions" appears to reflect Malthus' own position (see below) on an unchecked population "increasing in geometric progression" by doubling every 25 years, it has also missed the point that Malthus makes regarding general result which "Practically, it would sometimes be slower, and sometimes faster." But (inferred) it would still increase in geometric progression.

Hence, like everyone since Darwin, Smith misses the extension of the Malthusian Growth Model beyond fixed rate compound interest to variable rate compound interest, and beyond constant doubling periods to variable doubling periods. 

The Selfish Gene (1976, 1989 - resumed)

Dawkins then discusses the natural world's answer to exponential growth, and cites the example of the Canadian lynx whose population has historically been in a state of dynamic equilibrium (Dawkins, 1976, p.112):

"...It is an obvious fact that wild animal populations remain rather stable, with birth-rates and death-rates roughly keeping pace with one another. In many cases, lemmings being a famous example, the population fluctuates wildly, with violent explosion alternating with crashes and near extinction. Occasionally the result is outright extinction, at least of the population in a local area. Sometimes, as in the case of the Canadian lynx - where estimates are obtained from the numbers of pelts sold by the Hudson's Bay Company in successive years - the population seems to oscillate rhythmically. The only thing that animal populations do not do is go on increasing indefinitely."

Although Dawkins is correct to say that animal populations (or, indeed, any populations) do not "go on increasing indefinitely", it also true that all populations take every opportunity to do so. That is the nature of any species of replicator. It is only the action of Malthusian checks on population which restrain the Lynx population. As the Lynx population rises so they act as a Malthusian check (Death, one of the Four Riders of The Apocalypse) upon various prey populations whose numbers then fall. This results in famine (another of the Four Riders) reducing the Lynx population, whose numbers then decline. A fall in the Lynx population reduces the checks on the prey populations, whose numbers then rise, and the whole cycle starts over again. 

However, the point is that removal of checks on population typically encourages the exponential growth of that population (or, at the very worst, slows down negative population growth). As Darwin (see Darwin - An Exponentialist View) put it in Origin Of Species (Darwin, 1859):

"Hence we may confidently assert, that all plants and animals are tending to increase at a geometric ratio, that would most rapidly stock every station in which they could any how exist, and that the geometric tendency to increase must be checked by destruction at some period of life."

Dawkins correctly points out that humanity has attained an unnatural state, in which many of us simply die of old age (Dawkins, 1976, p.112):

"Wild animals almost never die of old age: starvation, disease, or predators catch up with them long before they become really senile. Until recently, this was true of man too."

Dawkins' Law

In the chapter Memes: the new replicators Dawkins compares biology with physics (Dawkins, 1976, p191):

"The laws of physics are supposed to be true all over the accessible universe. Are there any principles of biology that are likely to have similar universal validity?

...will there still be any general principle that is true of all life?

Dawkins states that he doesn't know, but that he is prepared to put his money on the following fundamental principle which I have termed Dawkins' Law (Dawkins, 1976, pp191-192):

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

I believe that Dawkins is right, as was Malthus when he wrote (in A Summary View, 1830) of populations in more mathematical terms:

"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."

To complete the picture, one must also state the complimentary law (I have turned Malthus' phrase on its head):

"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."

And again from A Summary View (1830), Malthus makes his case plain:

"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."

Dawkins believes that the DNA molecule is the prevailing replicating entity on Earth. However, it is possible to show mathematically that all populations of all replicators (yes DNA, but also bacteria, viruses, cells, animals, plants and fungi) do in fact adhere to Malthus' universal Principle Of Population.

The instructions on how to build a human are encoded in DNA replicators. These instructions cause cell replicators to differentiate and so build a human. In Malthusian terms, populations of humans can themselves be viewed as populations of replicators. After all, our children are clearly human copies based on the original replicators (i.e. parents). This three-fold view of human life is the exponentialist view of human life. It is truly an exponentialist view for Malthus' Exponential Law can be proven to apply at each level. 

River Out Of Eden. Chapter 5 - The Replication Bomb (Dawkins - 1995) 

Information Supernova

The replication bomb that Dawkins refers to in his chapter heading is life. In one of my favourite passages from Dawkins, he explains why humanity is so important (Dawkins, 1995, p.136):

"We humans are an extremely important manifestation of the replication bomb, because it is through us - through our brains, our symbolic culture and our technology - that the explosion may proceed to the next stage and reverberate through deep space."

To Dawkins this replication bomb, fuelled by the energy of our Sun, will help our Sun "go information" in much the same way that stars are said to "go supernova". I found a similar argument to Dawkins' memetic supernova in The Fabric Of Reality (in the chapter The Significance Of Life) by physicist David Deutsch, a contemporary of Dawkins at Oxford University. Deutsch (1997) states that "Genes embody knowledge about their niches" and concludes that it is the "survival of knowledge" which is the key attribute of successful replicators. 

Dawkins goes on to note that ours is the only replication bomb in the universe, of which we are aware. Dawkins describes the cause of the replication bomb, which started with the very first replicator billions of years ago (Dawkins, 1995, p.137):

 "...exponential growth: the more you have, the more you get.

Memes

Dawkins, who coined the term meme in his book The Selfish Gene, uses the example of a postcard "chain letter" as a form of replicator meme. The instructions for the postcard are (Dawkins, 1995, p.146):

"Make six copies of this card and then mail them to six friends within a week. If you do not do this, a spell will be cast upon you and you will die in horrible agony within a month.

Dawkins assumes that each week only a third of all recipients would follow the instructions. Hence, we have a weekly replication rate of 6, of which 4 die off (assuming 4 out of 6 recipients destroy their postcards), with 2 survivors for each set of 6 cards sent. Each recipient of a surviving card thus repeats the instructions, thus creating 2 more surviving cards, and so on. The "population" of postcards in circulation would therefore double weekly. In fact, the same would be true if each person that received such a postcard sent out 2 of their own and all survived. 

Dawkins points out that after 52 population doublings (one year) there would be around 4,000 trillion cards in circulation (Dawkins, 1995, p.146):

"Enough postcards to smother every man, woman, and child in the world."

Pops (individuals) 1 2 4 8 16 32 64 128 256 512 1024
A-pops 1 2 4 8 16 32 64 128 256 512 1024
B-pops 1 2 4 8 16 32 64 128 256 512 1024
C-pops 1 2 4 8 16 32 64 128 256 512 1024
D-pops 1 2 4 8 16 32 64 128 256 512 1024
E-pops 1 2 4 8 16 32 64 128 256 512 1024

Table B. Using the New Malthusian Scale to demonstrate meme replication via postcard

Note:  1A-pop = 1024 pops, 1B-pop = 1024 A-pops, 1 C-pop = 1024 B-pops etc.

After 52 weeks there would be 4 E-pops, which equals 4,503,599,627,370,496 postcards (roughly 4,500 trillion) postcards in circulation. 

Meme.gif (13980 bytes)

If you click on this image you will see the number of postcards in circulation (in green) for the first three weeks (2, 4 and 8). So, the number of postcards in circulation for a week clearly corresponds to the number of postcards in that postcard generation. As Dawkins suggests, this number doubles weekly (as demonstrated in Table B). Note that recipients of postcards from earlier weeks are not expected to send any more postcards beyond the first week. 

However, those who forward 6 postcards of their own are under no obligation to destroy the postcard that they themselves received. Thus the actual postcard population accumulates from week to week. This starts at 2 surviving postcards in week one, then 4 are added in the second week to make 6 and then 8 more are added in week three to make a total postcard population of 14. In week four another 48 postcards would be sent (8 * 6), of which 32 would be destroyed and 16 would survive. Hence, those 16 must be added to the week three population of 14 to derive the population of 30 in week four (14 + 16 = 30). 

Perhaps thinking of his own concept of "selfish genes" as "DNA chain letters", Dawkins makes the following observation on differential reproduction (Dawkins, 1995, p.147):

"In the competition for resources, variants of the replicator may arise that happen to be more efficient at getting themselves duplicated. These more efficient replicators will tend to replace their less efficient rivals. It is important to understand that none of these replicating entities is consciously interested in getting itself duplicated. But it will just happen that the world becomes filled with replicators that are more efficient."

Dawkins concludes this chapter with an intriguing attempt to define the typical thresholds through which replicators pass over geological time, from the first replicators to the colonisation of space. Once a population of replicators has succeeded in colonising space, I would model the progress of such a technological civilization in terms of the Kardashev levels, as discussed in Drexler - An Exponentialist View.

Climbing Mount Improbable. Chapter 9 - The Robot Repeater. (Dawkins - 1996)

Replicators and Populations

Here Dawkins discuss the deep but simple connection between a replicators, population and exponential growth (Dawkins, 1996)

"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. I'll develop the idea of exponential growth in a moment. 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."

The belief that exponential growth requires a constant growth rate, or "the population doubles at regular intervals", is widely held. It is a fallacy; a myth. Malthus was one of the first to attempt to explain the exponential growth of populations, and may have started this myth. Malthus made regular use of constant growth rates (through regular doubling) in his examples, but also clearly understood that growth rates vary (and yet the population will still grow exponentially). I'm sure that Dawkins understands this too, but I've had too many arguments over this issue to let that one go without comment.

Still, at least Dawkins clearly understands the principle of population growth, even if he never specifically uses the Malthusian Growth Model to demonstrate what he means. Instead, Dawkins regularly takes the usual generational approach to population modelling (and then - see below - correctly points out that such an approach is flawed!).

The only examples that I can think of for actual constant exponential growth are financial ones (e.g. fixed interest rates) and the decay rate from physics (i.e. negative exponential growth of radioactive material). Yet even in the financial world it is possible to show that variable compound interest will result in true exponential growth.

Constant growth rates do not normally apply to populations of replicators in the real world. Instead, variable growth rates are the normal case. As clearly stated by Malthus, population doubling times vary with growth rates.

Differential Replication

Here Dawkins discusses differential replication in terms of population genetics (Dawkins, 1996):

"...Other variants happened to have some property that caused them to be duplicated more rapidly or efficiently. They became consequently more numerous in the population. Since they would have been competing for the same raw materials as rival replicators, as time went by the average, typical replicator type in the population would continually have been supplanted by a new and a better average type."

This passage should be uncontroversial, for people have been saying something like this since Darwin. The implication is that Natural Selection alone is responsible for creating different rates of reproduction. However, Malthus was the first to mathematically investigate the rates of differential replication (or differential reproduction, if you prefer) for discrete populations of our own species. See Malthus - An Exponentialist View for more. Taking the different nations around the world today, it is still possible to demonstrate (as Malthus did) that different rates of growth exist today. Given the comparative lack of genetic or even phenotypic differences between human populations, the only possible conclusion is that other factors are at play. 

I also believe that Dawkins and others underestimate the significance of non-genetic factors on the differential replication of populations of replicators. In particular, the effect of the environment and non-instinctive behaviour on the rate of population growth is understated. I call this Malthusian Selection. The Malthusian Selection / Natural Selection debate to come is similar to the Nature / Nurture debate on the influences on individual humans, and I suspect that a serious study into the matter will reveal a similar balance (60/40), though I'm prepared to accept the balance could be revealed to be either way (and may in fact vary depending upon the species in question). For more see my articles Differential Reproduction - An Exponentialist View and Evolution - An Exponentialist View.

Cell Replication

In discussing the ancient evolutionary history of the eukaryotic cell from primitive bacterial cells, Dawkins provides a couple of examples of familiar animals made from eukaryotic cells (Dawkins, 1996):

"A mouse is a large edifice of perhaps a billion cells. An elephant is a colony of about 1,000 trillion (1015) cells, and each one of those cells is itself a colony of bacteria."

Along similar lines to other authors, Dawkins explains the power of exponential growth that applies within each individual, through the local doubling of cells (Dawkins, 1996):

"...The special way of growing that real living things employ is exponential growth. Another way of saying this is that living things grow by local doubling.

We start with a single cell which is very small. ...Perhaps the most remarkable property the cell has is the ability to divide into two daughter cells more or less like itself. Being like the parent cell, each daughter cell itself is capable of diving into two, making four granddaughter cells. Each one of the four, in turn, can double, making eight, and so on. This is exponential growth, or local doubling."

Here Dawkins (1996) uses paper folding as a means of explaining the shocking power of exponential growth (Dawkins, 1996):

People who are not used to it find the power of exponential growth surprising. As promised, I'll spend a little time on it, because it is important. There are many vivid ways of illustrating it. If you fold a piece of paper once, you have two thicknesses. Fold it again and it is four times as thick. Another fold and you have a wad, eight layers thick. But suppose that this mechanical stiffness were not a problem and that you could go on folding, say fifty times. How thick would the wad of paper be then? The answer is that it would be so thick that it would reach right outside the Earth's atmosphere and beyond the orbit of Mars.

Getting back to the local doubling of cells, Dawkins provides an explanation of how many cell generations it would take to grow a blue whale (Dawkins, 1996):

In the same way, by local doubling of cells all over the developing body, the number of cells very rapidly gets up into the astronomical range. A blue whale is made of about a hundred thousand trillion (1017) cells. But, such is the power of exponential growth, it would only take about fifty-seven cell generations, under ideal conditions, to produce such a leviathan. By cell generation, I mean a doubling. Remember that numbers of cells go up 1, 2, 4, 8, 16, 32. etc. ..."

Thankfully, unlike most authors who attempt to use a generational view to explain exponential growth, Dawkins responsibly provides a caveat - it is not realistic to use generations of cells to explain exponential growth because cells are not immortal (Dawkins, 1996):

"This way of calculating the number of cell generations is actually unrealistic because it only gives a minimum figure. It assumes that, after every cell generation, all cells do on to duplicate.  ...So a blue whale in fact consists of a number of cell lineages of different length, building different parts of the whale. Some of these lineages go on dividing for more than fifty-seven cell generations. Others stop dividing after fewer than fifty-seven cell generations."

Thus, Dawkins (1996) explains that his naive guide to calculating how many cell generations are required to grow any creature is based upon the weight of that creature (Dawkins, 1996):

"A naive calculation suggests that it would take a minimum of forty-seven cell-doubling generations to grow an adult human and only about ten more generations to grow a blue whale. These figures are certainly underestimates, for the reason I have given."

Dawkins is correct to call such estimates naive if they are regarded as cell generations. However, as Malthus taught us over 200 years ago, population doubling itself is not naive. In fact, the Malthusian Growth Model is the perfect way to model the population doubling of cells (see Cell Replicators - An Exponentialist View for more). Furthermore, Dawkins' naive estimates are only naive because he is thinking just like Darwin did - in generations. If Dawkins were to think as Malthus did, he would see that his naive estimates are actually fairly close to genuine population doubling estimates:

A-Pops 1 2 4 8   16 32   64 128 256 512   1024
B-Pops 1 2 4 8   16 32   64 128 256 512   1024
C-Pops 1 2 4 8   16 32   64 128 256 512 953.67 1024
D-Pops 1 2 4 8   16 32   64 128 256 512   1024
E-Pops 1 2 4 8   16 32 45.47 64 128 256 512   1024
F-Pops 1 2 4 8 8.88 16 32   64 128 256 512 888.18 1024

Table C. Naive cell-doubling generations equate to accurate Malthusian population doublings

Note:  1A-pop = 1024 pops, 1B-pop = 1024 A-pops, 1 C-pop = 1024 B-pops etc.

Table D below explains the highlighted entries in Table C above:

Animal New Malthusian Scale Decimal Scale Dawkins' Generations Population Doublings

Mouse 

 953.67 C-Pops

1 billion 

Not stated 30

Human 

 45.47 E-Pops

50,000 billion

47 46

Elephant 

 8.88 F-Pops

1,000 trillion

Not stated 54

Blue Whale 

 888.18 F-Pops

100,000 trillion

57 60

Table D. Dawkins' example creatures built using cell replication

For an exploration of how generations work in reverse (think of your family tree), please read my article Generations and Population Doublings.

Molecular Nanotechnology (MNT)

Dawkins briefly touches on the emerging field of MNT, stating (Dawkins, 1996):

 "Nanotechnology seems to us very strange, scarcely believable."

He correctly concludes that the key power of MNT over current methods of construction would be (Dawkins, 1996) "exponential multiplication". Noting that it may not come to anything, he concludes that this "new" technology is really very "old", and has been used by life (through genes) for aeons. For more on MNT, read Drexler - An Exponentialist View and Grey Goo - An Exponentialist Explanation.

Interestingly, Dawkins' Law would also apply to MNT assemblers and replicators when they are invented. Would that make them alive?

Malthus - To sneer, or not to sneer 

As with my other exponentialist articles, it was not my intention to find fault with the scientist whose writings I explore. Rather,  I wish to demonstrate why "that great philosopher" Malthus (as Darwin called him) still deserves to be read and understood. Still, also like Darwin (see Darwin's Views on Malthus for more), I will try to take consolation from the discouraging example of Malthus if my own views are subsequently misrepresented and misunderstood: 

Letter to C. Lyell, 6th June, 1860

"...It consoles me that -- sneers at Malthus, for that clearly shows, mathematician though he may be, he cannot understand common reasoning. By the way what a discouraging example Malthus is, to show during what long years the plainest case may be misrepresented and misunderstood."

Since Darwin's day, many respected scientists have failed to agree with Darwin's assessment of Malthus (see Paul R. Ehrlich - An Exponentialist View for the views of modern demographers). Evolutionist Elliot Sober states (Sober, 1984):

"Malthusianism is not a proper starting point for the theory of natural selection for reasons made abundantly clear by Fisher (1930, pp.46-47)."

Sober also notes the following quote from evolutionist R. A Fisher (Sober, 1984):

"There is something like a relic of creationist philosophy in arguing from the observation, let us say, that a cod spawns a million eggs, that therefore its offspring are subject to Natural Selection; and it has the disadvantage of excluding fecundity from the class of characteristics of which we may attempt to appreciate the aptitude... The historical fact that both Darwin and Wallace were led through reading Malthus's essay on population to appreciate the efficacy of selection, thought extremely instructive as to the philosophy of their age, should no longer confuse the consequences of that principle with its foundations."

Perhaps Fisher "sneers" at Malthus because Malthus was a creationist, and a Reverend. Yes, it is true that Natural Selection can and does affect the rate of growth for any replicator population (and so the growth rate is partially a consequence of Natural Selection). But Malthusian Selection can also influence the growth rate. For competing human populations it would seem that Malthusian Selection is the greater contributor, for how else can we explain the variation in growth rates between nations over time and even now? In discussing the benefits of the Human Genome Diversity Project and human genetic variance between individuals or races, Dawkins notes (Dawkins, 2003):

"Incidentally, a surprisingly small proportion of that variance  consists of between-race variance..."

For Natural Selection to be a factor in affecting human population growth rates (which is termed the Malthusian Parameter by Fisher), we would have to conclude some significant genetic advantage of one human population over another. This might be conceivable to a racist, but not to me. As noted by Malthus, for human populations what differs is behavioural factors and environment.

Furthermore, the Malthusian Growth Model and the Exponential Law have stood the test of time, and can be demonstrated to apply approximately to all replicator populations at all times. As extended by the Couttsian Growth Model, a universal law of population growth based on variable rates of compound interest is revealed. This model demonstrates the inherent nature of all replicators, and Darwin was right to think that the exponential growth of replicator populations thus drove natural selection (and so the growth rate is partially a foundation of Natural Selection). This model can also be used to demonstrate differential replication, irrespective of the factors involved in origin of the growth rate for a population. Please take your time to read through this exponentialist site, and see for yourself.

The Unit Of Selection (again)

Dawkins again clearly expresses his disdain for the idea of populations as the unit of selection for Natural Selection (Dawkins, 1982):

"Populations may last a long while, but they are constantly blending with other populations and so losing their identity. They are also subject to evolutionary change from within. A population is not a discrete enough entity to be a unit of natural selection, not stable and unitary enough to be selected in preference to another population."

As discussed by Dawkins himself in Chapter 7 of The Selfish Gene, it is in fact easy enough to factor in immigration and emigration into a population growth model. It is also easy enough to distinguish Population A from Population B (which may have evolved from Population A). As soon as you find it is meaningful and useful enough to distinguish between Population A and Population B then the population growth model can discretely apply to both. This is, in fact, precisely what demographers do all the time - see Human Replicators, An Exponentialist View for more.

Fisher's Malthusian Parameter is intended to be a measure of the fitness of a genome in relation to its environment. However, a higher growth rate leads to faster population growth (and population doublings). Thus, differential reproduction works at the level of population, via the Malthusian Parameter. In mathematical terms, a population can be selected for (with higher and higher rates of growth) or selected against (with lower and lower rates of growth).

However, from the point of view of the species (as opposed to the population), I believe Dawkins is still correct to reduce the unit of Natural Selection to the gene. After all, it is our genes which set us apart. But having done so, I believe it is a relatively easy proposition to mathematically model Natural Selection using my Couttsian Growth Model (via my New Malthusian Scale). Furthermore, and most significantly, I believe that doing so highlights the fact that Malthusian Selection is thus revealed as a major - and frequently overlooked - factor in determining rates of differential replication in the eternal struggle for existence first revealed by Malthus. 

Conclusion

I am convinced that if and when Dawkins does mention Malthus in some future work, Dawkins will not sneer at him1. Rather, I feel that Dawkins (like Darwin) would be capable of looking beyond the moralistic nature Malthus' essay to see the science contained therein. My hope is that this series of exponentialist articles might throw a different light on Malthus, and thus illuminate the implications of Malthus' work on populations for evolutionary theory as well as the pressing issue of human overpopulation.

 

Footnotes:

1 - Dawkins has finally mentioned Malthus - twice - in his new book The Greatest Show On Earth - The Evidence For Evolution (2009). The first reference is just the usual, extremely common reference to Darwin's having read Malthus "for amusement" (p.17). The second reference to Malthus is actually a quote - pointed out to Dawkins by his friend Matt Ridley. It's the same quote for which I pointed out Dawkins' Malthusian leanings at the end of the section on The Selfish Gene. I also happened to watch Dawkins' The Genius of Charles Darwin (yes, for amusement) in which he makes a passing reference to the influence on Darwin of Malthus' diatribe. Oh, well. It's not what I would have hoped for, but it's a start.

 

References:

Blackmore, Susan, The Meme Machine - with a foreword by Richard Dawkins. Oxford University Press. 1999.

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

Dawkins, Richard, The Selfish Gene. Oxford University Press. 1976, 1989. Permission to reproduce text from this book was granted by the Oxford University Press.

Dawkins, Richard, The Extended Phenotype. Oxford University Press. 1982.

Dawkins, Richard, A River Out Of Eden. Phoenix. 1995

Dawkins, Richard, Climbing Mount Improbable. Penguin. 1996.

Dawkins, Richard, A Devil's Chaplain. Weidenfield & Nicolson. 2003

Dawkins, Richard, The Genius of Charles Darwin (DVD). Channel 4 in the UK.

Dawkins, Richard, The Greatest Show On Earth - The Evidence For Evolution. Bantam Press. 2009.

Dennett, Daniel C., Darwin's Dangerous Idea - Evolution and the Meanings of Life. Penguin.1995.

Deutsch, David. The Fabric Of Reality. Penguin. 1997.

Drexler, K. Eric. Engines Of Creation - The Coming Era of Nanotechnology. Oxford University Press. 1990.

Freitas, Robert, A., Merkle, Ralph, C., Kinematic Self-Replicating Machines. Landes Bioscience. 2004.

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

Sober, Elliot, The Nature Of Selection - Evolutionary Theory In Philosophical Focus. University of Chicago Press. 1984.

Smith, John Maynard, The Theory Of Evolution. Canto edition*, Cambridge University Press. 1958, 1966, 1975, 1993*.

Peterson, William, The Founder Of Modern Demography: Malthus. Transaction Publishers (1979, 1999)

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

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