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:
The Internet Public Library

Selected Quotations Relating To Alfred Russel Wallace

The Alfred Russel Wallace page from Professor Charles Smith

 Alfred Russel Wallace - An Exponentialist View

MY LIFE (1905)

Wallace credits Malthus' essay as critical to his theory of evolution. The following two quotes from Wallace's autobiography were noted by Professor Frey (1970) in his introduction to the Penguin Classics edition of Malthus' "An Essay On The Principle Of Population":

"...the most important book I read was Malthus' Principle Of Population...It was the first work I had yet read treating any of the problems of philosophical biology, and its main principles remained with me as a permanent possession, and twenty years later gave me the sought-after clue to the effective agent in the evolution of organic species."

Again from "My Life":

"One day something brought to my recollection Malthus' Principle Of Population, which I had read about twelve years before. I thought of his clear exposition of the 'positive checks' to increase ...which keep down the population. ...It then occurred to me that these causes or their equivalents are continually acting in the case of animals also; and, as animals usually breed much more rapidly than does mankind, the destructions every year from these causes must be enormous in order to keep down the numbers of each species, since they evidently do not increase regularly from year to year., as otherwise the world would long ago have become densely crowded with those that breed most quickly. ...Why do some die and some live? And the answer was clearly, that on the whole the best fitted live. From the effects of disease the most healthy escaped; from enemies the strongest, the swiftest, or the most cunning; from famine, the best hunters or those with the best digestion; and so on. Then it flashed upon me that this self-acting process would necessarily improve the race, because in every generation the inferior would inevitably be killed off and the superior would remain - that is, the fittest would survive. Then at once I seemed to see the whole effect of this. ...The more I thought it over the more I became convinced that I had at length found the long-sought-for law of nature that solved the problem of the origin of species."

Wallace was right to think that he had found a law of nature, and right to recognise that the fittest are those who "on the whole" escape the "continually acting" Malthusian checks on population. Of course, without Malthus' excellent explanation of the exponential nature of population growth, it would not have been so easy to see why such checks were even necessary. So, whilst Wallace (independently of Darwin) certainly did discover the law of "the survival of the fittest" that explained the "origin of species", this was only possible because Malthus had discovered another law of nature - exponential growth - and its dire consequences for all living things. 


Wallace defines the "struggle for existence", a term both he and Darwin may have derived from Malthus:

"The life of wild animals is a struggle for existence. The full  exertion of all their faculties and all their energies is required to preserve their own existence and provide for that of their infant  offspring. The possibility of procuring food during the least  favourable seasons, and of escaping the attacks of their most  dangerous enemies, are the primary conditions which determine the  existence both of individuals and of entire species. These conditions will also determine the population of a species; and by a careful consideration of all the circumstances we may be enabled to  comprehend, and in some degree to explain, what at first sight appears so inexplicable- the excessive abundance of some species, while others closely allied to them are very rare."

The Law Of Population Of Species

This section of Wallace's paper is, at best, this is an imperfect restatement of Malthus' Principle Of Population. However, Wallace's mention of "the influence of man" is fairly chilling (see Checks On Population below). Here, Wallace provides his sole attempt (in this paper) at modelling exponential growth:

"Even the least prolific of animals would increase rapidly if unchecked, whereas it is evident that the animal population of the globe must be stationary, or perhaps, through the influence of man, decreasing. Fluctuations there may be; but permanent increase, except in restricted localities, is almost impossible. Very few birds produce less than two young ones each year, while many have six, eight, or ten; four will certainly be below the average; and if we suppose that each pair produce young only four times in their life, that will also be below the average, supposing them not to die either by violence or want of food. Yet at this rate how tremendous would be the increase in a few years from a single pair! A simple calculation will show that in fifteen years each pair of birds would have increased to nearly ten millions! whereas we have no reason to believe that the number of the birds of any country increases at all in fifteen or in one hundred and fifty years. With such powers of increase the population must have reached its limits, and have become stationary, in a very few years after the origin of each species."

Wallace takes a modest assumption of 4 "young ones" each year, 4 times in the lives of each pair of birds. Here, I attempted to model just the annual quadrupling effect:

Birds 4 16 64 256 1024 4096 16384
Birds 65536 2612144 1048576 4194304 16777216 67108864 268435456

Table A. Annual quadrupling of bird population

I've quadrupled a starting population of 1 bird (the mother) 14 times (14 years) and it is clear that far too many birds will exist even after 12 years (16,777,216 birds). This "simple calculation" fails to project "nearly ten millions" after "fifteen years". Obviously Wallace had something else in mind, possibly some Fibonacci series rather than an exponential series.

The fact that Wallace starts with a pair of birds should be irrelevant. It is the first pregnant mother, and her brood of 4 hatchlings which is relevant. However, the fact that each pair will have 4 broods in a lifetime makes things even worse - the population will grow even quicker! Wallace implies that his hypothetical birds are immortal ("...supposing them not to die either by violence or want of food"), so it is hard to know what to make of the meaning of the term of a life-time ("...if we suppose that each pair produce young only four times in their life"). Is it 15 years? Is Wallace assuming that no birds die during the 15 year lives of the original pair? Or do they live for 4 years, given they average 4 young per year, and breed 4 times? Also, what assumptions are made with regard to the sex of the offspring? What is the ratio of females to males? 

Using Wallace's defined parameters of 10 million birds after 15 years breeding, I tried using my New Malthusian Scale to see where Wallace's prediction would fall:

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 9.53 16 32 64 128 256 512 1024

Table B. Global population of birds doubling using the Alphabet Option (constant rate) of The New Malthusian Scale.

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

Wallace states that his bird population will reach its target population in 15 years. Given that each row represents 10 doublings, and assuming a constant rate of doubling of 6 months, then each row on Table A. would represent 5 years. Hence, after 3 complete rows (15 years) you would have 1024 C-Pops (1024 x 1024 x 1024 = 1,073,741,824 birds). The number of years is right, but that is too many birds- something is wrong!

Trying a different approach, how many population doublings does it take to reach Wallace's target total of nearly 10 million birds from a breeding pair? You would reach 9.53 C-Pops (9.53 x 1024 x 1024 = 9,992,929.28 birds, "nearly ten millions") during the 24th doubling. With a 6 month doubling time, that occurs during the 12th year. But we need to reach 9.53 C-Pops in 15 years! Therefore, dividing 15 years by 24 doublings, we derive a slower population doubling time of 7.5 months

Using a population doubling time of 7.5 months we can bracket Wallace's 10 million birds nicely. A starting population of 1 would reach 8 C-Pops (8,388,608 birds) after 14 years and 4.5 months, and 16 C-Pops (16,777,216 birds) after 15 years. Clearly, by ignoring Wallace's "simple calculation" (whatever it might be), the New Malthusian Scale provides a simple method to accurately demonstrate true exponential growth. Note that it is naive to assume a constant growth rate. Life is not so neat in reality. However, it is perfectly valid to project growth using a constant rate (so long as it is made clear that a constant rate of growth is not required to have exponential growth).

Either Wallace's example is broken, or my arithmetic is wrong. We do not have the benefit of Wallace's mathematical workings, but you clearly have mine. There is no way to derive Wallace's target population in the target timeframe, using the birth rates given, with a zero death rate. Therefore, I conclude that Wallace's example of exponential growth is broken. 

Thus, whilst I am quite sure that Wallace understood the essence of what Malthus had written, he was not the exponentialist that Darwin was (and Darwin was not the exponentialist that Malthus was). His treatment of exponential growth (the key to Malthus' Principle Of Population, and the driving force behind the struggle for existence) was cursory, and inaccurate. For the sake of his example, Wallace naively assumed a constant rate of growth (without stating the naivety of that assumption). Like so many others since, Wallace (despite having read Malthus) took a generational view of population growth. 

Checks On Population

Malthus had stated that "want of room and nourishment" are the most important checks on population. Again Wallace restates Malthus' own views, using the example of the American passenger pigeon as his own contribution:

"Perhaps the most remarkable instance of an immense bird population is that of the passenger pigeon of the United States, which lays only one, or at most two eggs, and is said to rear generally but one young one. Why is this bird so extraordinarily abundant, while others producing two or three times as many young are much less plentiful? The explanation is not difficult. The food most congenial to this species, and on which it thrives best, is abundantly distributed over a very extensive region, offering such difference of soil and climate, that in one part or another of the area the supply never fails.

...This example strikingly shows us that the procuring a constant supply of wholesome food is almost the sole condition  requisite for ensuring the rapid increase of a given species, since neither the limited fecundity, nor the unrestrained attacks of birds of prey and of man are here sufficient to check it.

...It is, as we commenced by remarking, 'a struggle for existence,' in which the weakest and least perfectly organized must always succumb."

Despite Wallace's early inkling of the damaging effect of man on nature (see Wallace's Law Of Population of Species above) Wallace notes in this example that "the unrestrained attacks of birds of prey and of man" failed to check the population of the American passenger pigeon. 

It is therefore a tragic irony that the American passenger pigeon was hunted to extinction by men between 1814 and 1914. At one time, it could take up to 3 days for flocks numbering in their millions to pass overhead. Thus Wallace's assertion regarding the supply of food as being the most significant factor in the success of a species is clearly disproved. In fact, it could be stated that being food is often a more significant factor, as in this case.


In Chapter II of Darwinism, "The Struggle For Existence" Wallace was to later write:

"The passenger-pigeon (Ectopistes migratorius) is, or rather was, excessively abundant in a certain area in North America..."

So, even in his own lifetime, Wallace became aware of the dwindling population of this soon to be extinct bird.

Skipping back to the opening paragraph of Chapter II, Wallace explains the deceptive nature of the struggle for existence:

"There is perhaps no phenomenon of nature that is at once so important, so universal, and so little understood, as the struggle for existence continually going on among all organised beings. To most persons nature appears calm, orderly, and peaceful. They see the birds singing in the trees, the insects hovering over the flowers, the squirrel climbing among the treetops, and all living things in the possession of health and vigour, and in the enjoyment of a sunny existence. But they do not see, and hardly ever think of, the means by which this beauty and harmony is brought about, They do not see the constant and daily search after food, the failure of which means weakness and death; the constant effort to escape enemies; the ever-recurring struggle against the forces of nature. This daily and hourly struggle, this incessant warfare, is nevertheless the very means by which much of the beauty of nature is produced, and also affords one of the most important elements in bringing about the origin of species...."

Whilst Natural Selection is itself a gradual process, this passage effectively dispels the popular notion of evolution as a gradual process. The Malthusian struggle for existence has a timescale which is very much more immediate, and yet Wallace is right to say it is "little understood".

Having explored several accounts of the struggle for existence in action, Wallace then writes of the Increase of Organisms in a Geometric Ratio:

"The facts which have now been adduced, sufficiently prove that there is a continual competition, and struggle, and war going on in nature, and that each species of animal and plant affects many others in complex and often unexpected ways. We will now proceed to show the fundamental cause of this struggle, and to prove it is ever acting over the whole field of nature, and that no single species of animal or plant can possibly escape it. This results from the fact of the rapid increase, in geometric ratio, of all species of animals and plants...

It is a pity that Wallace does not credit Malthus with the discovery of this fundamental law of nature at this point in his explanation, though Wallace has done so unequivocally elsewhere. Wallace then provides a few examples of geometric (or exponential) growth, starting the rapid increase of the flesh-fly, and the assertion by the Swedish naturalist Linnaeus that:

"...a dead horse would be devoured by three of these flies as quickly as a lion."

What does Linnaeus mean? Well, the rate of increase per fly is given as ten thousand-fold in a fortnight. Wallace continues:

"Supposing they went on increasing at this rate during only three months of summer, there would result one hundred millions of millions for each fly at the commencement of summer,-a number greater probably than exists at any one time in the whole world..."

If one fly gives rise to 10,000 flies at the end of each fortnight, and we therefore apply a ten thousandfold rate of increase per fortnight for 6 fortnights (3 months), Wallace asserts that the result will be 100 millions of millions of flies:



 1st fortnight



 2nd fortnight



 3rd fortnight



 4th fortnight



 5th fortnight

1000000000000000000000000  10,0006  6th fortnight



Table C: Fortnightly ten thousand-fold increase of flesh flies 

Wallace is out by quite a large margin. 

Wallace's next example is any common bird with us all year round (e.g.. redbreast, sparrow, thrush or blackbird):

"These lay on an average six eggs, but, as several of them have two or three broods a year, ten will be below the average of the year's increase. Such birds as these often live from fifteen to twenty years in confinement, and we cannot suppose them to live shorter lives in a state of nature, if unmolested; but to avoid possible exaggeration we will only take ten years as the average duration of their lives. Now, if we start with a single pair, and these are allowed to live and breed, unmolested, till they die at the end of ten years,-as they might do if turned loose into a good-sized island with ample vegetable and insect food, but no other competing or destructive birds or quadrupeds - their numbers would amount to more than twenty millions."

Wallace asserts that a ten-fold increase per year for 10 years will result a population of over 20 million birds:



 1st year



 2nd year



 3rd year



 4th year



 5th year

1000000 106  6th year
10000000 107  7th year
100000000 108  8th year
1000000000 109  9th year
10000000000 1010  10th year



Table D:  Annual ten-fold increase of bird population

Again Wallace is out by a very large margin (even if we add in the 2 starting birds). Whilst these examples show that Wallace's mathematical ability was questionable, they do not invalidate the point that he is making - namely, the extraordinary increase possible in ordinary circumstances. 

It was not my intention to make Wallace look incompetent, but rather to illustrate the difficulty of explaining exponential growth simply and effectively. If you care to read my articles on other famous exponentialists, you will see that most suffer the same problem. All, that is, except Malthus. Using Malthus to explain the increase of a bird population, with the use of my own New Malthusian Scale, the following table illustrates the constant population doubling of one of Wallace's common birds:

Pops 1 2 4 8 16 32 64 128 256 512 1024
Kilopops 1 2 4 8 16 32 64 128 256 512 1024
Megapops 1 2 4 8 16 32 64 128 256 512 1024
Gigapops 1 2 4 8 16 32 64 128 256 512 1024

Table E:  Seasonal doubling of bird population

Note:  1KP = 1024 individuals, 1MP = 1024KP, 1GP = 1024 MP

This table assumes that a population will double 4 times a year, hence 10 years will have passed after 40 population doublings. The resultant population is 1024 Gigapops, or 1,099,511,627,776 birds. This is much faster than Wallace's intended growth rate, but it's a simple matter to slow it down. Just assume 3 population doublings per year, for 30 doublings in 10 years - the result is now 1024 Megapops, or 1,073,741,824 birds. A population which doubles three times a year experiences an eight-fold increase in population per year, which is less than Wallace's intended ten-fold rate of increase. Nonetheless, the result is over 1 billion birds (which is far more than Wallace's prediction of over 20 million).

The clear advantage of my Malthusian approach is that it is simple, accurate and can be applied universally to all populations of replicators. It also allows the use of non-scientific notation whilst still being able to handle large numbers. Time becomes a variable, forcing the question - how long does it take the population to double? To answer that question you must consider the birth rate for a given period and the death rate for a given period to derive the growth rate for that period. Once you have that, you can quickly derive a crude population doubling time by diving the percentage rate into 70. 

An Attack On Malthus?

The following quote is from "The Last of the Great Victorians - Special Interview with Dr. Alfred Russel Wallace" by Frederick Rockell printed in The Millgate Monthly August 1912 issue. (from The Alfred Russel Wallace page)

Here, Rockell conjures up the scene:

"With the magic of his personality, Dr. Wallace conjured up before me a Utopia, where noble men and beautiful women with sweet children lived in a state of refinement, harmony, and plenty. I did not like to break the spell of the vision by any untoward remark, but a sceptical vein that runs through my most optimistic moments prompted me to ask: 'Under such a scheme, where plenty reigned, would not the population so increase that poverty would eventually come in again?"

Then Rockell goes on to describe Wallace's reaction:

"This had the result of stirring up the great scientist to make a vigorous protest. "The theory propounded by Malthus," he said, "is the greatest of all delusions. As man develops towards a higher type; as he becomes more refined and more civilised, so his fecundity decreases. Low down in the scale of life, birth is only limited by available sustenance. But the higher grows the type, the less is the fecundity. This is true, not only of ascending types in the evolutionary scale, but it is also true of ascending man. The fecundity of the slums is much greater than that of Mayfair. As man progresses in comfort and refinement, he tends to have fewer progeny; as witness the millions of India and China, compared with the almost stationary population of England, and the declining native population of France. Besides, if young people continued at school until the age of twenty-five, early marriages would be discountenanced, for public opinion would not tolerate marriage during the educational period."

So, what Wallace appears to be protesting about was the interpretation of the relative reproductive power of barbarous and civilised nations. Wallace believed that Malthus was wrong to assert that civilised nations were more fecund. Hence, if Malthus was wrong, Wallace felt that his Utopian visions were safe (it should also be mentioned that Wallace was in fact reacting to the Malthusian arguments of the very strong eugenics movement of his day, as becomes apparent if you read the whole interview - Malthus himself did not favour eugenics). 

As usual in such disagreements, both parties are partially correct (and therefore partially wrong, too). The issue comes down to what is meant by fecund. To Wallace and Darwin, who saw things from a generational viewpoint, fecundity means the number of average number of offspring for an individual of the species. To Malthus, whose model of population growth still applies today, fecundity means deducing a growth rate (positive or negative) by taking the death rate for a population from its birth rate for the same period.

To settle the argument, look at Table B from Darwin's Views on Malthus. This table compares the population growth of Europe, the USA, China and India in a typical Malthusian way (through the proven mechanism of population doubling).

Allowing for the fact that Malthus wrote on population between 1798 and 1830, and Wallace and Darwin on evolution from 1858 (with the joint presentation of their papers), what do we find? 

Well, it took Europe's population 100 years (1725 to 1825) to double from 128 million to 256 million. Then it doubled to 512 million in the next 125 years.

It took the USA's population only 15 years (1815-1830) to double from 8 to 16 million, then it doubled twice more in 30 years each time (taking them to 64 million by 1890), 40 years later they'd doubled to 128 million and 70 years later (in 2000) they'd doubled to 256 million.

It took India's population 300 years (1575-1875) to double from 128 to 256 million, and only 75 years (1875-1950) to double to 512 million.

It took China's population 185 years (1725-1910) to double from 256 to 512 million, and only 75 years (1910-1985) to double to 1024 million.

So, at the time that Malthus wrote (even allowing for migrations, which interfere with the Malthusian population model), the "civilised nations" in Europe and the USA were clearly more fecund than China and India. This is what Malthus had asserted. At the time that Wallace wrote, both China and India (under the influence of advanced Western technology and medicine) were entering a period of great fecundity. This is the opposite of what Wallace had asserted.

So, allowing for the times in which they wrote, Malthus was correct and Wallace was wrong.

Of course, another way of looking at things is to consider absolute increase. Hence, the larger your population base, the greater number of people will be added with each population doubling. 

A - Thousands 1 2 4 8 16
B - Billions 1 2 4 8 16

Table F. Absolute Increase

Assuming the same growth rate for both populations A and B, it is obvious that the absolute increase for each doubling is greater for population B. This is an interpretation of the word fecundity which none of Malthus, Darwin nor Wallace used. Nonetheless, it is an important consideration.

Today, many would say that Wallace is now correct in having stated that "As man progresses in comfort and refinement, he tends to have fewer progeny". 

This accurately reflects the West's hope for Zero Population Growth, that the static (or slightly negative in some cases) growth rates of the rich West will be duplicated throughout the rest of world following their "demographic transition" from the "poverty trap". It remains to be seen if this hope is justified. 

The Third World today, with its high growth rates, must either reach the same standard of living as the West or face continued massive Malthusian disasters (the Four Riders Of The Apocalypse, War, Famine, Pestilence and Death). Hence, the two possibilities are that Wallace's Utopian vision will be proven right (and the whole world will be rich with a stable population) or that the Malthusian vision is right (a minority are rich with a growing population, and a majority are again poor with a relatively stable population). In the Malthusian Dystopia, the poor would be returned to a position of suffering worse than they suffer today. 

However, should the West (including Japan) develop new advanced technologies (such as biotechnology, nanotechnology, genetics, cloning or AI), then they will again be capable of sustaining further exponential growth just as the rest of the world's growth might be slowing down. The question is, what will the West do with such technologies? The West might continue to restrict their own population growth, despite being able to actually sustain further growth. Then, if the whole world gets hold of such technology, will all nations be as responsible as the West like to think they will be? In other words, is Zero Population Growth sustainable? It's going to be matter of political will.

Also, should any nation learn to sustain population growth in space, their numbers would soon dwarf those of Earth.

Utopia or Dystopia?

The following quotes are from "Human Selection by Alfred Wallace" published in Volume 48 of Fortnightly Review in September 1890 (from The Alfred Russel Wallace page)

Having just described a fairly Utopian future, Wallace states:

"At first sight it may appear that in any state of society whose essential features were at all like those here briefly outlined, all the usual restraints to early marriage as they now exist would be removed, and that a rate of increase of the population unexampled in any previous era would be the result, leading in a few generations to a difficulty in obtaining subsistence, which Malthus has shown to be the inevitable result of the normal rate of increase of mankind when all the positive as well as the preventive checks are removed. As the positive checks--which may be briefly summarised as war, pestilence, and famine--are supposed to be non-existent, what, it may be asked, are the preventive checks which are suggested as being capable of reducing the rate of increase within manageable limits? This very reasonable question I will now endeavour to answer."

Wallace goes on to conclude:

"But in a state of society in which all have their higher faculties fully cultivated and fully exercised throughout life, a slight general diminution of fertility would at once arise, and this diminution, added to that caused by the later average period of marriage, would at once bring the rate of increase of population within manageable limits. The same general principle enables us to look forward to that distant future when the world will be fully peopled, in perfect confidence that an equilibrium between birth and death rates will then be brought about by a combination of physical and social agencies, and the bugbear of over-population become finally extinct."

Wallace here is in agreement with Malthus' position on "moral restraint" through such measures as late marriage. Today, contraception, sterilisation and abortion would appear to be more likely means of voluntarily family planning than late marriage, or abstinence from sex. Homosexuality also plays a key role in reducing the birth rate for a population, but is rarely proposed as a partial solution to the problem of overpopulation..

Again, Wallace argues that social factors such as improved education will slow population growth. Today, as far as the West is concerned, the facts would appear to agree with him. However, we are a long way from Wallace's global Utopian vision. In fact, the world continues to be just as Malthus' essay on population described it - largely Dystopian. 

This is not to say that Malthus wanted the world to be Dystopian - he believed it was created that way deliberately, by God, to sort out the Good from the Bad. I'm sure Malthus (were he alive today) would be delighted to see Wallace's dream come true, but a dream it still remains...

Wallace described what most of us (still) want, rather than what is. The "bugbear of over-population" can never become extinct, because evolution constantly favours those populations best able to grow exponentially (for more, read Evolution - An Exponentialist View). This is the Malthusian lesson (which applies to all life) that Wallace seems to have failed to understand.

Wallace asserts that women outnumbered men in his day, largely because of the hazardous nature of male employments. Wallace envisions a world in which women play a greater part in shaping society:

"On the whole, then, it seems highly probable that in the society of the future the superior numbers of males at birth will be maintained throughout life, or, at all events, during what may be termed the marriageable period. This will greatly increase the influence of women in the improvement of the race. Being a minority they will be more sought after, and will have a real choice in marriage, which is rarely the case now."

I doubt that Wallace had in mind the kind of female emancipation that followed from two worlds wars, during which women also undertook hazardous occupations such as working in munitions factories. Still, one has to admire his very modern views. 

"Humanity--the essentially human emotion--has caused us to save the lives of the weak and suffering, of the maimed or imperfect in mind or body. This has to some extent been antagonistic to physical and even intellectual race-improvement; but it has improved us morally by the continuous development of the characteristic and crowning grace of our human, as distinguished from our animal, nature.

    In the society of the future this defect will be remedied, not by any diminution of our humanity, but by encouraging the activity of a still higher human characteristic--admiration of all that is beautiful and kindly and self-sacrificing, repugnance to all that is selfish, base, or cruel."

This is perhaps my favourite passage from Wallace, where he captures the essence of what it means to be human. For the moment, it is clear that Wallace's future has not yet arrived. It remains to be seen how much is optimism, how much pessimism, and how much is reality. 

Differential Replication

Here, Wallace gives his views on the principle of differential replication:

"It seems evident that what takes place among the individuals of a  species must also occur among the several allied species of a group,- viz., that those which are best adapted to obtain a regular supply of food, and to defend themselves against the attacks of their enemies and the vicissitudes of the seasons, must necessarily obtain and preserve a superiority in population; while those species which from some defect of power or organization are the least capable of counteracting the vicissitudes of food, supply, &c., must diminish in numbers, and, in extreme cases, become altogether extinct. Between these extremes the species will present various degrees of capacity for ensuring the means of preserving life; and it is thus we account for the abundance or rarity of species."

Wallace explores this theme more in the sections "Useful Variations will tend to Increase; useless or hurtful Variations to Diminish." and "Superior Varieties will ultimately Extirpate the original Species". In Malthusian terms, this is what Wallace meant:

Year 0 20 40 60 80 100 120 140 160 180 200
1 1 1 1 1024 256 128 64 32 16 32
Normal Antelopes (1000s) 1 2 4 8 16 16 16 16 8 16 8
Weedy- Legged Antelopes (1000s) 32 64 128 256 512 256 128 64 32 16 8
Fit-legged Antelopes (1000s) 1 1 1 1 1 2 4 8 16 32 64

Table G. Population figures for three varieties of Antelope, over time.

Graphed, it looks like this:

Graph A. After 200 years the "Fit" emerge from the struggle of existence

In Table G you must imagine time stretching back for thousands of years, then come forward again to our starting position at Year Zero. In our simple example, taken from Wallace's own musings, there are three related species of antelope. I have called these varieties Weedy-Legged, Fit-Legged and Normal. Weedy-Legged antelopes have short, weak legs. Fit-legged antelopes have long, strong legs. Normal antelopes have legs somewhere between Weedy-Legged and Fit-Legged antelopes. 

In the beginning, there are just two species of antelope, Normal and Weedy-Legged. There are no predators, and they live on a vast grassland. Only competition for food, droughts, and infectious disease keep their numbers down. Nevertheless, allowing for the prevailing conditions, both populations are capable of doubling every 20 years. Weedy Legged have been around for much longer, and so their population has grown larger than that of Normal antelopes.

Between Years 60 and 80, from the East, unusually wet weather has allowed a small (virtually extinct) population of a third species (Fit-Legged) to cross a normally impenetrable desert and enter the grassland. Unfortunately for the other two species of antelope, a predator species has followed the Fit-Legged Antelopes (upon which it preys) across the desert. 

Suddenly, due to the presence of a predator, the Weedy-Legged and Normal species are disadvantaged. From the predator's perspective, it makes sense to chase the Weedy-Legged antelopes whenever possible. They are the easiest to catch. Then, the Normal Antelopes would prove easier to catch than the Fit-Legged antelopes. Most Fit-Legged antelopes continue their migration West, leaving the predator population fixated on the remaining Weedy-Legged and Normal antelopes.

In just 20 years, faced with a seemingly endless supply of easy meat, predator numbers boom to 10,240. This would entail a much shorter doubling period for the predator population:

Year 0 2 4 6 8 10 12 14 16 18 20
1 2 4 8 16 32 64 128 256 512 1024

Table H. Booming predator population, doubling every 2 years

As you can see, a doubling period of 2 years just about covers it. Then, with one too many population doublings in the last 2 years, the Golden Years are over and starvation begins to halve the predator population. In fact, things are so bad between Years 80 and 100 that the predator population halves twice.

The seriously disadvantaged Weedy-Legged antelope population is nonetheless slowly wiped out. They are just too easy to kill. For a while, the Normal antelope population remains completely stable at 16,000. Then, as Weedy-Legged antelopes die out, the predator population faces starvation and the Normal-legged antelopes' numbers fall to 8,000. Meanwhile, freed from the predator's check on their population, the Fit-Legged antelopes enjoy their first sustained period of population doubling. 

The falling predator population allows the Normal antelope population to recover a little, creeping back up to 16,000. Sadly, this encourages a growth in the predator population, and together they enter a vicious cycle of life and death known as dynamic equilibrium.

Whilst I appreciate that my example is fairly crude, nonetheless I hope you can appreciate that the pace of evolution is sometimes dictated by how quickly a species goes extinct, and not necessarily how quickly new species evolve. For the sake of simplicity I made free use of equivalent population doubling and halving times for 4 different species. Imagine if you factor in different doubling times for each  of the three antelope species (assuming identical conditions, predator numbers etc). Of course you now have to factor in the reality that conditions, and predator numbers, are neither identical nor constant. If you can imagine all of that then you have some idea of the complexity of the real differential replication in practice.  

Much like the Nature versus Nurture debate over the influences on human lives, both Natural Selection (through inherent genetic factors, and instinctive behaviour) and Malthusian Selection (through environmental and non-instinctive behavioural factors) both affect rates of differential replication. Which contributes the most may vary with the species and the circumstances. Hopefully somebody that reads this will see a rich area of research just ripe for their investigation.

Restoring Scientific Reputations

It was not my intention to examine all aspects of Wallace's contribution to modern evolutionary theory, but rather to examine those aspects which pertain to the exponential growth of populations. For this subject is the basis of differential replication, and the driving force behind natural selection. Without the exponential growth of populations, there would soon be no struggle for existence.

If you are interested in learning more about Alfred Wallace, and understanding his contribution to evolutionary theory, I recommend that you read John G. Wilson's new book (see below). Wilson, a distant descendant of Wallace's, is keen to restore Wallace's scientific and personal reputations. Although I am no relation to Reverend Malthus, I am keen to restore Malthus' scientific reputation, and draw attention to the reasons why both Wallace and Darwin credit Malthus' essay as the key to their theories on evolution. See Darwin - An Exponentialist View and Darwin's Views On Malthus for more Darwin.

Wallace - The Forgotten Naturalist

There are several mentions of Malthus in Wilson's interesting account of Wallace. Wilson notes that Wallace first went to London in the mid-1830's, but soon after took a teaching job in Leicester. It was here that Wallace had first read Malthus' essay. 

Wilson states that Malthus had argued that the fittest survived the struggle for life, and goes on to mention Malthus' theory that wars, famine and disease prevented population growth from outstripping food production.

Wallace's time (in 1858) with the Dyak people of Sarawak (now part of Indonesia) is mentioned, together with Wallace's musings over why the Dyak were not greater in number. Wilson cites Malthus' checks on population - starvation, disease, infanticide, immorality and infertility - and concludes that Wallace felt that none of these applied in the case of the Dyak. Oddly, Wilson then states that Wallace concluded that infertility (already dismissed!) was the cause (because the women carried heavy loads, and exhausted themselves through work).

Indonesia's population in 1999 was 216,108,345, and the population growth rate was 1.46%. With such a growth rate, a population doubles roughly every 48 years. In fact, since Wallace's time, the population of the whole region (Indonesia, Malaysia and Singapore) has doubled repeatedly:

Year 1840 1875 1930 1970 2006?
Population (Millions) 16 32 64 128 256

Table I. Human population doubling, Malay Archipelago.

So, whilst I don't have the exact figures for the Dyak people, it is clear that the people of the region are not suffering from infertility!

Wilson quotes a long (but highly relevant) passage from Wallace in which Wallace realises that Malthus' Principle Of Population also applied to animals. This lead him to the idea of the survival of the fittest.

In Wilson's last mention of Malthus, he again quotes Wallace on the influence of Malthus' Principle Of Population, this time in relation to both himself and Darwin. Wallace had stated that Malthus' essay "led us immediately to the simple but universal law of the Survival of the Fittest".

Perhaps today we might realise Malthus has been The Forgotten Evolutionist for too long.

The Most Interesting Coincidence

In Francis Darwin's publication of his father's autobiography and letters (Dover edition, 1958) I came across a rarely quoted letter from Alfred Wallace to a Professor A. Newton, Dec. 3rd, 1887. This passage immediately caught my attention:

"The most interesting coincidence in the matter, I think, is, that I as well as Darwin, was led to the theory itself through Malthus - in my case it was his elaborate account of the action of "preventative checks" in keeping down the population of savage races to a tolerably fixed but scanty number. This had strongly impressed me, and it suddenly flashed upon me that all animals are necessarily thus kept down - "the struggle for existence" - while variations, on which I was thinking, must necessarily often be beneficial, and would then cause those varieties to increase while injurious variations diminished....I was lying on my bed (no hammocks in the East) in the hot fit of intermittent fever, when the idea suddenly came to me. I thought it almost all out before the fit was over, and the moment I got up began to write it down, and I believe finished the first draft the next day."

It should have been no surprise to Wallace that Malthus' Principle Of Population applied to the animal kingdom, as Malthus himself said so. Still, to be fair to Wallace, Malthus himself chose to emphasise how the Principle Of Population applied to humanity. 

More than two centuries have elapsed since Malthus wrote the first edition of his essay, and in all that time only Darwin and Wallace have come close to understanding how Malthus' Principle Of Population applied to all living things, and created the Malthusian "struggle for existence" which drives natural selection. If you are aware of any other author (whom I have not listed on my Famous Exponentialists page) who you feel has fared better than Darwin and Wallace in this regard, then I would appreciate hearing from you.  

Yes, as Wallace stated in his letter, it is " the most interesting coincidence". It is a coincidence which has waited far too long to be properly investigated. I hope that my series of exponentialist articles will, at least, point the way to a more thorough investigation in the future.


The Atlas Of The Living World - edited by David Attenborough.(1989)

The Forgotten Naturalist In Search of Alfred Russel Wallace - John G. Wilson (2000)

The Autobiography of Charles Darwin and Selected Letters - Edited by Francis Darwin (1892)

An Essay On The Principle Of Population - Thomas Malthus (1798)

Atlas Of World Population History - Colin McEvedy and Richard Jones (1978)

Robert Wallace

Scottish minister Robert Wallace (1697-1771) is mentioned in Chapter VIII of Malthus' essay, and in the end-notes of the Penguin Classics edition (1985) of Malthus' essay. Robert Wallace's works included:

I would be interested to know whether Robert Wallace, whose works Malthus had read, was in any way related to Alfred Wallace.

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Copyright 2001 David A. Coutts
Last modified: 04 February, 2010