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Linear Growth Versus Exponential Growth
Logistic Growth Versus Exponential Growth
Compound Growth Versus Exponential Growth
Exponential Brownian Motion
The Mechanism Of Population Doubling
Generations and Population Doublings
What Is Exponential?

Richard Dawkins - An Exponentialist View

New Malthusian Scale

External Links:
How Many People Have Ever Lived? - Carl Haub, Population Reference Bureau

How many people have ever lived? - Professor Tom Ramsey

Historical Estimates Of World Population from the US Census Bureau

The Malthusian parameter of ascents: What prevents the exponential increase of one's ancestors? - Susumu Ohno
(from the Proceedings of the National Academy of Science, USA)

'Most Recent Common Ancestor' Of All Living Humans Surprisingly Recent - Science Daily

Generations and Population Doublings

Introduction

In my article Richard Dawkins - An Exponentialist View I showed that population doubling is a simple but powerful means of explaining population growth, and does not suffer from the naivety of the generational approach still favoured in the scientific literature.

Recently, I happened to "read for amusement Bryson on Science" and amusing it certainly was. However, it triggered my thinking with respect to the meaning of the word generation. What does it mean, exactly?

One explanation (which leads to the expression "generation gap") from my trusty Collins (1998) dictionary defines it as:

"All the people of approximately the same age..."

This tied in nicely with my original exponentialist definition:

"...the members of a population born "around the same time".

Another definition from my Collins (1998), which I'll use later in this article, states:

"The normal or average time between two generations of a species: about 35 years for humans."

Yet another explanation from the dictionary related to expressions like first , second or third generation used by immigrants the world over. Myself, I am a first generation Australian. My wife's parents are also first generation, making my wife second generation Australian. So what does that make my daughter - first or second generation, or somewhere in between?

Bryson On Science - Your Line Is Not Pure

It was chapter 26 which reminded me of a conversation with a friend of mine regarding exponential growth. Bryson was launching into an explanation of the relative miracle of his reader's lineage, by going backwards in time to examine the reader's ancestral tree. Pushing beyond everyone's comfort zone of parents, grand-parents, great-grand-parents back to the 19th century, Bryson (2003) estimates the number of ancestors the reader's life depended upon:

"Go back just eight generations to about the time Darwin and Lincoln were born, and already there are over 250 people on whose timely couplings your existence depends."

Darwin and Lincoln were both born on the same day - 12th February, 1809. At the time of Shakespeare (1564 - 1616) and the Mayflower(1620), Bryson calculates his reader required 16,384 ancestors to be "earnestly exchanging genetic material". Going even further back in time, Bryson calculates the following milestones:

Finally, around the time of the Romans:

This is, as Bryson (2003) puts it (my emphasis):

"...several thousand times the total number of people who have ever lived.

Clearly something has gone wrong with our maths here. The answer, it may interest you to learn, is that your line is not pure. You wouldn't be here without a little incest - actually quite a lot of incest - albeit at a genetically discrete remove."

Of course, Bryson isn't the first to realise this. But he's probably the funniest. For example, Carl Sagan explained it in a straightforward manner (Sagan, 1997):

"The same ancestor is related to us by many different routes. We are repeatedly, multiply connected with each of our relatives - a huge number of times for the more distant relations.

Something like this is true of the whole human population. If we go far enough back, any two people on Earth have a common ancestor."

For a little more on Sagan's take on this see Our Common Ancestral Tree.

Bryson's "incest" is also known, more politely, as "The Law Of Sibling Interference" (Ohno, 1996). The best way to visualise this is with a real world example - the marriage of first cousins ("kissing cousins"). As it happens, two Famous Exponentialists married their first cousins - both Darwin & Malthus. Charles Darwin happened to marry a first cousin from another famous family - Emma Wedgewood of the famous Wedgewood China pottery family. So, we'll use Darwin for our example. Darwin's Mother (Susannah Wedgewood) and Emma's Father (Josiah Wedgewood II) were brother and sister. Hence, although Charles and Emma each had two unique parents, rather than sharing 8 grandparents between them (4 unique grandparents each) they only shared 6 grandparents as they both  included Josiah Wedgewood I and Sarah Wedgewood (parents of Susannah Wedgewood and Josiah Wedgewood II) as grandparents. That means that Charles and Emma were 1/8th genetically related.

Familial Calculations

Bryson's advise is, if someone boasts of descent from someone famous like Shakespeare or William The Conqueror, you should cry "Me too!". As Bryson says:

"In the most literal and fundamental sense we are all family."

Of course, as has been pointed out to me by an interested reader, incest is - strictly speaking - sex between close family members. Close means up to great great grandparents (of which you had 16). That's 4 generations. After that, you are roughly as closely related as any stranger off the street (which means that you're 99.9% the same in terms of DNA matching). This is because after 4 generations your genetic relatedness drops to just over 3% (0.54 = 3.13%). That a 3% difference of the 0.1% difference between any two random humans. 3% of 0.1% is...not worth bothering about.

However, Bryson stretches the definition to include sex "...at a genetically discrete remove." Given that Darwin married his own cousin, but was acutely aware of the dangers of incest, lets assume (for the purposes of this article) that Bryson includes sexual relationships more distant than just the borderline taboo "kissing cousins".

Here is a mathematical representation of what Bryson is saying using my new Malthusian Scale:

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 89.5

98.7

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   128 256 512 1024

Table A. Bryson's milestone generations are indicated in red, with the estimate for the total number of people who have ever lived on Earth is around 89.5 to 98.7 C-Pops

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

Let's look at Bryson's usage of the standard decimal scale with bits of English thrown in, and compare that with my exponentialist approach using the New Malthusian Scale:

Milestone Years ago 2 to the power of  New Malthusian Scale Bryson's use of Decimal Scale / English

8 generations

280 8

256 Pops

"over 250"  (actually 256)

14 generations

490 14

16 A-Pops

16,384

20 generations

700 20

1 B-Pop

1,048,576

25 generations 875 25 32 B-Pops 33,554,432
30 generations 1,050 30 1 C-Pop 1,073,741,824

64 generations

2,240 64

16 F-Pops

"approximately 1 million trillion"
Writing that out, we get: 
1,000,000,000,000,000,000

This should actually be:
18,446,744,073,709,551,616

Table B. Comparison of Bryson's use of decimal scale and English with the New Malthusian Scale. 
A generation is assumed to be an average of 35 years.

Thus, just as Dawkins proved that generations are a naive concept going forwards in time, we can see also that generations as a concept are naive if we go backwards in time through our ancestral trees. As a concept, generations are rather ambiguous. So what do we know for sure?

We Are Family

The total number of people who have ever lived has been calculated at around 96.1 billion and 106.5 billion. 

We know for an absolute certainty that, 64 generations ago, you had 18,446,744,073,709,551,616 ancestors on your family tree. Allowing for an average generation time of 35 years, that would be a mere 2,240 years ago. This is indeed a little odd as all estimates indicate a population of only 150,000,000 people were alive around that time. If everyone alive at that time appeared in your family tree your ancestors would appear to outnumber the living at that time by a factor of :

1,22,978,293,825 to 1

Even if we play with the average span between generations, your exponentially growing ancestral tree would soon surpass the total number of people that have ever lived. The answer to the puzzle is that many of your ancestors must appear in your family tree more than once. 

As Bill Bryson has pointed out, this would require an enormous amount of incest in your family. However, don't worry, for the same is true of everyone alive today (that's over 6.2 billion people). We are one family. It's interesting to note, as we project backwards in time, how small "a genetically discrete remove" it appears to take to demonstrate that your family tree is, in fact, also my family tree.

Years ago Milestone Number of your ancestors People alive 
(millions)
Basic odds that anyone alive was your ancestor (one in...)
35 1 generation 2 3,558 1,779,000,000
70 2 generations 4 2,070 517,500,000
105 3 generations 8 1,600 200,000,000
140 4 generations 16 1,200 75,000,000
175 5 generations 32 1,100 34,375,000
210 6 generations 64 900 14,062,500
245 7 generations 128 770 6,015,625
280

8 generations

256

700 2,734,375
490

14 generations

16,384

500 30,518
700

20 generations

1,048,576

350 334
875 25 generations 33,554,432 300 9
1,050 30 generations 1,073,741,824 250 Certain
2,240

64 generations

18,446,744,073,709,551,616

150 Certain

Table C. Within just one or two millennia ago, the odds are that most people alive appear in your family tree.
Given that the same is true for anybody's family tree, this proves that we are all one family. (A generation is assumed to be an average of 35 years).

Of course, it is worth noting that not everyone who has ever lived had children. Thus, some of the people alive 2,240 years ago (and during  the years from then to now) died childless. Hence, you cannot possibly be descended from them. But at some point you will share common ancestors in humanity's one big family tree with every childless person who has ever lived.You are literally related to every human alive today, and every human that has ever lived. 

And we know for sure is that 64 generations ago your 18,446,744,073,709,551616 ancestors in your family tree must be contained within a number of people fewer than the total number of people then alive. The same number of ancestors applies to each person alive today, making 6,200,000,000 x 18,446,744,073,709,551616 (114,369,813,256,999,220,019,200,000,000) ancestors in total for the human race, all contained within a number of people fewer than the total number of people then alive.

According to Science Daily (2004), our most recent common ancestor (MRCA) may have lived just 3,000 years ago. This is not the same thing as saying that most people alive appear in your family tree, but it does reinforce roughly the same sort of timeframe as I've proposed above.

Mitochondria - The Molecular Clock

Recent scientific articles, using a molecular clock based on differences in mitochondrial DNA, put the African origin of modern humans around 120,000 to 200,000 years ago. In this case I have gone back 100,000 years and assumed a founding population of 1 kilopop (or 1 A-pop). Then I've added some estimated milestones using Historical Estimates Of World Population from the US Census Bureau and the "Atlas Of World Population History" by Colin McEvedy and Richard Jones (1978):

Time 98,000BC                      
A-pops 1 2 4   8 16 32 64 128 256 512 1024
Time         10000BC   1500BC 800BC 450BC 1000AD 1600  
B-pops 1 2 4   8 16 32 64 128 256 512 1024
Time 1805 1935 1977   2040?              
C-pops 1 2 4 6 8 16 32 64 128 256 512 1024

Table D. The complete history of global human population doubling on the New Malthusian Scale. The current human population of the Earth is roughly 6 C-pops.

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

This would put everyone alive today somewhere between the 22nd and 23rd population doubling (which is considerably less than the number of generations that have passed in this time). Even if the parameters change (say, starting at 200,000 years ago with 1,000 individuals), it would be easy and yet accurate to use the New Malthusian Scale to quickly model the required population doubling. It would also be possible to accommodate temporary reductions in the global population during these population doublings. For an example 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)

Note also that the time varies between population doublings. This is to be expected given that the annual rate of growth itself varies. Refer to my article The Scales Of 70 for an explanation.

I hope this article has been useful in demonstrating why the generational model should be discarded in favour of the population doubling model first proposed by Malthus (1798).

References

Bryson, Bill. A Short History Of Nearly Everything. Doubleday. 2003

Collins English Dictionary Fourth Australian Edition. Harper-Collins

McEvedy, Colin and Jones, Richard . Atlas Of World Population History. 1978

Ohno, Susumu. The Malthusian parameter of ascents: What prevents the exponential increase of one's ancestors?. Proceedings of the National Academy of Science. 1996. (web site accessed 2nd September, 2009).

Sagan, Carl. Billions and Billions: Thoughts on Life and Death at the Brink of the Millennium. Headline Publishing, 1997

Scientific American.  An Evolution Special.

Science Daily. 'Most Recent Common Ancestor' Of All Living Humans Surprisingly Recent. 2004 (web site accessed 2nd September, 2009).

US Census Bureau. Historical Estimates Of World Population (web site accessed 2nd September, 2009).

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Copyright 2003 David A. Coutts
Last modified: 29 June, 2012