Wallace's Thought-Bird Experiment

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Posted this in the math section but maybe some Biologists have read the paper & would know what Wallace means.

I've been reading Alfred Russell Wallace's original paper
on evolution that he published with Darwin in 1858 & he presents
an argument about birds multiplying that I've been thinking about.

From the paper;

Very few birds produce less than two young ones each year,
while many have six, eight or ten; four will certainly be
below 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 million!

Starting with two birds, they have 4x4=16 children in the first four years,
i.e. four sets of 4.

Then these 16 birds split into 8 pairs & in all the glory of

bird incest will have 8x16 = 128 children all together,

i.e. 64 pairs or 2^6 pairs.

Then, these 64 pairs of birds will have children of their own.
64x16 = 1024 = 2^10 children.
i.e. 512 pairs or 2^9 pairs.

Then, these 512 pairs of birds will have children of their own.
512x16 = 8192 = 2^13 children
4096 pairs or 2^12 pairs.


The formula I have for this is;

2^0 pairs (YEAR 0)--> 2^3 pairs (YEAR 4) --> 2^6 Pairs (YEAR 8) --> 2^9 Pairs (YEAR 12) 2^12 pairs (YEAR 16)
2^1 Birds (YEAR 0)--> 2^4 Birds (YEAR 4) --> 2^7 Birds (YEAR 8) --> 2^10 Birds (YEAR 12) 2^13 Birds (YEAR 16)

After going up to 16 years, over Wallace's 15! I'm getting a few million less
than Wallace and I'd like to know what I'm doing wrong?

I'd also really appreciate a faster way to do this sum,
obviously I was able to predict the pattern it followed
after doing the first two bird calculations & did it just
to be sure but I would guess there is a way to predict the
pattern in seconds.

Maybe I'm reading Wallace's passage wrong...

(This is a kind of thought experiment similar to how bacteria multiply in a
petri dish if left unmoderated by food & waste, it's just a hypothetical
situation & I'm mainly concerned with how he set up the equation based
on the quoted passage, how he reached 10 million when I reach 8192 &
how I would mathematically set this equation up without all the labourious
steps I have posted).

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Added info on how I view the situation;


The point of this calculation is to show that if progeny rates were an important factor in species survival then by a simple thought experiment for just one pair of birds we should have trillions of birds if every single set of birds that do exist follow the model for just one pair laid out here.

Obviously he is showing this is a totally flawed perspective but this was 1858 & he is just building a case in this part of the paper.

As far as I understand the passage I've quoted we are to take the fact that these birds will have 16 children in four years (4 a year for 4 years) - which is conservatively 4 years though in reality is much higher, that or high progeny rates makes levels much higher.

We also assume that these birds all survive and reproduce - in actuality the levels are varied, he talks about it straight after this part in more detail but it is irrelevant to the hypothetical calculation.

Also, seeing as this is a model of how big a population would grow from just a single pair of birds I see no way to avoid the incest, the fact that the offspring would be genetically malformed and have less chance at survival is irrelevant because our assumption is that they do survive.

The whole calculation is to see how big the level of birds grows assuming
this pattern worked in reality. Think of it as bacteria in a petri dish if
that makes it easier - it's the same principle just with a different rate of
population increase.

The assumption about the 50:50 ratio of male to female is acceptable in that it is an estimate for the high progeny level that does take place in reality.

Seeing as we take all these factors into account how does Wallace reach 10 million when I only get 8192?

From reading that passage maybe I've assumed something that limits my numbers...

Capt'n Midnight

use a spread sheet - put the titles in row one
then skip 3 rows

column A for the years
column B for Parents
column C for children

cell B5 is 2 initial number of parents


cell C5 is =B5*2 2 children per parent

cell B6 is =C5+B5-B2
Add the parents and children and subtract parents from 4 years ago , this is why you skipped the rows


generation 15 is 16,615,008

generation 27 is 7,562,070,811,260 which is roughly the human population

Capt'n Midnight

Check out Fibonacci too