The Fate of Birds

sponsoredwalk

Hi 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...

Delphi91

sponsoredwalk wrote: »

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...

First of all, you're assuming that the 16 birds will be male:female, 50:50. There's no guarantee of that. If it did work out that way and they did go off and have incestuous sex, the chances of all the offspring surviving would not be too high.

So if anything, I'd say your results are conservatively high.

Sid_Justice

I must admit I miss the point of your post. Is his maths wrong? I hope I'm not pointing out the obvious in that only a small proportion of juveniles reach adult hood and not adults reproduce successfully in their life time.

sponsoredwalk

I would guess that my maths is wrong, if he says nearly ten million after 15 years and I'm ending up with 8192 birds then more than likely I am wrong by math or misunderstanding how he did his calculation.

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...

The Big Red Button

sponsoredwalk wrote: »

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.

As far as I can see, that's not quite right ... OK, so you have two birds in Year One. They have four children in Year Two, and these children will have (between the two incestuous couples) eight children in Year Three, while their parents are having their second set of four children. Therefore twelve babies will be born altogether in Year Three, giving you a total of eighteen birds.

And so on and so forth over the fifteen years ... not just one batch a year in years Zero, Four, Eight and Twelve, as per your formula.

The Big Red Button

For what it's worth, I've attached how I'd work it out ... I don't think it's fully right, but hopefully not far off! I'm assuming the birds only live for four years each, by the way.

Fremen

If we assume that there is exactly a 50/50 male/female split between birds to simplify things, the calculation boils down to solving a difference equation.

Let [latex]B_n[/latex] be the number of birds which are alive at time n.

Each pair of birds produces four offspring. This makes

[latex]4 \left(B_n / 2 \right)[/latex] offspring total.

The number of birds in year n+1 is:

(Number of birds in year n) + (number of new offspring) - (number of birds which die after a four year lifecycle).

The corresponding recurrence relation is

[latex]B_{n+1} = B_{n} + 4 \left(B_n / 2 \right) - B_{n-4} [/latex]
i.e.

[latex]B_{n+1} = 3 B_n - B_{n-4} [/latex]

We start with one pair of birds which only die at year 4:

[latex]B_0 = 2[/latex]
[latex]B_{-1} = 0[/latex]
[latex]B_{-2} = 0[/latex]
[latex]B_{-3} = 0[/latex]

You can solve this exactly for year k by summing
[latex]B_{n+1} = 3 B_n - B_{n-4} [/latex]
from n=0 to k, then re-arranging the sums to give you a system of linear equations.

Here's wolfram alpha's plot of the solution, you can see that the Y axis is a log scale, which means that the number of birds is growing exponentially and that soon we will be overrun by our new avian overlords.

Here's a great book on methods for solving problems like this:
http://www.math.upenn.edu/~wilf/DownldGF.html

It's very readable, and quite famous.