Thursday, June 01, 2006

As I was going to St Ives: How many kids could Pale Male have?

Pale Male with twig, the Carlyle hotel in backiground
Photo by Lincoln Karim - May 31, 2006

Steve Watson, our California friend via Kestrel Cam , lent his agile brain to my Hawk Progeny challenge which was, more or less:

Given the number of successful offspring that came out of the Fifth Avenue nest from 1995 to 2004, and supposing that all offspring, and offspring of offspring were to survive
[I KNOW that's impossible, as Blakeman points out in a recent note--but that's the way the challenge goes] how many hawks could there now be in the world with Pale Male's genes?

In the wee hours of the morning [a fellow insomniac or is it just California time?] Watson did the following calculations:

I'll take a shot at's late at night and I reserve the right to say "ooooppps...THAT was a dumb mistake", but here goes...

Using the numbers posted for Pale Male's offspring for each year (prolific fella that he is), apply the following assumptions:

1) Each set of PM's offsprings heads off into the wild to mate
2) All offspring survive
3) All offspring mate
4) No consanguinuity of any generations or offspring (all offspring in any generation find a non-PM descendant to mate with)
5) Each pairing results in 3 eyasses
6) All eyasses of each pairing survive
7) Each pair mates once and then never again (this avoids having to sum over generations)

For 1995 (3 offspring) you get the following geometric progression
3, 9, 27, 81, 243...etc., until in 2006 you have 531441 in generation 12 (3 to the 12th power).

To generalize, let n_i be the number of offspring of PM in year i, i = 0..11 (that is, 1995 to 2006).
The total potential offspring in year 11 is (and this is hard to write without using real mathematical symbols):

Sum_i=0,11 ((n_i * 3)*3^(11-i))

which yields 792,477 potential offspring. This is ridiculous, of course, so you'd have start applying real assumptions, such as probabilities of pairing, probabilities for clutch size, etc. Oh, and pairs don't just breed once and then die or stop breeding. And so on.

Anyway, I'm now reminded of why compound interest is such a good thing, and that I need to go make an appointment with my stockbroker to re-evaluate my retirement funds! :) The number 3^n gets big FAST as n grows. It would be interesting to try this with empirical data...

PS from Marie

It's the following throwaway line in Watson's note that makes me think the number he came up with could be hugely, hugely bigger -- after all, think of how many years our pair -- Pale Male and his various mates --sent hawklets out into the world:

Oh, and pairs don't just breed once and then die or stop breeding.