GENEALOGY-DNA-L ArchivesArchiver > GENEALOGY-DNA > 2005-06 > 1117664832
Subject: Re: [DNA] Middle Eastern ancestral markers on new Euro 1.0 test
Date: Wed, 1 Jun 2005 18:27:12 EDT
I can't resist jumping in here a bit: recently, John, Ken Nordvedt & I spent
some time writing & running hundreds of population growth simulations using
this equation. It proved to be quite an instuctive exercise.
We started with a simple version: X = P(0) + X*P(1) + X^2*P(2), where P(i)
represents the probability of a man having i or more sons living to maturity,
and making the simplfying assumption that P(3), P(4), ... = 0.
It's possible to derive some analytical solutions to this equation: There's
a solution where X<1, if the sum P(1) + 2P(2) > 1. The growth rate is P(1)
+ 2P(2) -1. The extinction rate is P(0)/P(2).
John's mathematics are right-on.
In a message dated 6/1/2005 12:57:13 PM Pacific Standard Time,
> > If all you want is an example that has a 99% individual extinction
> here's one: suppose the likelihood is 40% that a man will have no sons
> who live to maturity, 30% that he will have one son who lives, 20% two
> sons, 9.5% three sons, 0.5% four sons, and no chance at all of more
> than four sons. This example will have a 99% extinction probability.
> The population will be almost stable: growing at a rate of 0.5% per
> How do you get to 99% from these numbers? For example, shouldn't you give
> some parameters with this number like 99% within x generations?
I still think you don't really want to see the details, but you asked for
them, so here they are. The probability of eventual extinction X is
equal to the following infinite sum:
P(0) + X P(1) + X^2 P(2) + X^3 P(3) + ...
where P(i) is the probability that a man will have "i" sons who live
to maturity. Note that X appears on both sides of the equation. Note
also that the sum of all the P(i) must be 1. Therefore, one solution
to this equation is X=1, which is to say that the entire population
is going to die out in the end. That is not an interesting solution.
There is another solution. To find it, make an initial guess at X
that is strictly between 0 and 1, say 0.5. Plug that into the
expression above. The resulting value will be different from the
initial guess. Take the result and call it the second guess and
plug it into the expression above. The new result will be different
from the previous guesses. Keep on repeating this process until it
converges. (Remember, you asked for this!) When it converges, you
have the interesting solution to the equation, and the value of the
extinction probability corresponding to the chosen P's. If you
repeat this process a few hundred times, it will converge, and the
answer for the case I gave above is 99%.
And, no, you shouldn't specify "within N generations" because that
wouldn't be the extinction probability. That would be the
N-generation extinction probability, and you would have to tabulate
these probabilities for all values of N in order to satisfy everybody.
The limiting value of that tabulation is, in fact, the extinction
probability as I defined it.
> Sorry. I should have said something like accumulate.
Well, that depends. If you mean that the N-generation probability of
extinction grows with N, then of course that's true, but it grows only
short of the limiting value. By definition, the limiting value is
fixed and does *not* grow.
> My point, given what I
> am responding to, is that within any population, being a majority Y-type
> mt-type or surname) does not make it necessary that there must be a
> selective advantage - which is what you said.
I never came close to saying any such thing. In fact, I said exactly
the opposite. I pointed out most emphatically that there are two
"juggernauts" in operation and that you must *not* confuse them. One
is the random extinction of rare haplotypes. The other is natural
selection. As I said then, natural selection is the more powerful,
since it operates quickly and completely. I don't think I mentioned
the other side of the story, which is that natural selection operates
in either direction with equal thoroughness. An advantageous trait
will come to dominate the population, but a disadvantageous trait will
be eliminated. The random elimination of rare traits is just a
statistical accident. It happens a lot because mutations are
constantly supplying new rare traits. It takes only a short run of
bad luck to wipe out a rare haplotype.
> John, you are talking about Y DNA or surnames, not about real extinction
> like in most evolutionary biology. So what you say is not correct, or at
> least seems to be wrongly written.
Not at all. The extinction of an individual or a group is exactly the
same thing as the extinction of a species or a genus. The only difference
is the number of individuals involved.
Now, here's the key point that you keep refusing to hear because you
"read" something sometime in the past. The probability of random
extinction for a group is just the Nth power of the individual
extinction probability, where N is the number of individuals in the
group. The example I gave with 99% individual probability was made up
quite arbitrarily, but it was subjected to a very important constraint
to guarantee plausibility: the population growth rate is positive, but
modest. Under such circumstances, the individual extinction probability
is bound to be quite high, so 99% is not at all surprising. The important
point is that 99% is less than 100%, and therefore the probability goes
inevitably down as the group size expands. For a small group, the
likelihood is high for eventual extinction. For a large group, the
probability is low. In this example, the extinction probability for
a group of just 2000 males is so small that you would call it zero.
That is the qualitative difference you keep asking for.
One more point. Yes, the extinction I'm talking about is the real
thing. The logical extension of your claim is that it's only a
matter of time until all life on Earth randomly winks out of
existence. This is separate from the potential catastrohpic failure
of "nuclear winter" or an asteroid collision, which may be just as
inevitable as you say, but it's not what we're talking about.
> Just to remind you of the subject: you say that any surname which is in a
> majority can be assumed, at least within reasonable probability, to have
> more than chance on its side - that it must mean that, something, for
> example the Y DNA of that family, must give better ability to breed males?
Again, I must correct you. I never said or implied that, and I can't
imagine where you got that notion. What I said, and continue to say,
is that common haplotypes do not randomly disappear. The conclusion
is: if they do disappear, it is not an accident.
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