Re: Hamilton's rule
- From: "Perplexed in Peoria" <jimmenegay@xxxxxxxxxxxxx>
- Date: Mon, 21 Nov 2005 11:38:47 -0500 (EST)
"Jim McGinn" <jimmcginn@xxxxxxxxx> wrote in message news:dlrpsa$1p1j$1@xxxxxxxxxxxxxxxxxxxxxx
>
> Perplexed in Peoria wrote:
> > "Jim McGinn" <jimmcginn@xxxxxxxxx> wrote in message news:dll6as$1qn2$1@xxxxxxxxxxxxxxxxxxxxxx
> > >
> > > Perplexed in Peoria wrote:
> > > > "Guy Hoelzer" <hoelzer@xxxxxxx> wrote in message news:dlh6sj$274$1@xxxxxxxxxxxxxxxxxxxxxx
>
>
> Guy asked:
> > > > > I don't see how your graph justifies HR at all.
> > > >
>
> PIP replied:
> > > > Well, the graph doesn't exactly justify HR. The relationship R = P +(D_P)r
> > > > plus some simple reasoning justifies HR.
> > >
>
> JMc asked:
> > > Simple reasoning?
>
> PIP, responding to Guy, continues:
> > > > The logic is that the definition
> > > > of 'r' (IBD) and the assumption of random mating leads to the relationship
> > > > R = P + (D-P)r. That relationship, leads (by a different line of reasoning)
> > > > to Hamilton's rule. We use the IBD definition of 'r' because it leads
> > > > to a nice (and correct) rule.
>
> JMc asked:
> > > Can you explain to us the intellectual mechanics that brought you to
> > > this conclusion?
> >
>
> PIP responds:
> > Part 1. The recipient receives a fraction r of his genes from the donor,
> > and the remaining (1-r) fraction from the general population.
>
> You lost me already. Sorry. I'm thinking maybe you skipped a step.
> If not then it seems you are employing a skyhook.
Not a skyhook, but I was employing shorthand language. Sorry, I will try
to be more precise below.
> I thought all
> organisms recieved all their genes from their parents. Now your
> saying that some of them come from, "the general population." What
> do you mean by this, and how would you apply it, ...
A nicely asked question. Ok. A fraction 'r' of the typical recipient's
genes (on average) come from the common ancestors that the recipient
shares with the donor AND they are identical (by descent) with the
genes that the donor received from the same ancestors. I described this
using the shorthand language that these genes 'come from' the donor.
The remainder (fraction (1-r)) of the recipient's genes come from
sources that don't provide genes to the donor. These genes are expected
(on average) to have the same allele frequencies as the general population.
I used the shorthand language of saying that these genes come from
the general population.
> you know, for
> example, how do we know which ones are from the donor and which
> are from the general population?
Well, we don't know which come from each source. We only know the
fractions, on average. And natural selection doesn't 'know' the
specifics either. So if an individual acts altruistically toward
an individual it 'knows' is its brother, it does not 'know' for sure
whether that brother has inherited the gene for altruism IBD. But
it does 'know' the probabilities. If rb>c then the act of altruism
is a favorable gamble from both the individuals perspective and from
the altruism allele's perspective.
> > Hence
> > R = rD + (1-r)P
> > = P + (D-P)r
> >
> > Part 2. The donor loses Dc copies of the allele, but the recipient gains
> > Rb copies. The population loses (Pb-Pc) copies, using the Woodgold trick
> > for density dependent population regulation. So alleles increase if
> > Rb > Dc + Pb - Pc
> > or (substituting the above value for R)
> > Pb + (D-P)rb > Dc + Pb - Pc
> > or (subtracting Pb from both sides)
> > (D-P)rb > (D-P)c
> > or (dividing through by D-P)
> > rb>c
> >
> > Send a quarter of the $10,000 to me, a quarter to Woodgold, and a half to NAS.
> > It really would have been cheaper and easier if you had followed my
> > advice and found this stuff in textbooks. (Less fun for you, though).
>
> Assuming you successfully address the issue I raised above and
> supply the clarification and assuming the rest of it checks out,
> yes, I will gladly send the money and divide it as you describe.
>
> I'd thought it was obvious but maybe it wasn't, you can't have any
> skyhooks in the proof. Sorry if this wasn't clear to you previously.
>
> Back to the drawing board.
Nah! I'll just try to clarify what I have already written. Incidentally,
in case you are worried that this excercise might cost you money, let
me make it clear that I never took your 'prize' promise seriously. But
if you remain as polite as in this exchange, I am willing to keep
supplying clarifications.
.
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