"John Edser"  wrote in message
news:...

I'll start with a slight nit-pick: if you are really writing a book,
and this is really an excerpt from it, I suggest you learn when to use
and when not to use apostrophes.  For instance, "selections" not
"selectons [sic.]", "JBS Haldane, one of population genetics' founding
fathers" not "JBS Haldane, one of population genetics founding
fathers" and "Trivers" not "Triver's".  That out
of the way, let us
move on to the biology:

> Hamilton's model indicates that an,
> organism fitness altruism gene can 
> spread in a population when:
> 
> 		rb>c  ...(1)
> 
> where:
> 	r = relatedness IBD of the organism helped
> 	    to the organism being helped.

No!  r is not identity by descent.  Hamilton approximated his r to 
Wright's correlation coefficient, also r.  But even in 1963 was aware
of the regression nature of relatedness.  r is emphatically not the
identity by descent of the organism helped to the organism being
helped,
whatever that is meant to mean!

> 	b = number of organism's provided with help

No! b is the total increment of direct fitness accrued by the
recipient
which can be attributed to the altruistic act.  

> The inequality (1) above is known as "Hamilton's 
> rule". Note that when rb=c the altruistic gene 
> freq. remains static but when rb The term "inclusive fitness" refers to the degree 
> to which a trait is passed from generation to 
> generation 

No! Inclusive fitness is the sum of all the direct fitness
consequences
for all individuals affected by the actor's behaviour, each increment 
weighted by that individual's relatedness to the actor, and so
transformed
into actor fitness.

> Hamilton's multi level view allowed organism 
> fitness altruism but Darwin's single level view 
> prohibited it, i.e. any fitness altruism observed 
> within nature refuted Darwin's view since any 
> selected lowering of fitness is always prohibited. 

No! Darwin discussed familial selection, what is essentially kin
selection,
in relation to the social insects, in the first edition of Origin.  

> In Hamilton's reasoning only a random 
> selection of recipients can be chosen by 
> a donor and the altruistic phenotype was 
> supposed as being coded for by just one 
> dominant gene, removing all genetic epistasis
> (many genes coding for one phenotype).
> 

No! Hamilton assumed no dominance. 

> Relatedness IBD is just the probability that
> any random organism may be carrying a gene 
> replicate from a parent organism. For a 
> normal sexual parent this probability is 
> reduced geometrically by 50% each generation. 
> In general, offspring are related 50% to 
> their parents using normal sex because parental 
> genes only have a 50% chance of being replicated 
> into their offspring using normal (random) meiosis. 
> Also, they have a 0.25 (0.5^2) chance of reaching 
> their grandchildren and 0.125 chance (0.5^3) of 
> being found in great grand children etc. In general, 
> the IBD relatedness of parents to offspring is 
> 0.5^n where n = the generation number. Note that 
> each parent views descendants as _equal_ investments,
> e.g. each child carries 50% of each parent's genes. 
> However, each descendant need not view itself as being 
> equal to its sibs. From the point of view of the child, 
> each sib carries 100% of its _own_ genes, whereas each 
> brother and sister only carries 50% of the child's 
> genes when the child is regarded as "the parent". This 
> fact highlights the duplicity of relatedness IBD which 
> can validly restart every subsequent organism generation.
> Note that the highest relatedness number per recipient 
> is always the IBD relatedness relative to the _closest_ 
> organism parent( r=0.5) so it would seem the most proximal 
> recipient, i.e. your own children should have kin selection 
> priority but kin selecting yourself where you are
> related r=1 to yourself, is always the best option,
> where such as event is just ordinary reproduction.
> 

This makes no sense!  If the game is zero sum (b=c), then yes, 
unless your recipient has relatedness 1 to yourself, r b < c,
and you will do best to plough that fitness into yourself rather
than the recipient.  But typically, kin selection is applied to 
situations where b > c.  

> Unlimited numbers of recipients cannot be provided
> with unlimited help just to satisfy a hypothetical 
> condition within Hamilton's Rule.  Helping 100 
> recipients related r=0.01 (rb=100*0.01=1) 
> may be equivalent to reproducing yourself twice in 
> the normal way (2*0.5 =1) i.e. moving a replication
> of your set of genomic genes into the next organism
> generation but it may not be equivalent in resource 
> units consumed. It may be more expensive, or just 
> physically impossible, to help that many recipients.

Hamilton's rule is only meant to be applied to behaviours
which we assume are possible.  We can assume a strategy
set, and work out which of the possible strategies is the
best for the gene to take.  You can't argue that the 
winning strategy might not be possible if you have already
opted to include it in the strategy set!  



> 
> 2) The Search For A Missing Constant Within
> Hamilton's Rule.



> the terms have no consistent relationship 
> to each other so their relationships can vary 
> every time the inequality is evaluated. This is 
> the case for Hamilton's rule since r,b and c are 
> all just variables. Thus Hamilton's rule, as it
> stands, is just arbitrary. 

r, b and c are not variables.  They are functions, containing
parameters
and the variable or variables under optimization.  If an altruist act
costs c and helps a relative (related to the actor by r) by b, then
fitness
can be written as

w = a - c x + r b x

where a is baseline fitness.  Differentiating w with respect to x, we 
obtain marginal fitness

r b - c

since the condition for x to increase through selection is that the 
marginal fitness is positive, we can say that increased altruism is s
selected for when

r b > c

Now, it could be that b and c, and even r, are all functions of x. 
Now
we have

w = a - c[x] x + r[x] b[x] x

We can differentiate this if we want to obtain the marginal fitness,
and
ascertain the direction of selection.  You complain that dishing out
more
and more altruism could mean accelerating costs for the actor.  This
can
be implemented by considering c'[x] > 0.



> This was because all these numbers = 1, i.e. in simple,
> additive gene terms 2 brothers or 8 cousins have 
> a probability of 1 for having a genomic set of genes 
> replicated from yourself. 

No! there is some nonzero probability that they will not carry any of
your
genes.  I suppose that 4 brothers and 16 cousins have probability 2 of
having
a genomic set of genes replicated from yourself?!?!

> A probability of 1 means
> this genomic set of genes is equivalent to your
> own set but only as far as just the simple _addition_ 
> of genes are concerned. Given this over simplification,
> Haldane's comment neglected the biological fact that if 
> he remained alive long enough to reproduce himself say, 
> 3 times he would be fitter than any of his other options 
> since rb=0.5*3 = 1.5 representing the gene fitness of 
> the 3 organisms normally reproduced.

This nonsense you are spouting stems from your gross misunderstanding
of
what b is actually representative of in Hamilton's rule, and perhaps
also
an unfamiliarity with the concept of 'reproductive value'.

> I am sure Haldane would not choose to die for 2 
> brothers (a), or 8 cousins (b), compared to reproducing 
> himself 3 times in the normal way before he dies.


Assume all reproductive values are equal (they need not be, but this
is
an illustration, so i don't want to make things too complicated for
you).
If Haldane doesn't do the reproductive act, he will have three extra
offspring.  If he does do it, he won't have these offspring.  Then c =
3.
Because we assume equal reproductive values, then each of the
recipients
is expected to have three extra offspring if they are saved by an
altruistic
Haldane.  This means either 

A) 2 brothers with relatedness 1/2 to Haldane each have three more
kids, i.e.
b = 6, r = 1/2

r b = 3

r b  = c = 3, so Haldane breaks even.  Or

B) 8 cousins, r = 1/4, each have 3 extra kids => b = 3*8 = 24

r b = 1/4 * 24 = 6

r b > 3 because 6 > 3; Haldane benefits from the altruistic act.

>Biology 
> does not limit Haldane to just helping other's
> to reproduce his genes unless he stipulates he
> always was, and will always remain, sterile.

Hence the 'c' component.

> Hamilton argued that in _every_ case the kin fitness 
> (rb) is greater than its cost (c) in organism 
> fitness, then the freq. of the altruistic gene 
> must increase. However, this is not the case. If 
> the missing total fitness of the donor 
> is now included within Hamilton's rule as
> the maximum K, for his altruistic gene 
> to spread, the kin fitness of the donor must 
> be greater than the donor's total fitness minus 
> the organism cost of kin fitness:
> 
> 		rb > K-c	
> and:
> 
> 		rb+c > K
> 

This is ridiculous.  Baseline fitness is a constant and hence
vanishes from Hamilton's rule, which is a measure of marginal
fitness.


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