John Edser wrote:
>>>>>JE:-
>>>>>Zero is not a constant it is just
>>>>>another number that may or may not
>>>>>be used to represent a constant
>>>>>term within a mathematical expression.
>>>>
>  
> 
>>>>BOH:- 
>>>>What?  Do you mean zero can vary?
>>>
> 
>>>>JE:-
>>>>OK, zero _is_ a constant as a number 
>>>>but what a number is used to represent 
>>>>can _vary_ since other numbers may be 
>>>>added/subtracted from it, if and only if, 
>>>>it represents a variable. A variable
>>>>is just a relative concept so you are
>>>>required to ask: relative to what and
>>>>then, supply an answer! The only possible
>>>>answer is: a constant. 
>>>
> 
>>>BOH:-
>>>Good.  So, I'll re-write Hamilton's Rule as rb-c>0, and
there you have 
>>>it.  rb-c is a variable, and 0 is your missing constant.
>>
> 
>>>JE:-
>>>This zero state has to represent
>>>something within a testable theory
>>>of nature. 
>>
>  
> 
>>BOH:-
>>Yes.  In this case, no difference between the costs and benefits of an 
>>altruistic behaviour.
> 
> 
>>JE:-
>>Please replace the number "0" with a new 
>>defined term, within Hamilton's rule.
> 
> 
> BOH:-
> Why?  Zero is zero in Hamilton's rule.  It's a constant.  
> 
> JE:-
> This is _not_ a difficult ask. If you really
> think you understand what zero represents
> within the rule, then go ahead and replace it with 
> an actual algebraic term that represents something 
> within the science of biology! 

Why?  The zero does represent something biological.  It's the point in 
the b/c tradeoff where there is no difference in fitness between the 
altruistic and non-altruistic phenotypes.


> BOH:-
> If the 
> benefits are greater than the costs (i.e. if rb>c), then the altruistic 
> trait can evolve.  Zero is the critical value here - it's this that rb-c 
> is compared to.
> 
> JE:-
> "If the benefits are greater than the costs" COMPARED
> TO WHAT MISSING MAXIMAL VALUE? 

They are compared to each other.  There is no missing maximal value.

If kin fitness only makes
> a relative benefit at just the gene level but decreases the 
> number of organisms reproduced by doing so, then yes, the
> gene spreads _relatively_, but also, yes, the gene 
> becomes _absolutely_ extinct as the population 
> constantly shrinks to zero. 

That's a very big "if" at the start, and Hamilton's rule says nothing 
about the absolute numbers in the population.  the scenario you're 
suggesting is indeed possible (its an example of evolutionary suicide), 
but it is certainly not an inevitable consequence of Hamilton's rule. 
It can, indeed increase the overall growth rate.  Whether it does this 
or not, hamilton's rule still holds.  There is no need to force 
demographics into the rule (although putting demographics in may be 
desirable for looking at particular instances - it's one area where 
there has been quite a bit of work recently).

Bob

-- 
Bob O'Hara

Rolf Nevanlinna Institute
P.O. Box 4 (Yliopistonkatu 5)
FIN-00014 University of Helsinki
Finland
Telephone: +358-9-191 23743
Mobile: +358 50 599 0540
Fax:  +358-9-191 22 779
WWW:  http://www.RNI.Helsinki.FI/~boh/
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