jimmenegay{at}sbcglobal.net (Jim Menegay) wrote in
news:c5f7am$k3n$1{at}darwin.ediacara.org: 

> William Morse  wrote in message
> news:... 


>> > We have the defining equation:
>> >   r = g - e
>> > where
>> >   g = genetic relatedness (this is the old r)
>> >   e = ecological competitiveness. 


> Thanks.  But, so far, I can't get the idea to work.  I need a theory
> explaining how "e" is conceptually computed, and some ideas as to how
> it can be empirically measured.  Plus, I need a variant of Fisher
> showing how "e" ties in to the dynamics of gene frequencies.  Only
> with all this ugly math in place can I then wave my magic wand and
> reveal the simple rule (g-e)b > c.
> 
> It seems pretty clear that I have to somehow combine a Fisher-type
> model with a Lotka-Volterra type model.  This has almost certainly
> been done by someone, though I have never seen the result.  There also
> probably needs to be some kind of analysis of population fluctuations
> in space and time.  This may involve tying in Kimura and maybe even
> (Dr. Holtzer, are you reading?) the renormalization group.  It would
> be very funny if the model that results from this were to incorporate
> Fisher and Kimura into a cooperative framework, rather than having
> them compete.  I would love to see a pop gen paper that
> straight-facedly cites "John Edser, personal communication".
> 
> Unfortunately, I don't think I have the mathematical and pop gen 
> background to carry out this project.  Maybe someone else can do so,
> if the idea can be made to work at all.

(snip)

I don't have the math either, but I would like to explore it further. 
First let me note that, after some further thought, while I like the idea 
of an e term I don't think it should be incorporated into r, as one can 
have an e interaction without any relatedness. In this case we can go 
back to r as relatedness, with the equation now: (r-e)b>c

To try to be as explicit as is possible for me:

b is not necessarily a benefit - it is the effect on recipients of the 
behavior - but by convention, the sign will be positive if it increases 
the fitness of the recipient.

c is not necessarily a cost - it is the effect on the originator of the 
behavior - but by convention, the sign will be positive if it decreases 
the fitness of the originator.

while c is an individual effect, rb and eb are sums over all the 
recipients of the behavior.


I don't really like the idea of negative r, and might like to switch it 
to absolute relatedness, but there are certainly points to be made in 
favor of the classic definition so I will leave it alone for now.

e is - well I guess that's one of the key questions.

Now I propose we widen the scope of the interactions to include not just 
conspecifics, but also all the major interactants of an individual. To 
this extent the conspecifics are not the species as a whole, but only the 
local population, while the other interactants expand to include the 
entire local ecosystem. This may create a measurement problem. AFAIK the 
Hamiltonian equation is based on fitness - a measure of reproductive 
success. The usual ecosystem metric is energy. So, as you suggested, we 
have to try to merge Fisher and Lotka-Volterra.This is not a new problem, 
so there should be some guidance out there.

  
One of the reasons I like the idea of e  is that it helped me understand 
how symbiosis can evolve, which is something I have puzzled over for a 
while. Obviously mutualism and commensalism is easy - there is a benefit 
for one or both parties with no cost for the other. But symbiosis seems 
to requires a cost from both parties. The cost is more than made up for 
by the benefit, but how does one start this process? There is no 
possibility of kin selection, and probably no possibility of reciprocal 
altruism  since the two are different species with no common 
communication pathways. 

This is probably a new idea only to me, but the addition of ecological 
competitiveness in the equation helped me to see how symbiosis could 
evolve in sort of a reverse red queen race. If there is a behavior that 
has a cost to the individual, but tends to induce a  mutualistic species 
to preferentially favor the individual over ecological competitors, then 
-eb > c. For the mutualistic species e is 0, while for the ecological 
competitor b is negative. It's sort of like the Yankees offering more 
money for a player than the Red Sox. Having the player both helps the 
Yankess and hurts the Red Sox. 


Of course so far the equation does not explicitly contain the thought 
that a benefit to one party is usually a cost to a competitor. It would 
be nice to include the overall carrying capacity in the equation, but I 
don't yet see how to do that.


Yours,

Bill Morse
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