"Guy Hoelzer"  wrote in message
news:cabf2v$l9m$1{at}darwin.ediacara.org...
> in article ca9vg0$4m0$1{at}darwin.ediacara.org, Perplexed in Peoria at
> jimmenegay{at}sbcglobal.net wrote on 6/10/04 8:41 AM:
>
> > "Guy Hoelzer"  wrote in message
> > news:ca5kl0$1pcj$1{at}darwin.ediacara.org...
> >> in article ca4lgr$1fqh$1{at}darwin.ediacara.org, Perplexed in Peoria at
> >> jimmenegay{at}sbcglobal.net wrote on 6/8/04 8:20 AM:
> >>> You are wrong about Hamilton's rule being frequency
sensitive in its
> >>> applicability.  The rule applies equally well at all frequencies.
> >>> If rb > c, then it is advantageous to have the
"altruistic allele".
> >>
> >> This statement makes so many mean field approximations, I am not sure
that
> >> it retains any meaningful relationship to reality.
> >
> > Two questions:
> > 1. What is a "mean field approximation"?
>
> A "mean field approximation" is merely the act of
simplifying a model by
> assuming that a mean value of something is the actual value for all parts
of
> the system.  So, parameters in equation-based models are almost always
"mean
> field approximations".  If your model includes a parameter representing
the
> variance in some measure, this is actually just another form of "mean
field
> approximation" because your model is asserting that the same variance is
> embodied by every element of the system.  For example, variance estimates
> often represent probabilities of events, which are implicitly assumed to
be
> the same everywhere.  From my own area of interest, the Island Model of
> Migration provides an excellent illustration of the use of "mean field
> approximations" because it makes many of them explicit.  For example, the
> Island Model assumes that the immigration and emigration rates are equal
and
> the same for every deme.  It also assumes that the allele frequencies
among
> immigrants are the same for every deme, and that every deme is equally
> connected to every other deme.  Mean field approximations like this allow
us
> to write equations using a small number of parameters, but the validity of
> such a model always depends on its robustness to violations of these "mean
> field approximations", which is rarely considered and even more rarely
> explored.  Symmetry breaking generically occurs in physical (real) systems
> when "mean field approximations" break down.

Thank you.  An excellent explanation.

> > 2. Where do they appear in my statement (or Hamilton's)?
>
> All the parameters in this "Rule" (r, b, and c) are examples
of "mean
field
> approximations."  In reality, the values attached to these parameters vary
> tremendously among manifested social interactions.  Whether the allele is
> advantageous under the kin selection model (the model that underlies
> Hamilton's Rule) depends strongly on these variances and the correlations
> among them.

I can see how one might attempt to use the rule by making mean field
approximations, but one is not required to do so.  For example, assume
that we are interested in studying a species with wide geographic
distribution through ranges of climate.  It is clear that r, b, and c
will all vary as functions (assume continuous) of the geographical
location "g", with g being two dimensional.  And, of course, the frequency
of the altruistic allele "p" is also a function of g.  So we have the
following Rule that applies at all positions p:
  p(g) increases iff r(g)*b(g)>c(g)
It is still the case that the Rule is independent of p(g).

> >> ... a more extensive analysis will show you
> >> that Hamilton's Rule fails as a quantitative predictor of allele
frequency
> >> change, except when the allele is in a low frequency range (but not too
> >> low),
> >
> > Hamilton's rule "rb>c" does not even attempt to be a
quantitative
predictor
> > of allele frequency change.  I would be very curious to see how it could
> > be used as one in the low frequency range.
>
> Good point.  The Rule is a very simple threshold extracted out of
Hamilton's
> kin selection model.  However, it is as error-prone as every other
> quantitative prediction of allele frequency change that could be derived
> from the general model of kin selection.  I stand by my challenge in this
> regard.  If you actually do a numerical model of a population subjected to
> kin selection, the Rule will frequently make false predictions regarding
the
> direction of allele frequency change in a frequency-dependent fashion,
even
> in a large but finite population.

Finite population.  Hmmm.  Yes, I agree that the inclusive fitness selection
coefficient becomes arbitrarily small when the frequency of the "altruistic
allele" approaches 1.  So, yes, I suppose the likelihood that drift will
overwhelm selection is frequency dependent.

> >> simply because "r" becomes an increasingly
inaccurate indicator of the
> >> probability that another individual shares the allele.
> >
> > How does this matter?  Do you believe that the rate of allele frequency
> > change is somehow dependent on the probability that another individual
> > shares the allele?  You will have to explain how.
>
> I am at a bit of a loss to understand your position.  [snip]
> The essence of Hamilton's
> argument was that "r" ("the probability that another
individual shares the
> allele") provided the link between the behavioral interactions among
> individuals and the evolutionary dynamics of allele frequency change.

Ah.  I see the problem.  You believe that "r" is the probability that
the altruism recipient shares the allele.  I believe that it is not.  It
is approximately equal to that probability only in the limit when the
frequency of the allele in the population is small.  However, I don't
see much hope of convincing you of this myself.  I will have to hope
that one of the recognized authorities does it for me (or rather, for you).
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