Date: Wed, 21 Jul 1999 12:07:46 +0200
From: "Gatherer, D. (Derek)" <D.Gatherer@organon.nhe.akzonobel.nl>
Subject: RE: Terminology and Quantification
To: "'memetics@mmu.ac.uk'" <memetics@mmu.ac.uk>
Aaron:
Note that this is not the only approach to defining memetic "fitness."
Francis Heylighen defines fitness in a manner more analogous to Fisher as
"...the average number of instances of that system that can be expected at
the next time step or "generation", divided by the present number of
instances." (Heylighen, 1996).
Derek:
Yes, but not quite. W also has to be normalised to relative to the number
of instances of those phenotypes in compeition. W can never be more than 1,
just as s can never be more than one.
Aaron:
My own model does not contain discrete generations, but allows for a
realistic mix of all different life phases to exist simultaneously in a
population. (Time steps are left to the empirical investigator to set.)
Under these circumstances, the closest thing to a Fisher-like definition of
"fitness" (for meme 1) is simply dN1(t)/dt and dN2(t)/dt (for meme 2).
Derek:
Normalise that and you have fitness. BUT, and this is a very important but,
you need to know the rates of increase or decrease, and relative abundance,
of _all_ competing allomemes (to use Durham's phrase). d/dt N1 doesn't by
itself give fitness.
Aaron:
N1(t) and N2(t) would be taken as integrals from a = 0 to a = infinity of
N1(a, t) and N2(a, t) respectively. As with Fisher and Heylighen, these are
quantities that may vary dramatically over time and over changing
population sizes.
Derek:
But by normalising, Fisher cancels out any population size change effect.
It's d/dt N1 _relative_ to d/dt N2 that's important.
Aaron:
Hence my preference for a matrix of propagation
parameters if one insists on having a mathematical construct called
"fitness."
Derek:
Unnecessarily complicated, as is the following:
Aaron:
This allows one to say that a meme is "highly fit" even though
its propagation rate is 0 if that growth rate of 0 happens at an
equilibrium prevalence of, say, 99%.
Derek:
Fisher's theory already allows for that simply by normalising max[w] to 1 in
every generation. You don't need any matrices. All these things you
propose have already been done, and in a more concise manner.
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