The response piece I wrote for the latest JASSS mentions the mathematics in
my 1998 Journal of Memetics paper "Units, Events, and Dynamics in Memetic
Evolution," published at Journal of Memetics - Evolutionary Models of
Information Transmission, 2.
http://www.cpm.mmu.ac.uk/jom-emit/1998/vol2/lynch_a.html. My current
newsletter (non-peer-reviewed) contains some further commentary on why the
mathematics is so "hard," as some have said. The full newsletter is at
http://www.thoughtcontagion.com/Mnemon1999a.htm and a version of the
post-publication comments are also at
http://www.mcs.net/~aaron/UEDerrata_addenda.htm.  



EXCERPT:

Frequent Question: Why does the math have to be so hard?


This question is directed at the population memetic equations in my 1991
and 1998 technical papers. Some very smart people have honestly confessed
that the equations are just "too hard." ... While it is possible for an
author to use extremely abstract mathematical operators that defy empirical
determination while bewildering all but the most mathematically
sophisticated readers, my own system of equations was not written for
abstruseness or escape from real world data. Rather, their complexity
arises from the complexity of the social phenomena they are modeling. For
instance, equations 1 and 2 unify the influences of horizontal and vertical
transmission in a continuous model that does not impose arbitrary
generations on the population. The first 4 terms in each equation are
particularly elaborate, but they can handle such phenomena as the
progression of a religion from mainly peer-to-peer transmission at low
prevalence to mainly parent-to-child transmission at high prevalence. One
of the common misconceptions about memes is that horizontal transmission
should always be faster than vertical transmission. At its fastest,
horizontal is in fact much faster. But with large, complex belief systems,
horizontal transmission takes a great deal of time and effort and may also
encounter resistance due to cognitive, emotional factors and competing meme
systems. As a movement wins more converts, it also starts to exhaust its
supply of persuadable non-converts, forcing it to slow down even more.
(Persuadability corresponds to "receptivity" in the 1998 paper, a subfactor
of the beta and gamma parameters of equations 1 and 2.)

Available data bear this out: Gallup polls have found that evangelical
Christianity had a 34% prevalence in the USA during August 27-30, 1976, and
a 44% of a 32% larger adult USA population 21.83 years later during June
22-23, 1998. That works out to a geometric mean growth rate of 2.479% per
year, or one above-replacement convert made per adherent per 40.3 years.
And that rate is a combination of parental and non-parental transmission,
with believers typically opposing abortion, homosexuality, and other
non-reproductive behaviors. It also includes growth from increased life
expectancy. Some of that results in more converts by or of senior
citizens--but some of the growth also comes from a non-conversion increase
in population due to long-time adherents saying alive longer. Even if we
assume that the 1976 figure was high by 5% of the adult USA population and
the 1998 figure was low by 5% of the adult USA population we still get only
a roughly 3.739% geometric mean growth rate in prevalence, or one
above-replacement convert made per adherent per 26.7 years. Either way, the
growth rate of the whole movement is comparable to growth rates for some
mainly parentally learned faiths. Thus, any equations that model the
phenomenon must model parental and non-parental transmission modes
simultaneously. Such a unified analysis of "apples and oranges" is what
leads to great complexity in the equations. One bonus, however, is that the
same equations apply during early phases of rapid horizontal spread
(abundant persuadable non-converts) and later phases of mainly parental
transmission that occur after most persuadable people are already
converted. This includes what might be called "slow horizontal"
transmission: after exhausting the supply of persuadable non-adherents in a
given decade, the movement must in effect "wait" for "old unpersuadables"
to die off while new non-adherents grow up and pass into phases of
*relatively* higher persuadablity decades later. Even improved
communication technology might make just a slight difference in this
process: the main effect could be to accelerate the depletion of
persuadable non-adherents without eliminating the long wait for new ones to
grow up. 

Such complex mixes of parental transmission, non-parental transmission, and
mortality lead to rather elaborate systems of equations. In fact, the
system of equations 1and 2 can even be expanded to deal with different
levels of persuadability, as by dividing into populations N1, N2, N3, etc.
corresponding to multiple memes with various levels of persuadability (or
persuasion immunity) to each other. The equations could actually have been
written in a more complex but general form with summation (sigma) signs,
subscripts i and j, as well as vertical and horizontal ellipses on top of
all the integrals and derivatives. Instead, they were written in their more
readable 2-equation kernel form. The good news, however, is that such
equations are mainly for specialists doing quantitative work. This includes
computational solutions of the equations or simulations using the
parameters in the equations. ...

The equations belong to a class that I have not seen used before: systems
of non-linear, partial differentio-integral difference equations with
integration over a difference parameter. Their distinct form makes it very
difficult to convincingly argue that population memetics is merely a
metaphor to an established biological field. The argument would need to
demonstrate mathematical isomorphism to some prior system of equations. As
the equations are for host population as a function of time using
measurable parameters, they also make it difficult to argue that memetics
is somehow tautological.



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