General Replicator Theory

Replicator Mathematics

Kevin Aylward B.Sc.


Index

Overview

This paper addresses the general mathematical approach of dealing with gene-meme Replicators. Although the mathematical difficulties are, in general, very significant, unlike other more prose based arguments, in principle, a mathematical solution is not prohibited.

See also meme/gene Competition Mathematics.

Mathematical Outline

The essentials:

There is a human Replicator that consists of gene and memes. Genes are physically passed on offspring. Memes are passed on to both offspring and the general population at large. The general population additionally passes memes to offspring. The generation period of memes and genes are usually different. Thus, it should now be quite obvious, why the approach of this theory is trivially correct. It is obviously the way it is! 

Solving a typical problem, that is calculating the population distribution of Replicators, and the population distribution of the genes and memes within a Replicator is in general, a very, very difficult task.

The memes of a Replicator are a function of its existing memes and genes and the environment.

The genes of a Replicator are a function of its parent genes and memes and the environment.

The environment is a function of the memes and genes of Replicators and external non biological factors such as heat, light and water.

This is amazingly complicated. For example a mathematical model for the evolution of language will necessarily involve a very complicated meme-gene interaction:

General Equations

PopulationMemes = f(memes,genes, environment)

PopulationGenes = g(memes,genes, environment)

PopulationReplicators= h(PopulationMemes, PopulationGenes, environment)

environment= i(PopulationReplicators)

etc...

Game Plan

The information content of a Replicator's memes requires to be quantified, as does each memes replication rate as a function of its complete internal and external environment. For example, a set of vocal utterances like "ugg" "gug" "upu" would require to be related to their physical response, and what replication rate was obtained for a Replicator due to such response. Its further complicated by the fact that language makes use of the prisoners dilemma advantage, that it is to the advantage to each individual Replicator that language communication is archived for all replicators.

Just to make the problem more tractable some initial simplifications can be made, just to get a basic idea of approach. For example:

Rrr = Rrro(1 + kLn)

where Rrr is the Replicator net replication rate, Rrro is the raw Replication rate, k is a constant, and Ln is the number of word memes in the general population.

The assumption here is that there is a linear relation between the number of words in the population and the replication rate advantage such word memes provide to the Replicator. This further assumes that the population word memes are spread to all Replicators, and the memes mean the same for all Replicators.

With these simplifications, one can actually effectively treat some memes as non-Lamarckian, even though they are Lamarckian. One can therefore simply plug their replication rates and values into the existing formulations of genetic evolution. That is, for some special cases, we can just pretend that memes get passed on as if there are genetic genes and still get valid results.

So, whilst the sociologist and geneticist might debate about wherever its nature or nature, the Darwinianists don't care. The specific mechanism is a mere detail. To explain, for example, why some culture takes effort in their preference for male offspring, is simple a matter of noting that males can replicate their memes and genes faster than females, since they can mate with more females than females can with males. "we observe mostly, that which replicates the most". Its quite immaterial whether a culture meme or a genetic gene is responsible.  

(work in progress)


These papers may be freely copied only for non commercial use,

provided full credit is given to the author.

Kevin Aylward 2003 - all rights reserved