Appendix 7.1 Modeling Evolution

Fred Hoyle was a brilliant mathematician.. Professor of mathematics at Cambridge.. solved the nuclear synthesis of the heavy elements in the center of stars.. He was an anti-creationist and made a genuine attempt to math model the evolutionary algorithm analytically.. He differs from what he calls the "new believers" (in evolution by natural selection).. in one characteristic.. He told the truth in "The Mathematics of Evolution".. You need to hear what he had to say..

"Let us start naively with the feedback equation..
dx/dt = s.x    (t = time)  (1.1)

in which x is considered to be the fraction of some large population that possesses a particular property, 'A' say, the remaining fraction (1 - x) possessing a different property 'a', all the other individuals being otherwise similar to each other."

After integration to find x and some elaboration on the reproductive outcomes of this model for A being advantageous (s > 0) he gets..
x = xoexp(st)

"So it is agreed for s > 0, with A then a favorable property, that x rises to unity with all members of the population coming to possess it in a time span of ln xo/s generations..  ...  And if s < 0 the solution dies away in a time span of the order 1/s generations, thereby implying that if A is unfavorable it will be quickly rejected.

I am convinced it is this almost trivial simplicity that explains why the Darwinian theory of evolution is so widely accepted, why it has penetrated through the educational system so completely. As one student text puts it.. 'The theory is a two step process. First variation must exist in a population. Second the fittest members of the population have a selective advantage and are more likely to transmit their genes to the next generation.'

But what if individuals with a good gene A carry a bad gene B having the larger value of |s|. Does the bad gene not carry the good one down to disaster? What of the situation that bad mutations must enormously exceed good ones in number?"  (A fact acknowledged by all the research).. and so after some work he gets..
x  ~=  xo/( xo + exp(-st))      (1.6)

Unlike the solution to (1.1) for s > 0, x does not increase to unity ... but only to 1/2... Property A does not "fix" itself in the species in any finite number of generations. A residuum of individuals remain with the disadvantageous property 'a'."

My verification of this next..