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Mathematics of Evolution
On the "Mathematics of Evolution" by Fred Hoyle
(Acorn Enterprises LLC, Memphis TN, 1999)
I cannot follow the argument given on p.8. When I attempt to reproduce it, I get a different result. My attempt is as follows.
Let n(t) and N(t) denote the total numbers in the populations of types a and A, respectively, at time t.
The statement "individuals with properties A and a are considered to produce offspring in the ratio 1 + s : 1" seems to me to imply that
dN/N = ( 1 + s) dn/n.
From the definition on the line following Hoyle's eq. (1.1), we have
x = N/(n + N).
From the above equations, I obtain the result
dx/dt = sx(dn/dt)/(n + N),
which does not reduce to Hoyle's result
dx/dt = sx
unless one makes some further assumption which gives the dimensionally incorrect result
dn/dt = n + N.
The dimensionality question can perhaps be answered by noting that Hoyle takes the unit of time to equal a change of one unit in n or N, so his time variable is actually dimensionless.
I cannot think of any justification for assuming the validity of the last equation, however.
WalterRemine has posted a reply to my criticism at:
He states (in part) :"You are correct concerning the math, but incomplete in your understanding of what Dr. Hoyle was doing there.
On page 8, he says, "Let us start *naively* with ..." and then he gives the classic feedback equation (equation 1.1) which shows up frequently in the natural world, even outside genetics.
That easily integrates to give equation 1.2, which is an exponential equation -- which grows enormously large, given sufficient time. Clearly something gets unrealistic here, and he was tipping that off with his above word "naively".
In effect, Dr. Hoyle is giving a simple tutorial discussion here, intended to introduce and teach a few simple points. On that page, he uses this self-same simple approach to illuminate both beneficial and harmful mutations (positive, and negative s value)...."
I do not believe that a tutorial, no matter how simple or naive, is a proper place for an unrecognized error in mathematical analysis. Dr ReMine urges me to read the rest of Hoyle's book, though he admits that it is heavy going. Is it worth it? In a court of law, at least, a witness, shown to be unreliable in any one part of his testimony, may safely be assumed to be unreliable in all parts of his testimony.
Moreover, if I had questions about the rest of Hoyle's analysis, what would I do? Hoyle, alas, is no longer with us. I have tried and failed to make e-mail contact with Wickramasinghe.