Thank you for your response.

I had no doubt you were correct, but as an exercise in trying to understand the math, I tried to work derivations out on my own.

Plus I was trying to cull the essentials into a few pages for an educational essay and lesson I’m writing on Fisher’s theorem. Also I’m trying to translate the conventions in pop gen literature to the symbolic conventions more familiar to typical math, engineering and physics students.

This is what I came up with, and even then some of my notation is sloppy partly because latex rendering on this site doesn’t work like in MS word.

This is what I came up with. Your notation uses N, I use P (for population). N tends to have connotations for other things.

Let capital P be absolute population size and W absolute Darwinian fitness:

P_i\left(K\right)=P_i\left(0\right)W_i^K

I define:

P_{tot}\left(K\right) \equiv \sum_{i=1}^{N}{P_i\left(K\right)}

\LARGE p_i\left(K\right) \equiv \frac{P_i\left(K\right)}{P_{tot}\left(K\right)}

\LARGE \bar{W}\left(K\right)\equiv\sum_{i=1}^{N}{p_i\left(K\right)W_i}

Given the definition of W_i we can relate the population size of an allele in one generation with the population size in another generation:

P_i\left(K+1\right)=W_iP_i\left(K\right)

Given this we can state a similar result for allele frequnces across two generations:

\LARGE p_i\left(K+1\right)=\frac{P_i\left(K+1\right)}{P_{tot}\left(K+1\right)}=\frac{P_i\left(K+1\right)}{\sum_{i=1}^{N}{P_i\left(K+1\right)}}

Combining some of the above:

\LARGE p_i\left(K+1\right)=\frac{W_iP_i\left(K\right)}{\sum_{i=1}^{N}{W_iP_i\left(K\right)}}

Dividing both numerator and denominator simultaneousl by P_{tot} (since this is basically dividing by 1):

\LARGE p_i\left(K+1\right)=\frac{W_ip_i\left(K\right)}{\sum_{i=1}^{N}{W_ip_i\left(K\right)}}

which implies

\LARGE p_i\left(K+1\right)=\frac{W_ip_i\left(K\right)}{\bar{W}\left(K\right)}

The bar symbol to indicate a mean value doen’t show up to well unfortuantely with latex.

Let W_{ref} be the absolute fitness used to scale the other absolute fitnesses to relative fitness. Then given customary notions of relative fitness, we can scale the above as follows

\LARGE p_i\left(K+1\right)=\frac{\frac{W_ip_i\left(K\right)}{W_{ref}}}{\frac{\bar{W}\left(K\right)}{W_{ref}}}=\frac{w_ip_i\left(K\right)}{\bar{w}(K)}

which then connects to the proof you stated on page 93 with slightly different notation.

I know that the prime symbol corresponds to the K+1 generation, but the prime symbol can be mistaken for the first derivative! Also, some of the the commas in the equations look like prime symbols for terms in the denominator.

Since I’m writing for non-popgen types students of math (like me), I’m writing my essay with symbols and conventions they may be more familiar with. I want math students to be able to understand your textbook so I’m trying to connect the notations in my essay. For example, this was actually very helpful notion in your book, and I merely preserved it and amended it:

\LARGE p_A^{(t)}

and

\LARGE p_A^{(t+1)}

I used K instead since t might be interpreted as a continuous variable.

So my notation is

\LARGE p_i(K)

and

\LARGE p_i(K+1)

I’m not suggesting pop gen adopt new conventions, I’m merely trying to translate the specialized conventions in pop gen for typical students of math, engineering, physics so they can more easily understand pop gen literature and concepts without having to go through an entire semester.