# When and why was the symbol w introduced for biological fitness

$$w$$ the visually similar $$\omega$$ are often used to represent biological fitness in population genetics models. (Sometimes $$W$$ is used for absolute fitness, and $$w$$ for relative fitness.)

These symbols appear in numerous textbooks and journal articles. I could fill up the page with examples. Here are two well-known books that use $$w$$, an introductory textbook, and an advanced reference, respectively: John Gillespie's Population Genetics: A Concise Guide, 1st edition, 1998; Warren J. Ewens, Mathematical Population Genetics I: Theoretical Introduction, 2004.

Both "w" and "$$\omega$$" seem like odd choices, though, especially given that the people who first developed population genetics (Fisher, Wright, Dobzhansky, et al.) mainly published in English.

What was "w" (or "$$\omega$$") intended to symbolize, or was it an arbitrary choice? Who introduced it, and when?

(There are other symbols in use, such as $$s$$ for "selection cofficient", which generally has a different meaning, but is closely related to most uses of $$w$$.)

• I don't mind downvotes, but unless it's obvious what I've done wrong, a downvote without comment makes me curious about the reason, and about whether I can improve my question, or whether I should remove it. – Mars May 15 at 20:24
• 1. Wikipedia ( en.wikipedia.org/wiki/Fitness_(biology) ) seems to imply that some use $\omega$ (Greek omega) for this. 2. What have you found out? What's the earliest use of $W$ used this way you know about? – kimchi lover May 15 at 23:18
• @kimchilover, thanks. I wasn't thinking about $\omega$, but I do think it's related. I have added a brief remark about sources. (If people here think the question is illegitimate unless I first go through my library and all articles I can find searching for early uses, then I would just delete it. I don't expect anyone else to do that kind of legwork, but if someone knows the answer, I will be grateful.) – Mars May 16 at 2:09

It seems to be a game of broken telephone with Boltzmann's formula for the entropy at the beginning, and Fisher's Genetical Theory of Natural Selection (1930) in the middle. Fisher's statistical measures of fitness are the reproductive value denoted $$v$$, "the present value of the future offspring", by analogy to compound interest in economics, and what he calls the Malthusian parameter denoted $$m$$, "the relative rate of increase or decrease of a population when in the steady state". He then relates the latter to the gene frequencies $$p,q$$ (pp.34-35):

"The two groups of individuals bearing alternative genes, and consequently the genes themselves, will necessarily either have equal or unequal rates of increase, and the difference between the appropriate values of $$m$$ will be represented by $$a$$, similarly the average effect upon $$m$$ of the gene substitution will be represented by $$\alpha$$. Since $$m$$ measures fitness to survive by the objective fact of representation in future generations, the quantity $$pqa\alpha$$ will represent the contribution of each factor to the genetic variance in fitness; the total genetic variance in fitness being the sum of these contributions... whence it follows that, $$\alpha dp =pqa\alpha dt$$ and, taking all factors into consideration, the total increase in fitness,

$$\Sigma (\alpha dp) = \Sigma (pqa\alpha)dt = Wdt.$$

If therefore the time element $$dt$$ is positive, the total change of fitness $$Wdt$$ is also positive, and indeed the rate of increase in fitness due to all changes in gene ratio is exactly equal to the genetic variance of fitness $$W$$ which the population exhibits. We may consequently state the fundamental theorem of Natural Selection in the form

The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time."

So for Fisher $$W$$ is not the fitness itself, but the genetic variance of fitness. Why he chose $$W$$ Fisher does not say, but he later discusses the analogy to the second law of thermodynamics, as derived from the statistical mechanics. This, of course, reminds us of the Boltzmann formula for the entropy $$S=k\ln W$$. If $$p(t)=p_0e^{mt}$$, as in the Malthusian model, then $$p(t+1)=e^{m}p(t)$$, and with $$W$$ as defined today we have $$m=\ln W$$.

$$W$$ in the Boltzmann formula is already a case of the broken telephone. It was intended to stand for the Wahrscheinlichkeit (probability in German) of a macrostate for some distribution of microstates, but was misinterpreted early to stand for the number of microstates, as it does now. If this is what he had in mind, Fisher's use is more true to the original since $$W$$ is often described as the probability "that the individual will be included among the group selected as parents of the next generation" (Maynard Smith). Fisher writes (p.36):

"It will be noticed that the fundamental theorem proved above bears some remarkable resemblances to the second law of thermodynamics. Both are properties of populations, or aggregates, true irrespective of the nature of the units which compose them ; both are statistical laws ; each requires the constant increase of a measurable quantity, in the one case the entropy of a physical system and in the other the fitness, measured by $$m$$, of a biological population."