The left-hand side of the Fundamental Theorem of Natural Selection: A reply

In a recent paper, Grafen (2018) discussed the left-hand side in the equation stating Fisher's (1930, 1958) "Fundamental Theorem of Natural Selection" (FTNS). Fisher's original statement of the FTNS is, in effect, "The rate of increase in fitness of any organism is equal to its genetic variance in fitness at that time" with the rate of increase in fitness understood as the one "due to all changes in gene ratios" (Fisher, 1930, p. 35). For purposes of exposition, Grafen (2018) considered what is today called the analogous discrete-time model, and restated the FTNS on p. 181 as "The increase in population [mean fitness] due to changes in gene frequencies [is equal to the] additive genetic variance in fitness [divided by the] mean fitness". Allowing for the fact that Grafen's statement of the FTNS relates to a discrete-time model, his statement is in effect a discrete-time version of Fisher's. It has however been widely accepted for many years, ever since Price's (1972) deep analysis of the FTNS, that Fisher's wording does not correctly describe the content of the FTNS. The same is therefore true of Grafen's statement. The confusion caused by these misstatements is unfortunate and adds to a continuing misunderstanding of the FTNS, whose source can also be found in Fisher's (1941) own explanation. Our purpose is to review the detailed analysis of the calculations leading to the FTNS to clarify the points at issue.