One final entertaining aspect of relative fitness is Fisher’s Fundamental Theorem. In his 1930 book Fisher states “The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time.” This is a bit grandiose as will become clear below, but it is an interesting fact of algebra and the definition of relative fitness that this is in fact the case.
By definition the mean relative fitness before selection is 1 (don’t make me do the math!). The mean relative fitness after selection is:
where Pg and Pg’ is the frequency of the gth genotype before and after selection respectively, and wg is the relative fitness of the gth genotype. So, the change is relative fitness is:
At this point it is convenient to square the 1 (I am allowed to do that right?):
And, thus we see that for Haldane selection Fisher’s Fundamental theorem is easy to the point of being silly. I constantly forget how to derive FFT. Every time I re-derive it at some point it smacks you in the face as being almost bizarrely obvious.
Well, naturally the same thing can be done without genes. If we have a set of individuals that vary in relative fitness then the mean relative fitness is
Similarly, the mean of the population after selection, but before reproduction is
And the change in relative fitness
Note the important difference: in the Haldane equations we had the hubris to think that we exactly knew how the genetics affected fitness. Thus we could get a version of Fisher’s fundamental theorem that went from one generation to the next. In the phenotypic example I did not specify the patterning node or the transmission rules at all, thus I did not give enough information to make any statements about the next generation. Nevertheless, working within a generation we are able to derive a form of the fundamental theorem that works quite well.
If, for some reason terrorists lock you in a cell for three weeks with Fisher’s “The genetical theory of natural selection”, and you actually read the book (OK its only page 35, you can do it), as I said before, he states “The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time.” So he was actually talking about the change that occurs between generations. The genetic variance he was referring to is the “additive genetic variance”, and the fitness he was referring to is the relative fitness.
Much has been written about the fundamental theorem. Many have questioned its usefulness, with, for example, Lewontin calling it “Fisher’s not-so-fundamental theorem” (Lewontin, R. C. 1970. The Units of Selection. Annual Review of Ecology and Systematics 1: 1-18.). The issue is that while the fundamental theorem is a mathematical truism, the real world is not always so simple that it is actually useful. Fisher himself was obviously aware of this, as the last part of his chapter on the fundamental theorem is in fact about the deterioration of the environment, and what would eventually be called the Alice in Wonderland situation (Lerner 1954 genetic homeostasis — you didn’t think I would cite Van Valen did you?). If I could put words in his mouth I suspect that Fisher felt that in most situations the environment deteriorated at just about exactly the same rate at which fitness increased. I am certain that he felt that the situation in which there was a steady increase in the fitness of a population over many generations would be rare indeed.
Incidentally, I find it interesting that Fisher spoke of “deterioration of the environment” in 1930, which Lerner claimed in 1954 as the “Alice in Wonderland situation”, and Van Valen claimed in 1973 as the red queen hypothesis. It is clear that this idea of the environment deteriorating becomes available for naming rights every 20 years more or less. Since nobody has claimed it in over 20 years, I hear by claim the “Cheshire Cat hypothesis”, which is the idea that the environment deteriorates at a rate fast enough blah blah blah, you get the picture. From this point forward everybody should refer to this idea as “Goodnight’s Cheshire Cat hypothesis”. This naming right shall last for 20 years at which time somebody else can claim it since nothing will be left but the grin.
The most interesting variant of the fundamental theorem is probably the Price equation. This has already come up, and will come up again in the future. (Price is a very interesting fellow, and a close colleague of Hamilton of Hamilton’s rule fame).
First, it is worth pointing out that a variance is the covariance of a trait with itself. Thus, the variance in relative fitness can also be thought of as the covariance of relative fitness with itself. This immediately allows us to generalize Fisher’s fundamental theorem. That is, the change in a trait is equal to the covariance between that trait and relative fitness. To see this consider a trait z, that varies among individuals, thus zi is the state of the trait in the ith individual (note, as usual, no transition equations!). The change in z as a result of selection, but before reproduction is:
Note, I took the liberty of multiplying the right half of the first line by 1, but of course, a very special 1, which is the mean relative fitness. This turns out to be a special case of the Price equation. At the risk of repeating myself, Okasha (2006. Evolution and the levels of selection. New York, Oxford University Press.) gives a very nice quick derivation of the basic Price equation (Price himself uses non-standard and somewhat awkward notation).
So, as with Red Queen hypothesis, we learn that there is nothing new under the sun, and that the Price equation is essentially Fisher’s fundamental theorem. Given that the Price equation is over 20 years old, and also directly traceable to Fisher, perhaps I should claim this one as well. Anybody up for the Goodnight equation?
There are actually two caveats that make the Price equation different from Fisher’s fundamental theorem. First, Fisher assumed that the environment was constant and that (he probably actually assumed that Dzi = 0). Second, and significantly, notice that Price used absolute fitness rather than relative fitness. This is significant because Price was a Hamilton colleague, and because Hamilton developed optimality models he (and Price) would not have seen the importance of using relative fitness. For me as a starting graduate student this was a huge problem, and a big part of the reason that kin selection models were absolutely incomprehensible to me for many years. Although unimportant for optimality models where the choice of scale for fitness is irrelevant, for rate of evolution approaches (quantitative genetics) it is absolutely essential. Finally, it is important to point out that the term is environmental change, the 5th force of evolution I discussed earlier.
Of course, there is much more to do with the Price equation. For those who are not faint of heart check out Steve Franks work either his book or his paper on the subject (Frank, S. A. 1997. The Price equation, Fisher’s fundamental theorem, kin selection, and causal analysis). Evolution 51: 1712-1729.
A discussion of the Price equation is not complete unless we extend it to structured populations. Basically if we imagine that there I individuals in each of J demes, then the relative fitness and trait values of the ith individual in the jth deme is ij and zij respectively. And we can take the Price equation and further divide it into within and between group components. It’s a lot of silly hard algebra, and eventually nearly everything cancels out, so I will spare you the gory details. In the final analysis the Price equation can be partitioned out as:
In words, we can divide the total change in a trait z into an among group covariance with group mean relative fitness, the average within group covariance between the trait and individual fitness, and similar changes in the trait itself. Now a word of caution on a future topic. I once talked to Steve Frank about the Price equation, and if I remember correctly, he denied ever calling cov(w.j,z.j) group selection. It isn’t. He knows it, I know it, and now you know it. (At some point in the future I will show you why it is not group selection.)