# A phenotypic view of evolution Evolution in Structured Populations

## A one line proof of Fishers fundamental theorem

Lots of people get bent out of shape about Fisher’s Fundamental theorem, and spend lots of pages talking about it. The problem is that people tend to see the FFT as being magical. Theoreticians promote this because, well, the basic proof is so simple that you need to add some sort of complication to justify your existence (Ok, some of the complications are interesting). So without further adieu, here is my one line proof of Fishers Fundamental Theorem

Added in edit:  I completely forgot to tell you what Fisher’s Fundamental Theorem was!

Fisher (1930) stated that “The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time.”.  I would strike the word “genetic” from that sentence.

First, define terms:

The relative fitness of the ith individual =

The frequency of the ith individual = pi (usually that will be 1/N, but the truth is general, all we need is a frequency, and in reality it should be particle, not individual in the definition)

The frequency of the ith individual after selection, or put another way, the fitness weighted frequency of the ith individual =

The mean relative fitness is

I use the symbol to denote the mean fitness after selection

Some math you need to know:

12=1 (not sure what to call this other than a truism)

(A*B)*C = A*(B*C) (the associative property of multiplication)

SO the one line proof:

Q.E.D.

In a recent article, Andy Gardner (1015, J. of Evol. Biol, DOI:10.1111/jeb.12566)says:

“Today, disagreement still persists as to the correct interpretation of the fundamental theorem. For example, whereas Okasha and Ewens both regard the theorem as concerning the selection of genes, I regard it as concerning the selection of individuals.”

Are you serious? I don’t think it applies to Koalas. Of course FFT applies to genes, and to phenotypes, and yes, even to Koalas. Heck it applies to anything to which relative fitness can be applied. OK, maybe I will give this to Andy: there is really no meaningful way to assign fitnesses to genes, so maybe the fundamental theorem isn’t so useful in that situation.

This actually raises an important point. Yes, the FFT is a truism, but that doesn’t mean it is meaningful in all circumstances. When people refer to it as the “not so fundamental theorem” they are not complaining about the theorem itself, but about the application of that theorem to a particular biological situation.

Koalas don’t know or care about Fisher’s Fundamental Theorem. However, just because you don’t know or care about something doesn’t mean it doesn’t apply to you. (http://en.wikipedia.org/wiki/Koala)

## Join the Conversation

1. Bill;

Great to hear that you read my drivel. I am just taking Fisher at his word. He explicitly states that the change in fitness is equal to the variance in fitness, so that is what I did. There is an implicit rate which is change per generation, but since that is not explicitly in the statement I left it out. The other thing is that I dislike that Fisher argues for “genetic variance”. In fact, since it is a mathematical truism, it works equally well for change within a generation. That is, consider that you have a set of individuals with differing probabilities of survival. Thus, the “differential fitness” is survivorship. Then, if we measure the number alive at some point in the future, and assign them a fitness of 1 if they are alive, and 0 if they are dead, and convert that to relative fitness we will find that the change in relative fitness in our experiment is equal to the phenotypic variance in relative fitness. Note genes are not involved, and the rate is determined by us — it is the time between when we started the experiment and everybody was alive, and the point where we ended the experiment, where some were dead. Obviously this is a silly experiment, but it shows that the “rate” is really set by us, and the basic equations make no statement about whether or not fitness is heritable.

Now that I think about if you could figure out how to do it there is no reason it wouldn’t work for a multiple generation period between the start of the experiment, and when the final fitnesses were calculated. Of course I have no idea how you would do that with anything other than a clonally reproducing organism, but again we see that the “rate” is of our choosing.

2. Hi Charles,
I too enjoy simplicity, but I think Fisher was thinking in much broader terms, the selection theory of everything. My question is: why do you leave out “rate” in your treatment? This is of course a model for rates of change in fitness, and so many treatments explicitly include this (like the one Conrad Istock used in his lectures). Cheers

Bill

3. Here are a few interesting comments on facebook that I want to preserve here:

Ryan Calsbeek: 2 questions: 1. What is the 1-squared for? and 2. Koalas?

My response:

Well, the 1-squared — I thought about leaving it out, but in calculating a variance you are supposed to subtract off the mean squared. Hence I was trying to be formally correct.

The Koalas? Saying FFT is about individuals instead of genes, or genes instead of individuals is silly and arbitrary, just like excluding koalas is silly and arbitrary. Plus Koalas are cute, and they have been in the news recently because of the fires in Australia.

Ken Spitze: Hi Charles – I enjoy your posts. Why do you want to delete “genetic” from FFT?

My response: For the simple reason that it works perfectly well if we apply it to individuals after selection, and before reproduction. In this case the trait need not be heritable, and it might not have any evolutionary repercussions, but you can still apply the equation.