## Declaring war on the Fundamental Theorem of Natural Selection

If you read the theoretical and mathematical literature of evolutionary biology as much as I do, you’ll quickly notice that three-quarters of a century has been spent trying to verify a cryptic statement made by Ronald Fisher and declared the Fundamental Theorem of Natural Selection. The “theorem” states that when mutation and genetic drift are negligible (e.g. in a large population) the rate of increase of mean fitness is equal to the variance in fitness in the population. Most of the time in that case we can use mean fitness as a Lyapunov function to demonstrate asymptotic stability of equilibria.

Unfortunately, most of the work on the FTNS shows that it doesn’t apply in most interesting cases and Fisher’s original derivation had some serious problems. Many have concluded that Fisher’s motivation and conclusion were unclear. Despite that, I’ve recently read that the FTNS is comparable to Newton’s Second Law of Motion. I disagree. I remember using Newton’s Second Law to solve tons of problems, and I have *never* used the FTNS to solve a problem in evolutionary theory. Never.

On top of all this, we have a substitute: there happens to be an actual theorem whose proof is rigorous, applies to any set of aggregate quantities and includes all the evolutionary details. I have just rederived yet another set of famous evolutionary equations from The Price Equation in less than five minutes this morning. Try it sometime! You can derive everything, from the most basic equations of single-locus selection, to mutation-selection balance, and everything else, from the Price Equation. I use Price’s Equation the way I used the Fundamental Theorem of Calculus, and the Fundamental Theorem of Linear Programming. As a universal problem solver. I don’t use Fisher’s theorem that way.

Price’s Equation truly is the Fundamental Theorem of Evolution and I, for one, am going to make my efforts to reverse the typical ordering of things. Fisher’s “theorem” (it’s not even a theorem!) is called “Fundamental” and the selection component of Price’s Equation is called the “secondary theorem.” Dude, that’s bogus: you can derive Fisher’s FTNS from the Price Equation!

To see just how fundamental Price’s Equation is, use the full equation, with the transmission bias component to derive equations for

- Single-locus selection dynamics
- Two-locus-two-allele selection dynamics
- Equilibrium between forward and backward mutation
- The Breeder’s Equation

You’ll see what I mean.

I use newton’s second law of motion all the time. I used it this morning to flip my grilled cheese, just an example. Used it to connect Price’s equation with fluid dispersion equations favored by Kimura, also this morning while on a treadmill on the bus.

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Thanks for mentioning the diffusion approximation: it’s an approximation and Price’s equation is exact.

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