Fisher's Fundamental Theorem of Natural Selection

Suppose the phenotype whose evolution we're interested in following is fitness itself.¹² Then we can summarize the fitnesses as illustrated in Table 2.

Table 2: Fitnesses and additive fitness values used in deriving Fisher's Fundamental Theorem of Natural Selection.

Genotype	$A_1A_1$	$A_1A_2$	$A_2A_2$
Frequency	$p^2$	$2pq$	$q^2$
Fitness	$w_{11}$	$w_{12}$	$w_{22}$
Additive fitness value	$2\alpha_1$	$\alpha_1 + \alpha_2$	$2\alpha_2$

Although I didn't tell you this, a well-known fact about viability selection at one locus is that the change in allele frequency from one generation to the next can be written as

$$\Delta p = \left({{pq} \over 2{\bar w}}\right) \left({{d{\bar w}} \over {dp}}\right) \quad .$$

Using our new friend, Taylor's theorem, it follows immediately that

$${\bar w}' = {\bar w} + \left(\Delta p\right)\left({{d{\bar ... ...2}\right) \left({{d^2{\bar w}} \over {dp^2}}\right) \quad .$$

Or, equivalently

$$\Delta {\bar w} = \left(\Delta p\right)\left({{d{\bar w}} \... ...2}\right) \left({{d^2{\bar w}} \over {dp^2}}\right) \quad .$$

Recalling that ${\bar w} = p^2w_{11} + 2p(1-p)w_{12} + (1-p)^2w_{22}$ we find that

$$\begin{eqnarray*} \frac{d{\bar w}}{dp} &=& 2pw_{11} + 2(1-p)w_{12} - 2pw_{12}... .../2)] \\ &=& 2[\alpha_1 - \alpha_2] \\ &=& 2\alpha \quad , \end{eqnarray*}$$

where the last two steps use the definitions for $\alpha_1$ and $\alpha_2$, and we set $\alpha = \alpha_1 - \alpha_2$. Similarly,

$$\begin{eqnarray*} \frac{d^2{\bar w}}{dp^2} &=& 2w_{11} - 2w_{12} - 2w_{12} + 2w_{22} \\ &=& 2(w_{11} - 2w_{12} + w_{22}) \\ \end{eqnarray*}$$

Now we can plug these back into the equation for $\Delta\bar w$:

$$\begin{eqnarray*} \Delta {\bar w} &=& \left\{\left({{pq} \over {2{\bar w}}}\r... ...ver {2{\bar w}}}(w_{11} - 2w_{12} + w_{22})\right\} \quad , \end{eqnarray*}$$

where the last step follows from the observation that $V_a = 2pq\alpha^2$. The quantity ${{pq} \over {2{\bar w}}}(w_{11} - 2w_{12} + w_{22})$ is usually quite small, especially if selection is not too intense. So we are left with

$$\Delta {\bar w} \approx {V_a \over {\bar w}} \quad .$$