The Price equation dictates the evolution of the mean of a trait.

Given a trait $z$ in a population, the fitness $w$ and the variation between individuals and their offspring $δz$ :

Δ z = \frac{1}{w} co v (w_{i}, z_{i}) + \frac{1}{w} E (w_{i} δ z_{i})

The term $\frac{1}{w} co v (z_{i}, w_{i})$ encompasses Natural selection and Genetic drift, and the term $\frac{1}{w} E (w_{i} δ z_{i})$ encompasses Mutation and recombination.

The strength of this equation comes from the fact it is very general and makes very few hypothesis.

Proof

Let’s take a population that produces a new generation. Each individual $i$ has a value $z_{i}$ for the trait. Each parent $i$ has a number $w_{i}$ of children. Each child $ij$ inherits the trait $z_{i}$ from its parent $i$ with a small variation $δ z_{ij}$ .

Then the average trait of the new generation is:

z^{'} = \frac{i = 1 \sum N j = 1 \sum w _{i} z _{i} + δ z _{ij}}{i = 1 \sum N w _{i}}

z^{'} = \frac{i = 1 \sum N w _{i} z _{i} + i = 1 \sum N j = 1 \sum w _{i} δ z _{ij}}{i = 1 \sum N w _{i}}

z^{'} = \frac{i = 1 \sum N w _{i} z _{i} + i = 1 \sum N j = 1 \sum w _{i} δ z _{ij}}{Nw}

z^{'} = \frac{i = 1 \sum N w _{i} z _{i} + i = 1 \sum N w _{i} E ( δ z _{i} )}{Nw}

w z^{'} = \frac{i = 1 \sum N w _{i} z _{i} + i = 1 \sum N w _{i} E ( δ z _{i} )}{N}

w z^{'} = E (w_{i} z_{i}) + E (w_{i} δ z_{i})

Now by definition $Δ z = z^{'} - z$ , so:

w Δ z = E (w_{i} z_{i}) - w z + E (w_{i} δ z_{i})

w Δ z = E (w_{i} z_{i}) - E (w_{i}) E (z_{i}) + E (w_{i} δ z_{i})

w Δ z = co v (w_{i}, z_{i}) + E (w_{i} δ z_{i})

Which gives the Price equation:

Δ z = \frac{1}{w} co v (w_{i}, z_{i}) + \frac{1}{w} E (w_{i} δ z_{i})

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Price equation

Proof

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