陆小凤之幽灵山庄:Proof and Meaning of Price Equation

Proof and Meaning of Price Equation
20:56 2011/8/12
See also http://en.wikipedia.org/wiki/Price_equation
The Price equation (also known as Price's equation) is a covariance equation which is a mathematical description of evolution and natural selection. The Price equation was derived by George R. Price, working in London to rederive W.D. Hamilton's work on kin selection[].
covariance:In probability theory and statistics, covariance is a measure of how much two variables change together. Variance is a special case of the covariance when the two variables are identical.
Fitness (biology) and inclusive fitness:
Proof of the Price equation
To prove the Price equation, the following definitions are needed. If ni is the number of occurrences of a pair of real numbers xi and yi, then:The average or expected value of the xi values is:

The covariance between the xi and yi values is:

The notation  will also be used when convenient.
Suppose there is a population of organisms all of which have a genetic characteristic described by some real number. For example, high values of the number represent an increased visual acuity over some other organism with a lower value of the characteristic. Groups can be defined in the population which are characterized by having the same value of the characteristic. Let subscript i identify the group with characteristic zi and let ni be the number of organisms in that group. The total number of organisms is then n where:
The average value of the characteristic is z defined as:
Now suppose that the population reproduces, all parents are eliminated, and then there is a selection process on the children, by which less fit children are removed from the reproducing population. After reproduction and selection, the population numbers for the child groups will change to n′i. Primes will be used to denote child parameters, unprimed variables denote parent parameters.The total number of children is n' where:
The fitness of group i will be defined to be the ratio of children to parents:

with average fitness of the population being

The average value of the child characteristic will be z' where:

where z′i are the (possibly new) values of the characteristic in the child population. Equation (2) shows that:

Call the change in characteristic value from parent to child populations Δzi so that Δzi = z'i ? zi. As seen in Equation (1), the expected value operator is linear, so

Combining Equations (7) and (8) leads to

but from Equation (1) gives:
and from Equation (4) gives:

Applying Equations (5) and (6) to Equation (10) and then applying the result to Equation (9) gives the Price Equation:
Example 1: Evolution of sightAs an example of the simple Price equation, consider a model for the evolution of sight. Suppose zi is a real number measuring the visual acuity of an organism. An organism with a higher zi will have better sight than one with a lower value of zi. Let us say that the fitness of such an organism is wi=zi which means the more sighted it is, the fitter it is, that is, the more children it will produce. Beginning with the following description of a parent population composed of 3 types: (i = 0,1,2) with sightedness values zi = 3,2,1:i 0 1 2ni 10 20 30zi 3 2 1Using Equation (4), the child population (assuming the character zi doesn't change)i 0 1 2ni 30 40 30zi 3 2 1We would like to know how much average visual acuity has increased or decreased in the population. From Equation (3), the average sightedness of the parent population is z = 5/3. The average sightedness of the child population is z' = 2, so that the change in average sightedness is:
which indicates that the trait of sightedness is increasing in the population. Applying the Price equation we have (since z′i= zi):
2008年英国还拍了一部相关电影《要命法则：w delta z》