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# Introduction

So far we've studied only the evolution of a single trait, e.g., height or weight. But organisms have many traits, and they evolve at the same time. How can we understand their simultaneous evolution? The basic framework of the quantitative genetic approach was first outlined by Russ Lande and Steve Arnold [2].

Let , , ..., be the phenotype of each character that we are studying. We'll use to denote the vector of these characters before selection and to denote the vector after selection. The selection differential, , is also a vector given by

Suppose is the probability that any individual has phenotype , and let be the fitness (absolute viability) of an individual with phenotype . Then the mean absolute fitness is

The relative fitness of phenotype can be written as

Using relative fitnesses the mean relative fitness, , is 1. Now

Recall that . Consider

where the last step follows since meaning that . In short,

That should look familiar from our analysis of the evolution of a single phenotype.

If we assume that all genetic effects are additive, then the phenotype of an individual can be written as

where is the additive genotype and is the environmental effect. We'll denote by the matrix of genetic variances and covariances and by the matrix of environmental variances and covariances. The matrix of phenotype variances and covariances, , is then given by1

Now, if we're willing to assume that the regression of additive genetic effects on phenotype is linear2 and that the environmental variance is the same for every genotype, then we can predict how phenotypes will change from one generation to the next

is the multivariate version of . This equation is also the multivariate version of the breeders equation.

But we have already seen that . Thus,

is a set of partial regression coefficients of relative fitness on the characters, i.e., the dependence of relative fitness on that character alone holding all others constant.

Note:

is the total selective differential in character , including the indirect effects of selection on other characters.

Next: An example: selection in Up: Selection on multiple characters Previous: Selection on multiple characters
Kent Holsinger 2012-10-14