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We wish to derive an expression for the curvature of the adaptive landscape, H. Phillips and Arnold (1989) reported the multivariate result given in (6), but without derivation.
 

Summary: Appendix
We wish to derive an expression for the curvature of the adaptive landscape, H. Phillips
and Arnold (1989) reported the multivariate result given in (6), but without derivation.
The following derivation is for the univariate case (i.e., selection acts on a single
phenotypic trait). The first part follows Lande (1976). Let the mean value of a phenotypic
trait, z, before selection be
,)(= dzzzpz (A1)
where p(z) is the phenotypic distribution before selection. Let W(z) be the fitness of an
individual with phenotype z, so that the average fitness in the population is
.)()(= dzzWzpW (A2)
The individual selection function W(z) will shift the mean and variance P (=2
) of the
trait, so that after selection the trait mean is
,)()(
1
* = dzzWzzp
W
z (A3)
and the trait variance is
.)()()(

  

Source: Arnold, Stevan J. - Department of Zoology, Oregon State University

 

Collections: Biology and Medicine; Environmental Sciences and Ecology