## U.S. Department of Commerce

#### Date of this Version

2009

#### Citation

*Molecular Ecology* (2009) 18, 1834–1847; doi: 10.1111/j.1365-294X.2009.04157.x

#### Abstract

Evolutionary processes are routinely modeled using ‘ideal’ Wright–Fisher populations of constant size *N* in which each individual has an equal expectation of reproductive success. In a hypothetical ideal population, variance in reproductive success (*V*_{k}) is binomial and effective population size (*N _{e}*) =

*N*. However, in any actual implementation of the Wright– Fisher model (e.g., in a computer),

*V*is a random variable and it’s realized value in any given replicate generation (

_{k}*V*) only rarely equals the binomial variance. Realized effective size (

_{k}^{*}*N*) thus also varies randomly in modeled ideal populations, and the consequences of this have not been adequately explored in the literature. Analytical and numerical results show that random variation in

_{e}^{*}*V*and

_{k}^{*}*N*can seriously distort analyses that evaluate precision or otherwise depend on the assumption that is constant. We derive analytical expressions for Var(

_{e}^{*}*V*) [4(2

_{k}*N*– 1)(

*N*– 1)/

*N*

^{3}] and Var(

*N*) [

_{e}*N*(

*N*– 1)/(2

*N*– 1) ≈ N/2] in modeled ideal populations and show that, for a genetic metric

*G*=

*f*(

*N*), Var(

_{e}^{^}

*G*) has two components: Var

*(due to variance across replicate samples of genes, given a specific*

_{Gene}*N*) and Var

_{e}^{*}*(due to variance in*

_{Demo}*N*). Var(

_{e}^{*}^{^}

*G*) is higher than it would be with constant

*N*=

_{e}*N*, as implicitly assumed by many standard models. We illustrate this with empirical examples based on

*F*(standardized variance of allele frequency) and

*r*

^{2}(a measure of linkage disequilibrium). Results demonstrate that in computer models that track multilocus genotypes, methods of replication and data analysis can strongly affect consequences of variation in

*N*. These effects are more important when sampling error is small (large numbers of individuals, loci and alleles) and with relatively small populations (frequently modeled by those interested in conservation).

_{e}^{*}