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Modelling evolutionary processes in small populations: not as ideal as you think


  • This article is a US Government work and is in the public domain in the USA

Robin Waples, Fax: +1 206 860 3335; E-mail:


Evolutionary processes are routinely modelled 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 (Vk) is binomial and effective population size (Ne) = N. However, in any actual implementation of the Wright–Fisher model (e.g., in a computer), Vk is a random variable and its realized value in any given replicate generation (inline image) only rarely equals the binomial variance. Realized effective size (inline image) thus also varies randomly in modelled ideal populations, and the consequences of this have not been adequately explored in the literature. Analytical and numerical results show that random variation in inline image and inline image can seriously distort analyses that evaluate precision or otherwise depend on the assumption that inline image is constant. We derive analytical expressions for Var(Vk) [4(2N – 1)(N – 1)/N3] and Var(Ne) [N(N – 1)/(2N – 1) ≈ N/2] in modelled ideal populations and show that, for a genetic metric G = f(Ne), Var(Ĝ) has two components: VarGene (due to variance across replicate samples of genes, given a specific inline image) and VarDemo (due to variance in inline image). Var(Ĝ) is higher than it would be with constant Ne = N, as implicitly assumed by many standard models. We illustrate this with empirical examples based on F (standardized variance of allele frequency) and r2 (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 inline image. These effects are more important when sampling error is small (large numbers of individuals, loci and alleles) and with relatively small populations (frequently modelled by those interested in conservation).

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