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

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 (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(k) is a random variable and its realized value in any given replicate generation (V(k)*) only rarely equals the binomial variance. Realized effective size (N(e)*) 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 V(k)* and N(e)* can seriously distort analyses that evaluate precision or otherwise depend on the assumption that N(e)* is constant. We derive analytical expressions for Var(V(k)) 4(2N - 1)(N - 1)/N(3) and Var(N(e)) N(N - 1)/(2N - 1) approximately N/2 in modelled ideal populations and show that, for a genetic metric G = f(N(e)), Var(G) has two components: Var(Gene) (due to variance across replicate samples of genes, given a specific N(e)*) and Var(Demo) (due to variance in N(e)*). Var(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(e)*. 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).