The genotype and expected trait value
Each locus with phenotypic effects (within np ) has an equal and additive effect on the individuals expected trait value, E(z ). Two alleles are possible at each locus, with alleles having a value of either 0 or d , where d > 0. This additive effect size d is calculated as a function of the environmental variance (VE ), the trait heritability (h2 ), and is chosen such that the stated heritability is achieved at initialisation (givenVE , np , and the initial frequency of alleles with effect size d ,f0 ):
\begin{equation} h^{2}=\ \frac{V_{G}}{V_{G}+V_{E}}\nonumber \\ \end{equation}
or, equivalently,
\begin{equation} V_{G}=\ \frac{{h^{2}V}_{E}}{1-h^{2}}\nonumber \\ \end{equation}
With a binomial distribution, the expected genetic variance is then given as,
\begin{equation} V_{G}\ =\ 2d^{2}n_{p}f_{0}(1-f_{0})\nonumber \\ \end{equation}
Where f0 is the initial frequency of favourable alleles present (i.e. those with effect sizes of d ) at the start of the simulation. We vary this value of f0 in our sensitivity analysis. Our effect size, d can be calculated as:
\begin{equation} d=\sqrt{\frac{h^{2}V_{E}}{2n_{p}f_{0}(1-h^{2})(1-f_{0})}}\nonumber \\ \end{equation}
Each individual’s expected phenotype is given by:
\begin{equation} E\left(z_{i}\right)=d\sum_{j=i}^{n_{p}}{\sum_{k=1}^{2}a_{i,j,k}}\nonumber \\ \end{equation}
where aj,k references the allelic value k(either 0 or 1) of locus j . Here the individual is represented byi . In our reference case we set heritability of the trait in the recipient population (h2 ) to 0.1. We explore the impact of differing heritability values in the sensitivity analysis. We centre the mean phenotype such that maximum fitness is conferred at the start of the simulation.