The infinitesimal model is based on the assumption that, conditional on the pedigree, the joint distribution of trait values is multivariate normal, then, selecting parents does not alter the variance amongst offspring. We explain how the infinitesimal model extends to include dominance as well as epistasis. Then, the evolution of a population depends on just a few quantities, which define the components of genetic variance and the inbreeding depression. In practice, the main difficulty in applying the infinitesimal model in the presence of dominance is that one must calculate the probabilities of identity by descent amongst up to four genes, which means that very many identity coefficients must be traced. We show how these coefficients can be calculated and approximated, allowing the infinitesimal model to be applied to help understand the evolutionary consequences of inbreeding depression.[-]

In the infinitesimal model, one or several quantitative traits are described as the sum of a genetic and a non-genetic component, the first being distributed within families as a normal random variable centred at the average of the parental genetic components, and with a variance independent of the parental traits. The idea behind the normal distribution of the genetic component is that the genetic part of the trait of interest is the sum of the ‘infinitesimal' contributions of the allelic states at a very large number of loci. This model has been widely used in quantitative genetics, but less so in evolutionary biology and the precise conditionsunder which it holds has remained rather vague. In this talk, we shall provide a mathematical justification of the model as the limit as the number M of loci tends to infinity of a model with Mendelian inheritance, which includes different evolutionary processes (genetic drift, recombination, selection, mutation, population structure, ...). Generalisations of the simple version of the infinitesimal model presented here, as well as some applications, will be presented in the following talks by Nick Barton and Alison Etheridge.[-]