A recursive method for computing expected kinship and inbreeding in complex and dynamic breeding programmes.

In many livestock breeding programmes, the development of inbreeding is of critical importance. Thus, the assessment of the expected development of inbreeding should be an essential element in the design of breeding programmes. We propose a new method to deterministically predict the rate of inbreeding based on the gene-flow method in well-defined complex and dynamic breeding programmes. In the suggested approach, a breeding programme has to be structured in homogeneous age-sex-groups, so called cohorts, with a defined origin of genes. Starting from an initial setup (usually an unrelated and non-inbred base population), transition rules to calculate the kinship within and between cohorts originating from reproduction or ageing, respectively, are defined. Using this approach recursively provides the expected development of kinship within and between all cohorts over time, which can be combined into average kinships for the whole population or defined subsets. From these quantities, relevant parameters like the inbreeding rate or the effective population size are easily derived. We illustrate the method with a simple static example breeding programme in sheep. Based on this reference breeding programme, we demonstrate the use of our approach for dynamic breeding programmes, in which cohort sizes or vectors of gene origin change over time: here, we model the situation of exponential population growth and a bottleneck situation, respectively. The suggested approach does not account for the effect of selection on the development of inbreeding, but ideas to overcome this limitation are discussed.