| Summary: | Modern breeding programmes of livestock species have successfully led to increased genetic merit in traits of economic relevance through accurate and intense selection. However, concomitant increased levels of inbreeding have been also observed. Quadratic optimisation constitutes a general approach to the joint management of the rates of genetic gain (AG) and inbreeding (AF) in selected populations. The rate of inbreeding can be used as a measure of risk in the breeding programme. The method optimises the genetic contributions of selection candidates for maximising AG while restricting AF to a pre-defined value. The AF restriction is achieved by applying a quadratic constraint on the average co-ancestry of selection candidates weighted by their projected use. The general objectives of this thesis were: i) to implement and evaluate the potential benefits of quadratic optimisation in real livestock populations; ii) to develop a deterministic framework for predicting AG under constrained AF and iii) to evaluate the benefits of quadratic optimisation in multiple trait scenarios under mixed inheritance models. The application of quadratic optimisation in two populations of beef cattle (Aberdeen Angus) and sheep (Meatlinc) led to important increases in the expected AG. At the observed AF in each population, increments per year in AG of 17% for Meatlinc and 30% for Aberdeen Angus were found in comparison to the AG expected from conventional truncation BLUP selection. More relaxed constraints on A F allowed even higher increases in expected AG in both populations. Stochastic simulations have revealed that under quadratic optimisation the selective advantage of the candidates for selection is primarily their Mendelian sampling terms rather than their breeding values as under truncation selection. Thus, under quadratic optimisation, the contribution of candidates to the future genetic pool is decided upon the best information on their unique superiority or inferiority with respect to the parental mean. A self-contained and accurate deterministic approach for predicting AG for pre-defined AF has been developed. It requires only specification of the trait heritability, the number of selection candidates and the target AF. Benefits from quadratic optimisation were also evaluated in a two-trait scenario where the trait with lower heritability was affected by an identified quantitative trait loci (QTL). Extra gains in the breeding goal were observed throughout the whole selection process from the combined use of both optimised contributions and QTL information. In contrast, this scheme was not the most effective for improving each of the traits in the breeding objective. The design and operational tools developed in this thesis constitute a general framework for the evaluation and realisation of the benefits from quadratic optimisation tools in practical livestock breeding programmes.
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