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Abstract The development of inbreeding in rotation breeding schemes, sequentially using artificial insemination (AI) sires over generations, was investigated for a full AI scheme. Asymptotic prediction formulae of inbreeding coefficients were established when the first rotation list of AI sires (possibly related) was in use. Simulated annealing provided the optimal rotation order of sires within this list, when the sires were related. These methods were also used for subsequent rotation lists, needed by the exhaustion of semen stores for the first bulls. Simulation was carried out starting with groups of independent sires, with different sizes. To generate a yearly inbreeding rate substantially lower than 0.05% (considered to be within reach by conventional conservation schemes using frequent replacements), the results obtained showed that the number of sires should be at least 10–15 and that the same sires should be used during at least 50 years. The ultimate objective was to examine the relevance of implementing rotation in breeding schemes on the actual rare French cattle breeds under conservation. The best candidate for such a test was the Villard-de-Lans breed (27 bulls and 73 000 doses for only 340 females) and it turned out to be the best performer with an inbreeding coefficient of only 7.4% after 500 years and five different sire lists. Due to the strong requirements on semen stores and on the stability of population size, actual implementation of this kind of conservation scheme was recommended only in special ('niche') cattle populations.

Abstract Best linear unbiased prediction of genetic merits for a marked quantitative trait locus (QTL) using mixed model methodology includes the inverse of conditional gametic relationship matrix (G-1) for a marked QTL. When accounting for inbreeding, the conditional gametic relationships between two parents of individuals for a marked QTL are necessary to build G-1 directly. Up to now, the tabular method and its adaptations have been used to compute these relationships. In the present paper, an indirect method was implemented at the gametic level to compute these few relationships. Simulation results showed that the indirect method can perform faster with significantly less storage requirements than adaptation of the tabular method. The efficiency of the indirect method was mainly due to the use of the sparseness of G-1. The indirect method can also be applied to construct an approximate G-1 for populations with incomplete marker data, providing approximate probabilities of descent for QTL alleles for individuals with incomplete marker data.

Abstract According to the general approach developed in this paper, dynamic management of genetic variability in selected populations of dairy cattle is carried out for three simultaneous purposes: procreation of young bulls to be further progeny-tested, use of service bulls already selected and approval of recently progeny-tested bulls for use. At each step, the objective is to minimize the average pairwise relationship coefficient in the future population born from programmed matings and the existing population. As a common constraint, the average estimated breeding value of the new population, for a selection goal including many important traits, is set to a desired value. For the procreation of young bulls, breeding costs are additionally constrained. Optimization is fully analytical and directly considers matings. Corresponding algorithms are presented in detail. The efficiency of these procedures was tested on the current Norman population. Comparisons between optimized and real matings, clearly showed that optimization would have saved substantial genetic variability without reducing short-term genetic gains.

Abstract A method was described for calculating population statistics on relationship coefficients without using corresponding individual data. It relied on the structure of the inverse of the numerator relationship matrix between individuals under investigation and ancestors. Computation times were observed on simulated populations and were compared to those incurred with a conventional direct approach. The indirect approach turned out to be very efficient for multiplying the relationship matrix corresponding to planned matings (full design) by any vector. Efficiency was generally still good or very good for calculating statistics on these simulated populations. An extreme implementation of the method is the calculation of inbreeding coefficients themselves. Relative performances of the indirect method were good except when many full-sibs during many generations existed in the population.

Abstract Multivariate BLUPs can be derived when data are a mixture of continuous traits and observed discrete traits controlled by logistic latent traits. Algorithms were developed for predicting discrete responses to BLUP selection, and latent responses when the selection process included additional culling on scores. These algorithms were Taylor expansions using well-known expressions such as the probabilities and the two first moments of the truncated multinormal distribution, after appropriate re-parametrizations. They were compared to very accurate quadrature integrations. The test examples were suggested by a situation found in chickens where selection can involve body weight and leg deformity described by two logistic latent variates. Quadratic Taylor expansions generally provided a good accuracy. Therefore, they could be recommended when quadrature methods are too demanding, e.g., for complex breeding schemes.

Abstract Background Current research on quantitative genetics has provided efficient guidelines for the sustainable management of selected populations: genetic gain is maximized while the loss of genetic diversity is maintained at a reasonable rate. However, actual selection schemes are complex, especially for large domestic species, and they have to take into account many operational constraints. This paper deals with the actual selection of dairy goats where the challenge is to optimize diffusion of buck semen on the field. Three objectives are considered simultaneously: i) natural service buck replacement (NSR); ii) goat replacement (GR); iii) semen distribution of young bucks to be progeny-tested. An appropriate optimization method is developed, which involves five analytical steps. Solutions are obtained by simulated annealing and the corresponding algorithms are presented in detail. Results The whole procedure was tested on two French goat populations (Alpine and Saanen breeds) and the results presented in the abstract were based on the average of the two breeds. The procedure induced an immediate acceleration of genetic gain in comparison with the current annual genetic gain (0.15 genetic standard deviation unit), as shown by two facts. First, the genetic level of replacement natural service (NS) bucks was predicted, 1.5 years ahead at the moment of reproduction, to be equivalent to that of the progeny-tested bucks in service, born from the current breeding scheme. Second, the genetic level of replacement goats was much higher than that of their dams (0.86 unit), which represented 6 years of selection, although dams were only 3 years older than their replacement daughters. This improved genetic gain could be achieved while decreasing inbreeding coefficients substantially. Inbreeding coefficients (%) of NS bucks was lower than that of the progeny-tested bucks (-0.17). Goats were also less inbred than their dams (-0.67). Conclusions It was possible to account for complex operational constraints while developing goat selection schemes, both efficient and sustainable. Therefore, the recommended selection and mating decisions might receive attention from goat breeders using both AI and NS.

Selection in dairy cattle populations usually takes into account both the breed profiles for many traits and their overall estimated breeding values (EBV). This can result in effective contributions of breeding animals departing substantially from contributions optimised for saving future genetic variability. In this work, we propose a mating method that considers not only inbreeding but also the detailed EBV of progeny or the EBV of sires in reference to acceptance thresholds. Penalties were defined for inbreeding and for inadequate EBV profiles. Relative reductions of penalties yielded by any mating design were expressed on a scale ranging from 0 to 1. A value of 0 represented the average performance of random matings and a value of 1 represented the maximal reduction allowed by a specialized, single-penalty, mating design. The core of the method was an adaptative simulated annealing, where the maximized function was the average of both ratios, under the constraints that both relative penalty reductions should be equal and that the within-herd concentration criterion should be equal to a predefined reasonable value. The method was tested on two French dairy cattle populations originating from the same AI organization. The optimised mating design allowed substantial reductions of penalty: 70% and 64% for the Holstein and the Normandy populations, respectively. Thus, this mating method decreased inbreeding and met various demands from breeders.