Background: In the current context of climate change, livestock production faces many challenges to improve the sustainability of systems. Dairy farming, in particular, must find ways to select animals that will be able to achieve sufficient overall production while maintaining their reproductive ability in environments with increasing temperatures. With future forecasted climate conditions in mind, this study used data from Holstein and Montbeliarde dairy cattle to: (1) estimate the genetic-by-temperature-humidity index (THI) interactions for female fertility, and (2) evaluate the production-fertility trade-off with increasing values of THI. Results: Two-trait random regression models were fitted for conception rate (fertility) and test-day protein yield (production). For fertility, genetic correlations between different THI values were generally above 0.75, suggesting weak genotype-by-THI interactions for conception rate in both breeds. However, the genetic correlations between the conception rate breeding values at the current average THI (THI = 50, corresponding to a 24-h average temperature of 8 °C at 50% relative humidity) and their slopes (i.e., potential reranking) for heat stress scenarios (THI > 70), were different for each breed. For Montbeliarde, this correlation tended to be positive (i.e., overall the best reproducers are less affected by heat stress), whereas for Holstein it was approximately zero. Finally, our results indicated a weak antagonism between production and fertility, although for Montbeliarde this antagonism intensified with increasing THI. Conclusions: Within the range of weather conditions studied, increasing temperatures are not expected to exacerbate the fertility-production trade-off. However, our results indicated that the animals with the best breeding values for production today will be the most affected by temperature increases, both in terms of fertility and production. Nonetheless, these animals should remain among the most productive ones during heat waves. For Montbeliarde, the current selection program for fertility seems to be adequate for ensuring the adaptation of fertility traits to temperature increases, without adverse effects on production. Such a conclusion cannot be drawn for Holstein. In the future, the incorporation of a heat tolerance index into dairy cattle breeding programs would be valuable to promote the selection of animals adapted to future climate conditions.

Genomic models that link phenotypes to dense genotype information are increasingly being used for infering variance parameters in genetics studies. The variance parameters of these models can be inferred using restricted maximum likelihood, which produces consistent, asymptotically normal estimates of variance components under the true model. These properties are not guaranteed to hold when the covariance structure of the data specified by the genomic model differs substantially from the covariance structure specified by the true model, and in this case, the likelihood of the model is said to be misspecified. If the covariance structure specified by the genomic model provides a poor description of that specified by the true model, the likelihood misspecification may lead to incorrect inferences. This work provides a theoretical analysis of the genomic models based on splitting the misspecified likelihood equations into components, which isolate those that contribute to incorrect inferences, providing an informative measure, defined as $$\varvec{\kappa }$$ κ , to compare the covariance structure of the data specified by the genomic and the true models. This comparison of the covariance structures allows us to determine whether or not bias in the variance components estimates is expected to occur. The theory presented can be used to provide an explanation for the success of a number of recently reported approaches that are suggested to remove sources of bias of heritability estimates. Furthermore, however complex is the quantification of this bias, we can determine that, in genomic models that consider a single genomic component to estimate heritability (assuming SNP effects are all i.i.d.), the bias of the estimator tends to be downward, when it exists.

Genomic models that incorporate dense marker information have been widely used for predicting genomic breeding values since they were first introduced, and it is known that the relationship between individuals in the reference population and selection candidates affects the prediction accuracy. When genomic evaluation is performed over generations of the same population, prediction accuracy is expected to decay if the reference population is not updated. Therefore, the reference population must be updated in each generation, but little is known about the optimal way to do it. This study presents an empirical assessment of the prediction accuracy of genomic breeding values of production traits, across five generations in two Korean pig breeds. We verified the decay in prediction accuracy over time when the reference population was not updated. Additionally we compared the prediction accuracy using only the previous generation as the reference population, as opposed to using all previous generations as the reference population. Overall, the results suggested that, although there is a clear need to continuously update the reference population, it may not be necessary to keep all ancestral genotypes. Finally, comprehending how the accuracy of genomic prediction evolves over generations within a population adds relevant information to improve the performance of genomic selection.

Background: Genetic merit, or breeding values as referred to in livestock and crop breeding programs, is one of the keys to the successful selection of animals in commercial farming systems. The developments in statistical methods during the twentieth century and single nucleotide polymorphism (SNP) chip technologies in the twenty-first century have revolutionized agricultural production, by allowing highly accurate predictions of breeding values for selection candidates at a very early age. Nonetheless, for many breeding populations, realized accuracies of predicted breeding values (PBV) remain below the theoretical maximum, even when the reference population is sufficiently large, and SNPs included in the model are in sufficient linkage disequilibrium (LD) with the quantitative trait locus (QTL). This is particularly noticeable over generations, as we observe the so-called erosion of the effects of SNPs due to recombinations, accompanied by the erosion of the accuracy of prediction. While accurately quantifying the erosion at the individual SNP level is a difficult and unresolved task, quantifying the erosion of the accuracy of prediction is a more tractable problem. In this paper, we describe a method that uses the relationship between reference and target populations to calculate expected values for the accuracies of predicted breeding values for non-phenotyped individuals accounting for erosion. The accuracy of the expected values was evaluated through simulations, and a further evaluation was performed on real data. Results: Using simulations, we empirically confirmed that our expected values for the accuracy of PBV accounting for erosion were able to correctly determine the prediction accuracy of breeding values for non-phenotyped individuals. When comparing the expected to the realized accuracies of PBV with real data, only one out of the four traits evaluated presented accuracies that were significantly higher than the expected, approaching $$\sqrt{{{\text{h}}}^{2}}$$ h 2 . Conclusions: We defined an index of genetic correlation between reference and target populations, which summarizes the expected overall erosion due to differences in allele frequencies and LD patterns between populations. We used this correlation along with a trait’s heritability to derive expected values for the accuracy ( $${\text{R}}$$ R ) of PBV accounting for the erosion, and demonstrated that our derived $${\text{E}}\left[{\text{R}}|{\text{erosion}}\right]$$ E R | erosion is a reliable metric.

Genomic prediction (GP) aims to predict the breeding values of multiple complex traits, usually assumed to be multivariate normally distributed by the largely used statistical methods, thus imposing linear genetic relationships between traits. Although these methods are valuable for GP they do not account for potential nonlinear genetic relationships between traits in scenarios. For individual traits, this oversight may minimally affect prediction accuracy, but it can limit genetic progress when selection involves multiple traits. Deep learning (DL) offers a promising alternative for capturing nonlinear genetic relationships due to its ability to identify complex patterns without prior assumptions about the data structure. We proposed a novel hybrid DLGBLUP model which uses the output of the traditional GBLUP, and enhances its predicted genetic values (PGV) by accounting for nonlinear genetic relationships between traits using DL. We simulated data with linear and nonlinear genetic relationships between traits in order to verify whether DLGBLUP was able to identify nonlinearity when present and avoid inducing it when absent. We found that DLGBLUP consistently provided more accurate PGV for traits simulated with strong nonlinear genetic relationships, accurately identifying these relationships. Over 7 generations of selection, a greater genetic progress was achieved with PGV that accounted for nonlinear relationships (DLGBLUP), compared with GBLUP. When applied to a real dataset from the French Holstein dairy cattle population, DLGBLUP detected nonlinear genetic relationships between pairs of traits, such as conception rate and protein content, and somatic cell count and fat yield, although, no significant increase in prediction accuracy was observed. The integration of DL into GP enabled the modeling of nonlinear genetic relationships between traits, a possibility not previously discussed, given the linear nature of GBLUP. The detection of nonlinear genetic relationships between traits in the French Holstein population when using DLGBLUP indicates the presence of such relationships in real breeding data, suggesting that it may be relevant to further explore nonlinear relationships. This possibility of nonlinear genetic relationships between traits offers a different perspective into multitrait evaluations, with potential to further improve selection strategies in commercial livestock breeding programs. This is particularly relevant when integrating new traits into multitrait evaluations or incorporating new subpopulations, which may introduce different forms of nonlinearity. Finally, it is shown that DL can be used as a complement to the statistical methods deployed in routine genetic evaluations, rather than as an alternative, by enhancing their performance.

It is widely known that the environment influences phenotypic expression and that its effects must be accounted for in genetic evaluation programs. The most used method to account for environmental effects is to add herd and contemporary group to the model. Although generally informative, the herd effect treats different farms as independent units. However, if two farms are located physically close to each other, they potentially share correlated environmental factors. We introduce a method to model herd effects that uses the physical distances between farms based on the Global Positioning System (GPS) coordinates as a proxy for the correlation matrix of these effects that aims to account for similarities and differences between farms due to environmental factors. A population of Hanwoo Korean cattle was used to evaluate the impact of modelling herd effects as correlated, in comparison to assuming the farms as completely independent units, on the variance components and genomic prediction. The main result was an increase in the reliabilities of the predicted genomic breeding values compared to reliabilities obtained with traditional models (across four traits evaluated, reliabilities of prediction presented increases that ranged from 0.05 ± 0.01 to 0.33 ± 0.03), suggesting that these models may overestimate heritabilities. Although little to no significant gain was obtained in phenotypic prediction, the increased reliability of the predicted genomic breeding values is of practical relevance for genetic evaluation programs.