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A substantial positive curvature was discovered in offspring-parent regression for a bristle number in a laboratory population of Drosophila melanogaster. This initiated a theoretical investigation of the restrictions on the classical prediction equation in quantitative genetics. The main interest was in models of genetic and environmental variation which lead to asymmetrical responses in opposite directions in the first generation of selection, that is to non-linearity in offspring-parent regression. Non-linearity was assessed by fitting a quadratic regression in an infinite random mating population. A genetic model with a small number of loci, each with an arbitrary number of alleles, was used. The effects of dominance, multiplicative interaction and unequality of loci were studied. When there are no environmental deviations, apart from the case of complete dominance, single parent and mid-parent regression were found to show similar curvature, so that in general the offspring-parent regression between genotypic values has largest departures from linearity when there are rare, almost completely recessive alleles segregating at equal loci or in an analogous way directional recessivity and low averaged gene frequency over unequal loci. When the number of alleles is increased non-linearity decreases. For the number of loci making equal contributions to the variation the amount of non-linearity is roughly proportional to the number of loci. To study the effect of an additive independent environmental deviations various distributions for them were used. It was shown that when the offspring-parent regression between genotypic values is linear and H1 is the ratio of genotypic to phenotypic variance, the regression of offspring on parental phenotype is linear only if the skewness of the environmental distribution is a proportion JH'/(tH') of that of the genotypic distribution. The more the skewnesses depart from this equality or smaller H2 is, the larger the departures from linearity are. The genotypic non-linearity shows up only if H2 is very large. When the environmental deviations have a normal distribution, the largest departures from linearity are expected when there are rare and completely recessive alleles segregating at loci with large contributions. Models with dependence between genotypic and environmental distributions were also studied. Multiplicative interaction as such was shown to make only small contributions to nonlinearity, its effect being more substantial when there is a locus with a very large effect acting in a genetic background due to a very large number of loci with small effects. The use of Abplanalp's linear heritability estimates in checking the asymmetry of response were examined fitting a quadratic regression between sibs. Non-linearity in half-sib regression was found to be the same as in the regression of offspring on single parent. Dominance and common environmental effects were shown to cause biases in full-sib estimates. The effects of linkage on sib on sib regression were discussed when there is a large number of multiplicative loci contributing to the variation.

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The effects of spatial structure are explored using a simple model of migration between two populations, each in a balance between mutation and stabilising selection on an additive polygenic trait. Gene flow can maintain genetic variance within each population, albeit at low levels. If the optimum fluctuates in time, much higher levels of genetic variation can be maintained within populations, and in the presence of isolation divergence between the populations can be seen. Biotic interactions may be an important factor maintaining variation within and between populations, and these may lead to Red Queen dynamics even in constant environments. Here I define the Red Queen very broadly as "ongoing coevolution", and discuss the use of the terms 'coevolution' and 'ongoing'. I suggest that different Red Queen type processes can be classified the amount of novelty available in the system and by whether or not increasing sophistication occurs. In this thesis I model a specific biotic interaction: the coevolution of parasites with the acquired immunity of their hosts. High levels of linkage disequilibrium (strain structure) can be maintained in this system, and for a large range of parameters Red Queen type dynamics are observed. In some cases fluctuations in linkage disequilibrium and epistasis may result in a fairly large advantage to sexual over asexual recombination. However, this advantage is not large enough to outweigh the "two-fold cost of sex". This deterministic advantage arises primarily because recombination impedes the response to fluctuating epistasis rather than because it facilitates the response to directional selection. Sex may also be advantageous in the presence of Red Queen type dynamics because sexual genotypes that are stochastically lost can be recreated through recombination, unlike their asexual counterparts.

Continuous genetic improvement of growth and conformation traits in broiler populations has coincided with an increase in defects in heart and lung function. These defects have led to an increased incidence of metabolic disorders such as ascites (AS) (or pulmonary hypertension), a functional hypoxia. The incidence of AS in well-managed flocks is low, but it nonetheless causes important economic losses to the breeding industry and is an important issue from a welfare standpoint. The aim of this thesis was to study the genetics of AS-related quantitative traits. A low blood oxygen saturation (SaO) value is a good indicator of AS susceptibility. The existence of substantial genetic (polygenic) variation for SaO was demonstrated for four meat-type chicken lines. Estimates of heritabilities for SaO ranged from 0.1 to 0.2 and additive genetic correlations with production traits were not different from zero. SaO data from one of these lines were analysed using a mixed inheritance model (i.e. including a major locus (MG) and polygenes) and the results suggested that a MG with two alleles at intermediate frequencies affected SaO. The putative MG accounted for a difference of 13% SaO between homozygotes and the decreasing allele was recessive. The MG was also estimated to have an overdominant effect on weight and fleshing score. The mode of action of the putative MG on SaO and production traits would hinder manipulation of its allele frequency without the use of molecular markers. A population was designed to map this putative MG. Power studies were performed to select a number of sires and their half-sib progeny. Sires were selected on the basis of their probability of being heterozygous at the putative MG as estimated by the segregation analysis. Regions around the three ryanodine receptor loci (RYR1, RYR2 and RYR3), which are candidate genes for AS, were chosen to perform a linkage study. No evidence of linkage of any of the regions studied with SaO, as a predictor of AS, was detected.

This study covered two main areas in the methodology of Quantitative Trait Locus (QTL) detection with the use of genetic markers: i) the use of sib pair analysis for QTL detection in animal populations of large family size and ii) the use of selective genotyping in sib pair analysis for the reduction of genotyping costs. The regression of the squared phenotypic differences of sib pairs on the proportion of alleles they share Identical By Descent (IBD) at a marker locus can provide a test of linkage between a marker and a QTL. The t-values of the regression coefficient can provide a way to detect linkage. The regression coefficient can provide estimates of the QTL position and variance. The power of detection and parameter estimates (position and QTL variance) obtained from the analysis of progeny originating from populations with different family size, were compared using simulated data. Families of large size provide large numbers of sib pairs, since the number of sib pairs is related to the family size. The correlation of phenotypic differences between sib pairs of the same family has little effect on the test statistic and consequently on the power of detection. In order to decrease the amount of genotyping without any negative effects on the power of detection, a different approach to sib pair analysis was used. Instead of using the associations between phenotypic differences and proportion of alleles shared IBD at a marker locus for the detection of linkage, any changes in the proportion of alleles shared IBD at a marker locus in a selected sample of phenotypically discordant sib pairs could be used to identify linkage between the marker and a QTL. The t-values of the regression constant were used to test any deviations of the mean proportion of alleles shared IBD at a marker locus from the expected mean (0.5). It was found that this is the most powerful test statistic in samples of sib-pairs selected for high phenotypic differences.

In Chapter 2, a directed linkage scam for loci involved in body weight and carcass composition traits is performed in a commercial Charollais sheep population. Five chromosomes were investigated based on prior evidence for major genes effecting the studied traits in other breeds. A maximum likelihood variance component analysis using identity-by-descent values estimated by Markov chain Monte Carlo methods was performed on a complex pedigree containing a total of 570 sheep. Of the total of nine QTL detected, the estimated position of only one overlapped with the regions showing major genes that were used in chromosome selection. During the analysis of the Charollais sheep population, a region of the genome showing a significant deviation from the published sheep linkage map was detected. This region is examined in more detail in Chapter 3, with the addition of further microsatellite markers as well as the investigation of this region in two further sheep breeds. With the inclusion of the published linkage map, this demonstrated a total of three linkage maps across four populations. Such heterogeneity in linkage maps across sheep breeds has important consequences for the design and analysis fine-mapping studies. The significance of a QTL linkage peak is not readily evaluated with general pedigrees. The extension of permutation methodology that is commonly used with structured pedigrees to more general pedigrees is investigated in Chapter 4. Chapter 5 examines the population dynamics of a well studied wild Soay sheep population. A unified statistical framework is developed for all major aspects of the life cycle of the sheep. This forms the basis of a simulation model of the population that is used to predict the amount of linkage disequilibrium in the population (Chapter 6) and the effective population size of the population (Chapter 7). The examination of the linkage disequilibrium structure in a population is an important step in the design of studies with the aim of fine-mapping quantitative trait loci. The simulated population showed significant decline of linkage disequilibrium with genetic distance and low levels of background linkage disequilibrium, indicating that the Soay sheep population is a viable resource for linkage disequilibrium fine mapping. Through the use of the simulation model, the effective population size of the Soay sheep population was estimated to be approximately 0.17 of its census population size. This is approximately half the value obtained with the use of a general predictive equation.

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