The importance of carrying capacity,fertility, and viability for the competitive ability of populations

Summary A simple mathematical model is analyzed which describes the behaviour of a population under the influence of viability and fertility as well as carrying capacity and competition by another species. One locus with two alleles determining viability, fertility, and carrying capacity is considered. Throughout the analysis absolute rather than relative numbers are used. The following results have been obtained: 1. The species producing more zygotes wins the competition. 2. Absolute numbers must be used for viability and fertility in theoretical considerations, because the outcome of competition may be changed by using different absolute numbers of the same relation. 3. Differences in viability are amplified by limiting the number of individuals. 4. Differences in every parameter can be compensated by any other parameter.     

A symmetric two locus model with viability and fertility selection

A two locus deterministic population genetic model is analysed. One locus is under viability selection, the other under fertility selection with both forms of selection completely symmetric. It is shown that linkage equilibrium may occur at two different equilibrium points. For a two-locus polymorphism to be stable, it is necessary that the viability locus be overdominant but not necessary that the fertility locus, considered separately, be able to support a stable polymorphism. The overlaps in stability are not as complex as under two locus symmetric fertilities, but considerably more complex than with symmetric viabilities. Extensions of the analysis for the central linkage equilibrium point with multiple viability and fertility loci are indicated.Research supported in part by NIH grants GM 28106 and GM 10452     

Population genetics study of the differential fertility in urban populations

In an urban population with widespread birth control practice the distribution of the number of pregnancies, births and abortions was studied in a cohort of women of completed fertility. The mean number of pregnancies per woman was 4.03 +/- 0.08 (sigma = = 2.98); the mean number of births - 1.12 +/- 0.02 (sigma = 0.77). 7.4% of women which had completed their reproductive performance had no pregnancies and 19.5% - no births. The Crow's Index of the Opportunity for Selection and its components connected with differential fertility and differential mortality were estimated. In the population under study two components of selection - selection at the prenatal stages and selection associated with infertility - are shown to be still significant. Such type of selection is exemplified by investigation of couples suffering from repeated spontaneous abortions.     

Methods for predicting rates of inbreeding in selected populations

Summary In selected populations, families superior for the selected trait are likely to contribute more offspring to the next generation than inferior families and, as a consequence, the rate of inbreeding is likely to be higher in selected populations than in randomly mated populations of the same structure. Methods to predict rates of inbreeding in selected populations are discussed. The method of Burrows based on probabilities of coselection is reappraised in conjunction with the transition matrix method of Woolliams. The method of Latter based on variances and covariances of family size is also examined. These methods are one-generation approaches in the sense that they only account for selective advantage over a single generation, from parents to offspring. Two-generation methods are developed that account for selective advantage over two generations, from grandparent to grandoffspring as well as from parent to offspring. Predictions are compared to results from simulation. The best one-generation method was found to underpredict rates of inbreeding by 10–25%, and the two-generation methods were found to underpredict rates of inbreeding by 9–18%.     

Genetic evaluation methods for populations with dominance and inbreeding

Summary The effect of inbreeding on mean and genetic covariance matrix for a quantitative trait in a population with additive and dominance effects is shown. This genetic covariance matrix is a function of five relationship matrices and five genetic parameters describing the population. Elements of the relationship matrices are functions of Gillois (1964) identity coefficients for the four genes at a locus in two individuals. The equivalence of the path coefficient method (Jacquard 1966) and the tabular method (Smith and Mäki-Tanila 1990) to compute the covariance matrix of additive and dominance effects in a population with inbreeding is shown. The tabular method is modified to compute relationship matrices rather than the covariance matrix, which is trait dependent. Finally, approximate and exact Best Linear Unbiased Predictions (BLUP) of additive and dominance effects are compared using simulated data with inbreeding but no directional selection. The trait simulated was affected by 64 unlinked biallelic loci with equal effect and complete dominance. Simulated average inbreeding levels ranged from zero in generation one to 0.35 in generation five. The approximate method only accounted for the effect of inbreeding on mean and additive genetic covariance matrix, whereas the exact accounted for all of the changes in mean and genetic covariance matrix due to inbreeding. Approximate BLUP, which is computable for large populations where exact BLUP is not feasible, yielded unbiased predictions of additive and dominance effects in each generation with only slightly reduced accuracies relative to exact BLUP.     

A simulation study on detecting purging of inbreeding depression in captive populations

Inbreeding depression threatens the survival of small populations of both captive and wild outbreeding species. In order to fully understand this threat, it is necessary to investigate what role purging plays in reducing inbreeding depression. Ballou (1997) undertook such an investigation on 25 mammalian populations, using an ancestral inbreeding regression model to detect purging. He concluded that there was a small but highly significant trend of purging on neonatal survival across the populations. We tested the performance of the regression model that Ballou used to detect purging on independently simulated data. We found that the model has low statistical power when inbreeding depression is caused by the build-up of mildly deleterious alleles. It is therefore possible that Ballou's study may have underestimated the effects of ancestral inbreeding on the purging of inbreeding depression in captive populations if their inbreeding depression was caused mainly by mildly deleterious mutations. We also developed an alternative regression model to Ballou's, which showed an improvement in the detection of purging of mildly deleterious alleles but performed less well if deleterious alleles were of a large effect.     

Population viability analysis of the Danube sturgeon populations in a Vortex simulation model

Populations of six sturgeon species in the Danube River (beluga, Russian sturgeon, stellate sturgeon, sterlet, ship sturgeon and Atlantic sturgeon) have experienced severe decline during the last several decades, mostly due to the unsustainable fishery, river fragmentation and water pollution. Present lack of knowledge on basic sturgeon demography, life history and relative effects of different negative factors is further hindering implementation of efficient policy and management measures. In the present study, population viability analysis in a Vortex simulation model has been conducted in order to assess the state of the six Danube sturgeon species, their future risk of extinction and to determine the most suitable conservation and management measures. Population viability analysis has revealed a large sensitivity of the Danube sturgeon populations to changes in the natural mortality, fecundity, age at maturity and spawning frequency. It was also confirmed that the sturgeons are highly susceptible to even moderate levels of commercial fishery, and that their recovery is a multi-decadal affair. Stocking with adult individuals was shown to produce considerably greater effect on population persistence than stocking with juveniles, but the latter approach is probably still preferable since it avoids many inherent problems of aquaculture cultivation. This study represents the first population viability analysis of the Danube sturgeons.     

Predicting rates of inbreeding in populations undergoing selection

Tractable forms of predicting rates of inbreeding (DeltaF) in selected populations with general indices, nonrandom mating, and overlapping generations were developed, with the principal results assuming a period of equilibrium in the selection process. An existing theorem concerning the relationship between squared long-term genetic contributions and rates of inbreeding was extended to nonrandom mating and to overlapping generations. DeltaF was shown to be approximately (1)/(4)(1 - omega) times the expected sum of squared lifetime contributions, where omega is the deviation from Hardy-Weinberg proportions. This relationship cannot be used for prediction since it is based upon observed quantities. Therefore, the relationship was further developed to express DeltaF in terms of expected long-term contributions that are conditional on a set of selective advantages that relate the selection processes in two consecutive generations and are predictable quantities. With random mating, if selected family sizes are assumed to be independent Poisson variables then the expected long-term contribution could be substituted for the observed, providing (1)/(4) (since omega = 0) was increased to (1)/(2). Established theory was used to provide a correction term to account for deviations from the Poisson assumptions. The equations were successfully applied, using simple linear models, to the problem of predicting DeltaF with sib indices in discrete generations since previously published solutions had proved complex.     

Prediction of rates of inbreeding in selected populations

A method is presented for the prediction of rate of inbreeding for populations with discrete generations. The matrix of Wright's numerator relationships is partitioned into 'contribution' matrices which describe the contribution of the Mendelian sampling of genes of ancestors in a given generation to the relationship between individuals in later generations. These contributions stabilize with time and the value to which they stabilize is shown to be related to the asymptotic rate of inbreeding and therefore also the effective population size, Ne approximately 2N/(mu 2r + sigma 2r), where N is the number of individuals per generation and mu r and sigma 2r are the mean and variance of long-term relationships or long-term contributions. These stabilized values are then predicted using a recursive equation via the concept of selective advantage for populations with hierarchical mating structures undergoing mass selection. Account is taken of the change in genetic parameters as a consequence of selection and also the increasing 'competitiveness' of contemporaries as selection proceeds. Examples are given and predicted rates of inbreeding are compared to those calculated in simulations. For populations of 20 males and 20, 40, 100 or 200 females the rate of inbreeding was found to increase by as much as 75% over the rate of inbreeding in an unselected population depending on mating ratio, selection intensity and heritability of the selected trait. The prediction presented here estimated the rate of inbreeding usually within 5% of that calculated from simulation.     

Quantifying inbreeding in natural populations of hermaphroditic organisms

We review molecular methods for estimating selfing rates and inbreeding in populations. Two main approaches are available: the population structure approach (PSA) and progeny-array approach (PAA). The PSA approach relies on single-generation samples and produces estimates that integrate the inbreeding history over several generations, but is based on strong assumptions (for example, inbreeding equilibrium). The PSA has classically relied on single-locus inbreeding coefficients averaged over loci. Unfortunately PSA estimates are very sensitive to technical problems such as the occurrence of null alleles at one or more of the loci. Consequently inbreeding might be substantially overestimated, especially in outbred populations. However, the robustness of the PSA has recently been greatly improved by the development of multilocus methods free of such bias. The PAA, on the other hand, is based on the comparison between offspring and mother genotypes. As a consequence, PAA estimates do not reflect long-term inbreeding history but only recent mating events of the maternal individuals studied ('here and now' selfing). In addition to selfing rates, the PAA allows estimating other mating system parameters, including biparental inbreeding and the correlation of selfing among sibs. Although PAA estimates could also be biased by technical problems, incompatibilities between the mother's genotype and her offspring allow the identification and correction of such bias. For all methods, we provide guidelines on the required number of loci and sample sizes. We conclude that the PSA and PAA are equally robust, provided multilocus information is used. Although experimental constraints may make the PAA more demanding, especially in animals, the two methods provide complementary information, and can fruitfully be conducted together.     

Quantifying realized inbreeding in wild and captive animal populations

Most molecular measures of inbreeding do not measure inbreeding at the scale that is most relevant for understanding inbreeding depression—namely the proportion of the genome that is identical-by-descent (IBD). The inbreeding coefficient F_Ped obtained from pedigrees is a valuable estimator of IBD, but pedigrees are not always available, and cannot capture inbreeding loops that reach back in time further than the pedigree. We here propose a molecular approach to quantify the realized proportion of the genome that is IBD (propIBD), and we apply this method to a wild and a captive population of zebra finches (Taeniopygia guttata). In each of 948 wild and 1057 captive individuals we analyzed available single-nucleotide polymorphism (SNP) data (260 SNPs) spread over four different genomic regions in each population. This allowed us to determine whether any of these four regions was completely homozygous within an individual, which indicates IBD with high confidence. In the highly nomadic wild population, we did not find a single case of IBD, implying that inbreeding must be extremely rare (propIBD=0–0.00094, 95% CI). In the captive population, a five-generation pedigree strongly underestimated the average amount of realized inbreeding (F_Ped=0.013<propIBD=0.064), as expected given that pedigree founders were already related. We suggest that this SNP-based technique is generally useful for quantifying inbreeding at the individual or population level, and we show analytically that it can capture inbreeding loops that reach back up to a few hundred generations.     

Mathematical model for adaptive evolution of populations based on a complex domain

A mutation is ultimately essential for adaptive evolution in all populations. It arises all the time, but is mostly fixed by enzymes. Further, most do consider that the evolution mechanism is by a natural assortment of variations in organisms in line for random variations in their DNA, and the suggestions for this are overwhelming. The altering of the construction of a gene, causing a different form that may be communicated to succeeding generations, produced by the modification of single base units in DNA, or the deletion, insertion, or rearrangement of larger units of chromosomes or genes. This altering is called a mutation. In this paper, a mathematical model is introduced to this reality. The model describes the time and space for the evolution. The tool is based on a complex domain for the space. We show that the evolution is distributed with the hypergeometric function. The Boundedness of the evolution is imposed by utilizing the Koebe function.     

Effects of inbreeding on the genetic diversity of populations

The study of variability within species is important to all biologists who use genetic markers. Since the discovery of molecular variability among normal individuals, data have been collected from a wide range of organisms, and it is important to understand the major factors affecting diversity levels and patterns. Comparisons of inbreeding and outcrossing populations can contribute to this understanding, and therefore studying plant populations is important, because related species often have different breeding systems. DNA sequence data are now starting to become available from suitable plant and animal populations, to measure and compare variability levels and test predictions.     

Effects of inbreeding and other genetic components on equine fertility

The Finnish mating records of Standardbred trotters (SB; n = 33 679) and Finnhorses (FH; n = 32 731) were analysed to study the effect of the level of inbreeding on foaling rates and to estimate the heritability of foaling rate. A linear mixed model was assumed, with the outcome of the foaling (foal or no foal) as the trait of the study. A restricted maximum likelihood-based method was used to calculate the estimates of the variance components. Predictions of breeding values and estimates of fixed effects were also calculated. The average level of inbreeding was 9.9% in the SB and 3.6% in the FH. The average foaling rates were better in the SB (72.6%) than in the FH (66.3%), but within each breed intense inbreeding had a statistically significant negative effect on foaling rate (P < 0.05). Also, the mating type, the age and breeding type of the mare, and the age of the stallion had statistically significant effects on foaling rate (P < 0.001). The heritability of foaling rate was between 3.4% and 3.7% in SBs and between 5.5% and 9.8% in FHs, when the outcome of the foaling was considered to be a trait of the expected foal. With the same model, the estimates of maternal genetic effect were 4.7% for SBs and 3.2% for FHs, and the estimates of the permanent environmental effects of the stallion were between 1.3% and 1.7%. Avoiding matings with very high inbreeding coefficients would improve foaling rates. It would also be possible to devise a breeding program for better equine fertility, but because the heritability is low, improvement of environmental factors deserves special attention.     

Patterns of inbreeding depression and architecture of the load in subdivided populations

Inbreeding depression is a general phenomenon that is due mainly to recessive deleterious mutations, the so-called mutation load. It has been much studied theoretically. However, until very recently, population structure has not been taken into account, even though it can be an important factor in the evolution of populations. Population subdivision modifies the dynamics of deleterious mutations because the outcome of selection depends on processes both within populations (selection and drift) and between populations (migration). Here, we present a general model that permits us to gain insight into patterns of inbreeding depression, heterosis, and the load in subdivided populations. We show that they can be interpreted with reference to single-population theory, using an appropriate local effective population size that integrates the effects of drift, selection, and migration. We term this the "effective population size of selection" (NS(e)). For the infinite island model, for example, it is equal to NS(e) = N1 + m/hs, where N is the local population size, m the migration rate, and h and s the dominance and selection coefficients of deleterious mutation. Our results have implications for the estimation and interpretation of inbreeding depression in subdivided populations, especially regarding conservation issues. We also discuss the possible effects of migration and subdivision on the evolution of mating systems.     

Exact inbreeding coefficients in populations with overlapping generations

A theory is given that allows inbreeding coefficients to be calculated exactly for populations with overlapping generations. Emphasis is placed on providing equations well suited for computer iteration. Both monoecious and dioecious populations are considered and family size is not restricted to being Poisson. One-locus and two-locus inbreeding coefficients are evaluated, although the reader may omit the two-locus sections. The exact treatment is shown to be preferable to approximate treatments in that it applies to both early and late generations for all populations sizes. Inbreeding effective numbers found by the exact treatment are compared to various approximate numbers, and the approximate values are found to be generally very good.