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This Dissertation deals with statistical inference on the mutation rates α₁ and α₂ of a population genetic model introduced by Moran [Proc. Camb. Phil. Soc. 54 (1958), pp. 60-71]. The deductive theory by approximate methods of such models has reached an advanced stage but little has been done along the line of statistical inference. Moran's model is a model of the Markov chain type. It was selected for investigation because it is the only finite population genetic model for which the cteductive theory by exact methods is well enough established to stimulate an investigation of statistical inference. The first broad area of discussion of this dissertation deals with the simultaneous consideration of the mutation rates α₁ and α₂. Maximum likelihood estimates for α₁ and α₂ are obtained iteratively from the Newton-Raphson scheme for simultaneous solution of two equations in two unknowns. Several theorems are given which ensure that the log likelihood function involving α₁ and α₂ has a unique maximum in the parameter space of useful values. The transition matrix consists of conditional probability elements involving the unknown parameters α₁ and α₂. These elements are the probability of a transition from one state to another in at most unit steps. The eigenvalue expression along with the corresponding pre- and post-eigenvector matrices are given. The post-eigenvector matrix has elements consisting of Hahn polynomials. The pre-eigenvector matrix is obtained by inverting the post-eigenvector matrix for which an expression is given. The Hahn polynomials form a family of orthogonal polynomials. They were introduced by Hahn [Math. Nach. 2 (1949), pp. 4-34], and further discussed by Karlin and McGregor [Scripta Math, 26 (1961), pp. 33-46]. These polynomials form the foundation and are basic to many of the results of the dissertation. The expression for the expected value of the number of transitions from one state to another is given and this expression is also in terms of Hahn polynomials. Finally for this positively regular transition matrix involving both of the mutation rates α₁ and α₂, asymptotic multivariate normality of the maximum likelihood estimates α₁, α₂ is discussed along with hypothesis testing. Also discussed are large sample approximations., methods of designing and conducting experiments and replicated experiments. The second broad area of this dissertation deals with an absorbing Markov chain. That is, α₂ is set equal to zero and investigation on α₁ only is carried out. For this case the above transition matrix becomes an absorbing one (regular) and inferences are obtained from realizations on this absorbing chain whose peculiarities provide some unique difficulties. The eigenvalue expression with the corresponding post-eigenvector matrix whose elements are also Hahn polynomials and the expression (in terms of Hahn polynomials) for the expected number of transitions from one state to another are all given. Of particular interest are several postulated theorems on the maximum likelihood estimate α₁ of the mutation rate α₁ of the absorbing Markov chain in which an attempt is made at establishing the properties and normality of α₁. The estimate is again obtained iteratively. An outline of the proofs of the postulated theorems is presented. Gaps in the proof are a result of unresolved questions in positive regular Markov chain theory. In connection with the above theory and postulated theorems a simulation study on the IBM 650 was undertaken. This study substantiated many of the assumptions of the postulated theorems. The study, however, was not extensive enough to be conclusive. A further study is proposed. Replicated experiments are also discussed. Of particular interest here is a geometric type stopping rule in which the negative binomial is employed. Methods of conducting and designing experiments are discussed. An appendix discusses the Hahn polynomial system along with many of its important properties. / Ph. D.

This paper is divided into two parts. The first part is devoted to a study of absorption in a genetic population model and the second part to introducing a new method of matrix inversion. The author begins Part I by presenting the haploid population model under consideration, a constant-size population in which mutation is absent. Using the approach or Malecot, the problem is formulated in terms of an absorbing Markov chain. A brief review or earlier work by Maleoot, Fisher, Wright, and Feller is given before proceeding to the study of the distribution or time taken for the population to consist of only one gene type. The first method used by the author to determine the probability function for first passage time requires the eigenvectors or the transition matrix. A blocking transformation is used to show that only the eigenvectors corresponding to the even eigenvalues are needed to find the distributions. It is also shown that these eigenvectors are symmetric. In order to simplify computation of the needed vectors, a transformation which triangularizes the transition matrix is presented. This transformation also leads to a simple derivation of the eigenvalues and is used to derive the distributions for populations of size two through nine. Although the general solution is not obtained, expressions for the first seven eigenvectors are listed along with general results concerning the triangularized form. Next, the author attacks the problem by developing the theory or moments or the distribution and applies this theory to determine the means and variances. These are tabulated for population sizes 2(1)9 and 10(10)50. Then, by assuming a large population size, a diffusion process continuous in space and time is used to approximate the Markov chain. Expressions for the mean and variance are derived and are tabulated for the same values as in the moment approach. A comparison of the entries of these two tables gives support to the theory that the diffusion process works as the population size becomes infinite but with the gene frequency ratio kept fixed. Also, it demonstrates that diffusion theory works better for a gene frequency ratio near one-half than for values near zero or one and further that this theory consistently over-estimates the absorption time moments. The author notes, however, that the point of primary importance is the indication that diffusion theory works even with fixed gene number, that is, even in the tails of the distribution. He also states that the percentage error decreases at a faster rate for fixed gene frequency ratio than for fixed gene number and concludes the first part by suggesting a rate of decrease of percentage error for constant gene frequency. In Part II, the author begins by noting the importance of the inverse matrix in Markov chain theory. The new inversion technique given is basically one which generates the inverse with a minimum number of determinantal operations. The method is thoroughly described and it is applied to some problems encountered in the first part of the paper. The paper ends with a comparison of the new method with well-known techniques presently in use which shows that the method proposed is especially efficient for small order matrices. / Ph. D.

Paronychia virginica Spreng. (Caryophyllaceae) is a perennial evergreen herb of exposed, relatively xeric habitats. Approximately 10 mid-Appalachian populations remain in Virginia, West Virginia, and Maryland and are disjunct from populations located primarily in Texas, Oklahoma, and Arkansas. A study was conducted to test the hypothesis that eastern and western populations differ significantly and, therefore, represent at least two distinct taxa. Statistical analyses of 8 qualitative and 24 quantitative morphological characters indicated very highly significant (P < 0.001) variation between eastern and western populations of P. virginica. Characters differing most significantly included sepal pubescence, awn length, awn pubescence, awn curvature, length-width ratio of leaves, and shape of leaf apices. Starch gel electrophoresis was performed and six enzyme systems/nine loci (EST-2, EST-3, LAP, MDH-1, MDH-2, PGI, PGM-1, PGM-2, and SKDH) were identified as being consistently scorable and informative. Although gene flow between populations of P. virginica was shown to be restricted (mean FST = 0.353), populations are maintaining relatively high levels of genetic diversity. Genetic variability was quantified for each population and mean values for number of alleles per locus (A), percent loci polymorphic (P), and expected heterozygosity (HEXP) were found to be 1.95, 47.22%, and 0.204, respectively, exceeding those values reported for seed plants, widespread species, and endemic species. Hierarchical F statistics suggest higher levels of genetic variability within individual populations than among populations, regardless of geographic location. All statistically significant (P < 0.05) deviations from Hardy-Weinberg equilibrium indicated a deficiency in heterozygotes at the respective loci. Considering results from both the morphometric and allozyme analyses, the current author suggests recognizing two distinct subspecies, P. virginica subsp. virginica in the eastern U.S. and P. virginica subsp. scoparia in the south-central U.S. Conservation efforts should be focused on the maintenance of existing populations in both eastern and western regions of the U.S. in order to preserve the genetic and evolutionary potential of these taxa. / Master of Science

Michaux's sumac (Rhus michauxii) is a federally endangered rhizomatous shrub endemic to the southeastern United States, with two of the largest populations located at Maneuver Training Center--Fort Pickett, VA (Fort Pickett), and a nearby property, Deepwater. Michaux's sumac requires soil disturbance and fire to maintain healthy populations. Before being added to the endangered species list, 47% of populations were extirpated due to habitat loss, fire suppression, and hybridization with smooth sumac (Rhus glabra). Concerns with hybridization include hybrid swamping if hybrids are fertile, or outbreeding depression if hybrids display reduced fitness. I used genotyping-by-sequencing to estimate the extent of hybridization at Fort Pickett and Deepwater, and to assess how such hybridization may impact survival of Michaux's sumac as a distinct species at each of these locations. Additionally, population structure was examined using DAPC (discriminant analysis of principal components) and Admixture analyses to determine whether the colonies at Fort Pickett and Deepwater make up separate populations, meta-populations, or one large population. Analysis of 107,344 SNPs (single nucleotide polymorphisms) using Introgress and Admixture software suggested widespread hybridization at both Fort Pickett and Deepwater, with hybrids present in most of the sampled colonies. Population structure analyses revealed differentiation between the Fort Pickett and Deepwater populations, but little evidence of separate populations among the colonies sampled at Fort Pickett. These results are important for conservation planning to ensure the long-term survival of Michaux's sumac at Fort Pickett and Deepwater and can be used to help inform future management decisions. / Master of Science / Michaux's sumac (Rhus michauxii) is a small, federally endangered shrub endemic to the southeastern United States, with populations remaining in North Carolina, Virginia, and Georgia. To date, the largest known colonies of Michaux's sumac are in Virginia at Maneuver Training Center--Fort Pickett, VA (Fort Pickett), and at a nearby privately owned property called Deepwater. Michaux's sumac requires soil disturbance and fire to reduce competition and maintain healthy populations. It currently faces threats from habitat loss due to agricultural land use and fire suppression, and hybridization with a closely related species, smooth sumac (Rhus glabra). Hybridization is a threat to Michaux's sumac at Fort Pickett and Deepwater because it co-occurs with smooth sumac throughout the area. This study determined how much smooth sumac and Michaux's sumac are hybridizing in these locations and assessed whether hybridization is a threat to the long-term survival of the populations at each site. A secondary goal of the study was to gain a better understanding of how genetically similar the colonies within and between locations are to one another. Understanding the level of hybridization and the population structure of Michaux's sumac is important for making management decisions to protect the species. I found widespread hybridization between Michaux's and smooth sumac, with hybrid individuals at nearly all the colonies sampled. Additionally, there is evidence that Fort Pickett and Deepwater comprise two distinct populations, but the colonies inside each area are likely not separate populations. These results will inform future conservation management decisions for the species.

An investigation of the genetic and meristic patterns of coastal cutthroat trout (Oncorhynchus clarki clarki) in two coastal basins, one in Oregon and one in Alaska, revealed varying degrees of differences among populations within each site. Coastal cutthroat trout collected from seven sites above and below barriers from Elk River, Oregon were characterized by significant genetic structuring (Fst=0.095) based on allozymes and significant differences among nine meristic characters. This may reflect the geologic history and local conditions of the Elk River Basin. Despite this divergence, Elk River populations were relatively similar to each other (Nei's genetic identity> 0.0987), suggesting that these populations share common ancestry. Fewer differences were detected among the coastal cutthroat trout from four sites below and above a barrier within Vixen Inlet, Alaska. Genetic structuring among populations was low (Fst=0.016) and significant differences were detected in only two of nine meristic characters. This may reflect the recent glacial history of southeast Alaska. In both sites, there was concordance between allozyme data and meristic data. / Graduation date: 1996

The genetic structure of 49 natural populations of four species (P. attwateri, P. boylii, P. pectoralis, and P. polius) of the Peromyscus boylii species group was analyzed through application of chromosomal and electrophoretic techniques. Chromosomal variation within and among populations of the boylii species group was analyzed from 178 specimens. Electrophoretic techniques were utilized for the demonstration of variation in enzymes and other proteins encoded by structural loci and applied to the study of the evolution of the boylii species group by estimation of levels of genetic heterozygosity within populations, estimation of degree of genetic similarity between conspecific populations and between species, and determination of patterns of geographic variation in allelic frequencies and levels of heterozygosity. Six distinct chromosomal patterns were observed among the populations of the four species of the boylii species group. All specimens had a diploid number of 48 and the major difference in chromosomal morphology was in the number of pairs of large to medium biarmed autosomes. Little or no chromosomal variation was observed in three species (attwateri, pectoralis and polius), but considerable chromosomal variation occurred among populations of P. boylii. Generally, the chromosomal variation in P. bylii was between allopatric populations, with each chromosomal pattern limited to a recognized subspecies. Polymorphism was observed in two populations. The polymorphism observed in P. polius was the result of pericentric inversion involving the smallest pair of metacentric autosomes. The polymorphism observed in P. bolii cileus was interpreted at the result of gene flow between P. boylii rowleyi and P. boylii spicilegus. In addition to chromosomal evidence, analysis of electrophoretic data demonstrated and suggested effective gene flow between the chromosomal forms of P. boylii. Electrophoretically demonstrable variation was analyzed in 11 proteins encoded by 17 autosomal loci. Of the 17 structural loci, 11 were polymorphic in one or more populations . No more than five loci were observed to be polymorphic within a single population, with a mean number of polymorphic loci of 2.26. The levels of genetic variability, as measured by the proportion of loci in a heterozygous state in the average individual of a population, for populations of the boylii species group were within the range reported for other rodents. Populations of P. attwateri exhibited levels of heterozygosity considerably lower than those reported for other species of Peromyscus. The low level of genetic variability in P. attwateri was probably the result of founder effect of the original population which was isolated on the Edwards Plateau during the late Pleistocene and subsequent genetic drift which allowed fixation of genes within this isolated population. Geographic variation in levels of heterozygosity observed among populations of P. attwateri and P. boylii indicates a north to south cline of increasing heterozygosity with the most variable populations occurring in the southern portion of the range of each species. A wide range of the levels of heterozygosity and genetic similarity was observed among populations of P. pectoralis and is the result of genetic contribution from three Pleistocene refugia. Analysis of paired combinations of populations of P. boyli indicated that the various chromosomal forms of P. boylii. form a conspecific unit. Populations of P. attwateri were genetically distinct from other members of this group. Examination of specific loci for allelic homology suggested that P. attwateri arose from genetic divergence of a population of P. boylii isolated on the Edwards Plateau during the late Pleistocene.

Three populations of Southern flying squirrels were studied in the Ouachita Mountains of Arkansas to assess the impact of population subdivision-due to island formation--on the population genetics of Glaucomys volans. One island, one mainland, and one open population were investigated. A 367 nucleotide hypervariable region of mitochondrial DNA was sequenced in individuals from each population. Individuals and populations were compared to assess relatedness. Higher sequence diversity was detected in the open and island populations. One island individual shared characters with both the island and mainland populations. Results support the hypothesis that the mainland population may have reduced gene flow. Also, the island population may have been originally founded by at least two maternal lineages.