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The objective of this dissertation is to develop theoretical and computational tools for the study of qualitative changes in the dynamics of systems with discontinuities, also known as nonsmooth or hybrid dynamical systems, under parameter variations. Accordingly, this dissertation is divided into two parts. The analytical section of this dissertation discusses mathematical tools for the analysis of hybrid dynamical systems and their application to a series of model examples. Specifically, qualitative changes in the system dynamics from a nonimpacting to an impacting motion, referred to as grazing bifurcations, are studied in oscillators where the discontinuities are caused by impacts. Here, the study emphasizes the formulation of conditions for the persistence of a steady state motion in the immediate vicinity of periodic and quasiperiodic grazing trajectories in an impacting mechanical system. A local analysis based on the discontinuity-mapping approach is employed to derive a normal-form description of the dynamics near a grazing trajectory. Also, the results obtained using the discontinuity-mapping approach and direct numerical integration are found to be in good agreement. It is found that the instabilities caused by the presence of the square-root singularity in the normal-form description affect the grazing bifurcation scenario differently depending on the relative dimensionality of the state space and the steady state motion at the grazing contact. The computational section presents the structure and applications of a software program, TC-HAT, developed to study the bifurcation analysis of hybrid dynamical systems. Here, we present a general boundary value problem (BVP) approach to locate periodic trajectories corresponding to a hybrid dynamical system under parameter variations. A methodology to compute the eigenvalues of periodic trajectories when using the BVP formulation is illustrated using a model example. Finally, bifurcation analysis of four model hybrid dynamical systems is performed using TC-HAT. / Ph. D.

Previous research on grazing behavior has shown that ruminants will select a mixed diet. The use of adjacent monocultures is an essential tool for determining dietary preference of forages. Much of the work to date has been conducted with white clover (Trifolium repens L.) and perennial ryegrass (Lolium perenne L.). Partial preference for white clover over ryegrass has been reported consistently and partial preference for legumes is thought to occur regardless of the legume and grass species being evaluated. Two forage species, tall fescue (Festuca arundinacea Schreb. or Lolium arundinaceum (Schreb.) S.J. Darbyshire) and alfalfa (Medicago sativa subsp. sativa L.), which had not been evaluated together previously as adjacent monocultures were grazed by beef steers in the present set of experiments. Steers exhibited a partial preference for alfalfa of 61 to 65% when given a choice of grazing alfalfa or tall fescue as adjacent monocultures, regardless of the ground area proportion of the two forages offered. Steers grazing tall fescue monocultures spent more time ruminating (P = 0.02) and tended to graze less time (P = 0.06) than steers in adjacent monoculture treatments. Time spent idling, number of prehensions and mastications, and bite rate were similar (P > 0.05) among treatments. Steers grazing tall fescue monocultures spent less time standing, more time lying, were less active and took fewer steps (P ≤ 0.05) than steers in adjacent monoculture treatments. Grazing behavior was examined when alfalfa had not been in the previous diet of the steers. Cattle without previous experience grazing alfalfa spent 78% of the time grazing alfalfa, whereas after having experience grazing it they spent a lower (P = 0.04) proportion of their time grazing alfalfa (72%). Overall proportion of the day spent grazing both forages was lower (P = 0.0001) when alfalfa was novel (40%), compared to when steers were experienced grazing both forages (46%). Proportion of the day spent idling was greater (P < 0.0001) when alfalfa was novel (35%), compared to when both forages were familiar to the steers (26%). Previous research has reported that ruminants exhibit a diurnal pattern of preference by decreasing the proportion of white clover consumed from morning to late afternoon while increasing the proportion of perennial ryegrass in the diet. This is thought to be a strategy to increase fiber intake before nightfall or as a response to higher carbohydrate levels in grass in the afternoon. In the present study, proportion of grazing time in alfalfa was higher (P = 0.02) in the afternoon (76.8 %) than in the morning (72.1 %). While fiber concentration was higher in the tall fescue, carbohydrate concentrations were similar. Steers were not attempting to increase fiber intake in the afternoon in the present study. Dry matter intake of steers grazing adjacent monocultures of alfalfa and tall fescue was estimated with n-alkanes. Diet composition was estimated using n-alkanes and long chain alcohols (LCOH) in several different combinations. The use of LCOH added additional characterization of the forages, but diet composition estimates were not different (P ≥ 0.22) than when estimated using four different n-alkanes. Laboratory analysis costs may be reduced if n-alkanes alone can adequately characterize the forages being consumed, depending on the forage species in question. Meteorological conditions impacted DMI with intake being less in hotter conditions. Steers had similar partial preferences for alfalfa over tall fescue (P = 0.13, 79% and 70% alfalfa in yr 1 and 2, respectively) even though total DMI differed between years (P = 0.002, 9.4 kg d-1 and 4.5 kg d-1 in yr 1 and 2, respectively). Lower DMI in yr 2 was attributed to hotter air temperatures. When animals are consuming two different forages as adjacent monocultures such as in the current experiments, it is important to determine the proportion of each forage in the diet before calculating DMI using odd chain n-alkanes of the forage along with a dose even chained n-alkane. Dry matter intake can be overestimated if the proportion of the forages consumed is not estimated and accounted for in the equation. This would apply to other studies utilizing mixed swards or any diet containing multiple components that differ in concentration of the n-alkane being used for DMI estimation. Analysis of n-alkane concentration should be performed on each item in the diet and the proportion of each item in the diet estimated so that the right value can be used in the calculation. Differences in marker concentrations between years also indicate the importance of analyzing those concentrations in the feed or forage at the time of fecal collection and not using values reported from previous research. / Ph. D.

Endophyte infected Tall fescue (Festuca arundinacea Schreb.) is the most dominant grass used for pasture in the Southeastern U.S. As a result, fescue toxicosis is a major concern. Producers need alternative forages for grazing cattle that do not have this negative aspect. The objective of this experiment was to determine the grazing behavior of cattle grazing Lakota (L) prairie grass (Bromus catharticus Vahl.), endophyte infected (E+), endophyte free (E-), and novel endophyte (Q) tall fescues. Angus-crossbred steers (279±8 kg) steers wore electronic behavior data recorders in four sampling periods, and direct visual appraisals of behavior were taken in five sampling periods during the months of May to September, 2004. Overall, during the visual appraisal phase steers grazing L spent most time (P<0.05) grazing while E+ spent the least time grazing. Overall, steers grazing E+ spent more time (P<0.05) idling than those on L, E-, or Q. Steers grazing E+ spent more time (P<0.05) standing than steers grazing Q. Steers grazing Q and E- spent more time (P<0.05) lying than those grazing E+. During the data recorder phase there were no significant differences between treatments for time spent grazing. Steers grazing E+ spent less time (P<0.05) lying and ruminating than steers grazing Q or L. Conversely, time spent standing and idling for steers grazing E+ was higher (P<0.05) than for steers grazing Q or L. These results indicate that L, E-, and Q may offer benefits to producers due to more time spent in productive activities during summer months. / Master of Science

I investigated the baseline benthic macroinvertebrate community in relation to the environmental conditions in a section of Smith Creek, north of Harrisonburg, VA, prior to restoration. Quantitative benthic macroinvertebrate and environmental samples were collected in April and September 2006 from the Bruce Farm (BR) section of Smith Creek and the nearby Mixed Use (MU) section of Mountain Run. BR had been heavily used for cattle grazing for decades and suffered from sediment, nutrients, and lack of a forested riparian zone. MU had a forested riparian zone, but still received nutrient and sediment inputs from upstream cattle grazing. Visual habitat assessments were performed in September 2006 and were compared to quantitative measures. Benthic macroinvertebrate densities and taxa richness were greater at BR (total density for combined seasons = 52,438; taxa richness for both seasons = 84) than MU (total density for combined seasons = 3,982 and taxa richness for both seasons = 63). Biological environmental variables related to nutrients and growth of plants on rocks (ash-free dry mass, chlorophyll a, epilithic biomass) influenced the benthic macroinvertebrate assemblage more than physical environmental variables related to the substrate composition (% fines, % gravel, Trask's sorting coefficient). Visual habitat estimates were not as effective as quantitative measures of habitat for explaining the benthic macroinvertebrate assemblage. / Master of Science

Increasing feeding costs has led many dairy farmers to use pasture for lactating cows to reduce their production cost. Little is known about the efficiency of nutrient utilization by lactating cows under grazing conditions. The objective of this study, therefore, was to investigate ruminal turnover, intake and site and extent of nutrient digestion in grazing lactating cows. Four dual cannulated (rumen and proximal duodenum) cows were randomly assigned to two groups to graze permanent pasture with no supplement or with 6.4 kg/d corn-mineral mix in a switchback design with three periods of two wk each. Solid and liquid markers (chromic oxide and CoEDTA), used to estimate duodenal flow, fecal output, and ruminal turnover, were administered through the rumen cannula twice daily at 1100 and 2300 h. The supplemented cows had greater milk production (23.7 kg/d) than those on pasture only (19.5 kg/d) with an increase of .66 kg of milk production per kg of concentrate. Corn, however, depressed milk fat percentage resulting in a similar milk fat yield between the two diets. Supplemental corn reduced rumen ammonia-N (22 vs 17 mg/dl) and increased N recovery at the duodenum (86 vs 75% of N intake). True Nitrogen digestibility averaged 72% and microbial N flowing to the duodenum was 67% of the total N flow. This results indicated that grazing cows may benefit from concentrates containing undegradable protein. Daily OM intake was not different, but when cows were fed the supplemented diet, pasture OM was lower than when fed pasture only. Organic matter, NDF and ADF digestibility in the rumen and whole digestive tract were greater when cows were fed. Cellulolytic activity may have been reduced by grain supplementation due to decreased ruminal pH (6.4 vs 6.2). The reduction in fiber digestibility may explain the decrease in forage intake when cows were fed supplemental grain. Ruminal rates of passage (kp) for solid (7.3 %/h) and liquid (18.2 %/h) markers were similar for both diets. The kp values observed for solid and liquid indicate that grazing cows may have a faster ruminal turnover than cows fed diets containing primarily hay or silage. / Master of Science

The growing importance of grassland and all its products is becoming more evident. A grassland economy is the backbone of the sheep, dairy cattle, and beef cattle industries. Grassland farming makes for a more permanent type of agriculture. An increased acreage in sod crops results in less erosion, better soil structure and drainage, and higher organic matter. Soil productivity can be maintained or improved under grassland farming practices. A general appreciation of the value of pasture as a source of livestock feed, improvements in strains of grasses and legumes, and better knowledge of fertilizer usage have encouraged the increase in acreage of highly productive pastures. Well managed pastures are highly productive and supply nutritious herbage high in minerals and protein. The Northern Virginia Pasture Research Station at Middleburg, Virginia has experiments under way to measure quantity of production and the nutritional value of pasture herbage when grazed by dairy and beef cattle (1 and 2). These experiments are designed to measure the value of species, mixtures, methods of rotational grazing, and continuous grazing of one mixture as compared with grazing different mixtures in a rotational sequence. Data are being obtained on carrying capacity, livestock gains, carcass characteristics, palatability, and rate of milk production. This study was undertaken to gain supplementary information on the established grazing experiments at Middleburg. It was postulated that data on animal behavior would help interpret the animal responses observed. This report gives data on animal grazing frequency; however, data on other aspects of animal behavior were obtained. / Master of Science

Reclaimed coal-mined lands in the Appalachian region can be successful established and utilized for beef cattle production. Currently, these areas are underutilized partly due to an increase in invasive plant species, such as multiflora rose (Rosa multiflora Thunb. Ex Murr.), autumn olive (Elaeagnus umbellata Thunb.), and sericea lespedeza (Lespedeza cuneata (Dum.-Cours.) G. Don). The steep topography and low economic returns from beef cattle make conventional control methods inhibitive. Goats are effective browsing on invasive plant species. An experiment was conducted in 2006 and 2007 at the Powell River Research and Education Center near Wise, VA (77° 43' 30" west longitude, 38° 57' 30" north latitude, elevation 155.5 m) to determine the effects of an ungrazed control, cattle alone grazing, and mixed grazing goats with cattle on forage biomass, botanical composition, relative plant abundance, and animal performance. The three treatments included an ungrazed control, cattle grazing alone, and mixed grazing goats with cattle. Experimental design was a randomized complete block design with two replicates for the control and three replicates for the grazed treatments. Three times during the grazing season the following were measured, analyzed or assessed: nutritive values of pasture, autumn olive, multiflora rose, and sericea lespedeza were assessed; forage biomass was determined by clipping four 0.25 m² quadrants per control replicate and eight 0.25 m² quadrants per grazed replicate; botanical composition and relative abundance of plant species was assessed by the Double DAFOR method from five fixed points in each control replicate and ten fixed points in each grazed replicate; animals were weighed; autumn olive shrub height was measured with a clinometer from a distance of 10 m from the shrub. Branch length was measured with a tape measure from the base of the branch to the end tip. Shrub survival was measured by counting shrubs in each replicate and determining visually percent leaf-out. Each year, control and cattle alone treatments had greater (P < 0.05). Generally, grass content increased in the grazed treatments from spring to fall while weed content increased in the control treatment (P < 0.05). By the end of the two experimental years, the legume components of the pasture were low. This was more evident in the ungrazed control than the grazed treatments. The relative abundance of tall fescue and orchardgrass (P < 0.05) increased in grazed treatments while sericea lespedeza became a dominant weed in the control (P < 0.05). Goats showed high preference for sericea lespedeza and maintain this plant in a leafy, vegetative stage. This leafy, vegetative growth was found to acceptable to cattle. Cattle performance was not affected by treatment (P < 0.05) but total animal output was higher for mixed grazed compared to cattle alone treatments (P < 0.05). The nutritive values of multiflora rose, autumn olive, and sericea lespedeza were higher than pasture in most instances (P < 0.05). In our experiment, autumn olive was severely impacted by goat browsing. Shrub survival was lower in mixed grazing (61%) by the end of the experiment compared to over 90% for the control and cattle grazing treatments (P < 0.05). Overall, pastures were utilized more uniformly in mixed grazing compared to other treatments. Mixed grazing goats with cattle appear to be a viable option for livestock producers in the Appalachian coal mining region. / Master of Science

This investigation has had for its purpose the determination of first, all members of the Gramineae (Grass) family found in the county; second, the incidence of each species with the various types of soil; third, the grazing value of each species; fourth, the life span of the parent plants; fifth, the present grassland status; and sixth, the potentialities of developing desirable grassland.