In this thesis, I present a computational neural model that reproduces the high-level behavioural results of well-known fear conditioning experiments: first-order conditioning, second-order conditioning, sensory preconditioning, context conditioning, blocking, first-order extinction and renewal (AAB, ABC, ABA), and extinction and renewal after second-order conditioning and sensory preconditioning. The simulated neural populations used to account for the behaviour observed in these experiments correspond to known anatomical regions of the mammalian brain. Parts of the amygdala, periaqueductal gray, cortex and thalamus, and hippocampus are included and are connected to each other in a biologically plausible manner. The model was built using the principles of the Neural Engineering Framework (NEF): a mathematical framework that allows information to be encoded and manipulated in populations of neurons. Each population represents information via the spiking activity of simulated neurons, and is connected to one or more other populations; these connections allow computations to be performed on the information being represented. By specifying which populations are connected to which, and what functions these connections perform, I developed an information processing system that behaves analogously to the fear conditioning circuit in the brain.

Many commercially grown apple cultivars have a biennial cropping habit, producing many small fruit in one year and few or none in the following year. The production of fruits is known to inhibit flower initiation for the following year. This undesirable trait is frequently managed by removing (thinning) some flowers or young fruit in years of heavy flowering which improves the size of remaining fruits, but does not reliably improve flowering in the following year. The effect of thinning on flower initiation is not well understood. Two mathematical models are developed describing the relationship between flowering in one year and the next. The first models the effects of thinning on return bloom and attempts to define maximum repeatable flower number. The second models how proximity of growing points may impact biennial bearing and maximum annual flower number. This second model may be useful to advance research into biennial bearing in apple.

This thesis investigated the epidemiology and control of Cysticercus ovis infection on Canadian sheep farms. Canadian slaughter data indicated an increase in sheep carcass condemnations due to C. ovis in 2007 and 2008. Trace-back of 237 carcasses condemned in Ontario, between 2009 and 2011, revealed they originated from 133 farms across Canada. A case-control study was performed (n=40 cases, 56 controls) to identify farm-level risk factors for carcass condemnations. Farm dogs scavenging deadstock (OR=4.04; 95% CI: 1.16–14.04) and failing to dispose of deadstock properly (OR=11.78; 95% CI: 2.93–47.40) were significantly associated with condemnations in multivariable analyses. A transmission model for Taenia ovis was created and control options were assessed. Model simulations predicted cestocide treatment of guardian dogs every fifth week, and proper deadstock disposal would reduce the risk of C. ovis infection in lambs.

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This thesis explores the roles of physical and mathematical modelling in the prediction of temperature profiles in the composting process. A literature-based evaluation of the performance of laboratory- and pilot scale composting reactors, showed that physical models used in composting research frequently do not properly simulate the full-scale composting environment, and may therefore produce results which are not applicable at full scale. In particular, self-heating, laboratory-scale, reactors typically involve significant convective/conductive/radiative losses, even with insulation present. This problem can be overcome by using controlled temperature difference or controlled heat flux laboratory reactors, which allow convective/conductive/radiative heat fluxes to be controlled to levels close to those occurring in full-scale systems. A new method of assessing the simulation performance of composting systems is presented. This utilises the areas bounded by the temperature-time profile and reference temperatures of 40 and 55 ℃ (A₄₀ and A₅₅), the times for which these temperatures are exceeded (t₄₀ and t₅₅), and times to peak temperature. An evaluation of published temperature profiles showed a marked difference in these parameters when comparing many laboratory- and full-scale reactors. The impact of aeration is illustrated, and laboratory- and pilot-scale reactors able to provide good temperature profile simulation, both qualitatively and quantitatively, are identified. Mathematical models of the composting process are reviewed and their ability to predict temperature profiles assessed. The most successful models in predicting temperature profiles have incorporated either empirical kinetic expressions, or utilised a first-order model, with empirical corrections for temperature and moisture. However, no temperature models have been able to predict maximum, average and peak temperatures to within 5, 2 and 2 ℃ respectively, or to predict the times to reach peak temperatures to within 8 h, although many models were able to successfully predict temperature profile shape characteristics. An evaluation of published constant-temperature and varying-temperature substrate degradation profiles revealed very limited evidence to support the application of single exponential, double exponential or non-logarithmic Gompertz functions in modelling substrate degradation kinetics, and this was identified as a potential weakness in the temperature prediction model. A new procedure for correcting substrate degradation profiles generated at varying temperature to a constant temperature of 40 ℃ was developed and applied in this analysis, and on experimental data generated in the present work. A new approach to the estimation of substrate degradation profiles in the composting process, based on a re-arrangement of the heat balance around a reactor, was developed, and implemented with both a simulated data set, and data from composting experiments conducted in a laboratory-scale constant temperature difference (CTD) reactor. A new simulated composting feedstock for use in these experiments was prepared from ostrich feed pellets, office paper, finished compost and woodchips. The new modelling approach successfully predicted the generic shape of experimental substrate degradation profiles obtained from CO2 measurements, but under the conditions and assumptions of the experiment, the profiles were quantitatively different. Both measured CO2-carbon (CO2-C) and predicted biodegradable volatile solids carbon (BVS-C) profiles were moderately to well fitted by single exponential functions with similar rate coefficients. When corrected to a constant temperature of 40 ℃, these profiles gave either multi-phase or double exponential profiles, depending upon the cardinal temperatures used in the temperature correction procedure. If it is assumed that the double exponential model generated is correct, this work provides strong evidence that a substrate degradation curve generated under appropriate laboratory conditions at 40 ℃ would, given the correct cardinal temperatures, generate a correct substrate degradation profile under varying temperature conditions, and that this in turn would enable an accurate and precise prediction of the temperature profile using a heat and mass balance approach. This finding opens the door for the development of a simple laboratory test for composting raw material characterisation, but underlines the need for accurate estimates of the physical cardinal temperatures. Experimental factors appear to be the likely cause of the dysfunction between previously reported substrate degradation patterns and existing substrate degradation models, and suggestions for further research are provided in order to more precisely and accurately quantify these factors.

The kidneys are amazingly versatile organs that perform a wide range of vital bodily functions. This thesis provides an analysis into a range of mathematical models of the tubuloglomerular feedback (TGF) mechanism. The TGF mechanism is an autoregulatory mechanism unique to the kidney that maintains approximately constant blood flow to the organ despite wide fluctuations in pressure. Oscillations in pressure, flow, and sodium chloride concentration have been attributed to the action of the TGF mechanism through a number of experimental studies. These oscillations appear spontaneously or in response to a natural or artificial pressure step or microperfusion. The reason for sustained oscillatory behaviour in nephrons is not immediately clear. Significant research has gone into experimentally determining the signal to the TGF mechanism, but the physiological significance is not mentioned in the literature. Considerable modelling of the oscillations attributed to the TGF mechanism has also been undertaken. However, this modelling uses models that are inherently oscillatory, such as a second-order differential equation or delay differential equations. While these models can be fitted to closely approximate the experimental results they do not address the physiological factors that contribute to sustained oscillations. This thesis aims to determine the contributing factors to the sustained oscillations. By understanding these factors a better hypothesis of the physiological role of the oscillations should be possible. Chapter 3 presents a mathematical model by Holstein-Rathlou and Marsh [28] that uses a partial differential equation (PDE) model for the tubule and a second-order differential equation for the TGF feedback. The remainder of this chapter shows that oscillations occur without an inherently oscillatory second-order differential equation due to the delays in the system. Tubular compliance was also shown to be necessary for sustained oscillations. Sustained oscillations were not exhibited in the TGF model with a noncompliant tubule. Although damped oscillations were exhibited for a wide range of parameter space. Adding compliance to the tubule increased the delay around the loop of Henle. This additional delay elicited sustained oscillations. The computationally expensive PDE model of 3 was simplified to an ordinary differential equation (ODE) model in Chapter 4 by assuming a spatial profile. This model exhibits much of the same qualitative behaviour as the PDE model including sustained oscillations for similar ranges of parameter space. Compliance was also found to be important in the generation of sustained oscillations in agreement with the PDE tubule model. This model is less computationally expensive than the PDE model and allows analysis that was unfeasible with the PDE model. Significant natural and artificial blood pressure fluctuation occur in experimental rat models. Chapter 5 examines the effect of inlet pressure forcing on a nonoscillatory and an oscillatory model. The inherently nonoscillatory noncompliant model becomes oscillatory with a physiologically realistic pressure forcing. The oscillatory compliant model remains oscillatory with the addition of a inlet pressure forcing. Pressure fluctuations were hypothesised to contribute to sustained oscillations and could be validated experimentally. Two extensions to the single nephron TGF models are presented in Chapter 6. A realistic juxtaglomerular delay is added to the single nephron models with both the ODE and PDE tubular models. Physiologically realistic juxtaglomerular delays induce sustained oscillations in the otherwise nonoscillatory noncompliant models. The remainder of this chapter presents a different model for a variable interstitial sodium chloride concentration profile. This model demonstrates experimentally observed function of the countercurrent mechanism by which a concentration gradient is set up and maintained in the interstitium. Two single nephron models with ODE tubular models are coupled in Chapter 7. The coupling is modelled through the effect on the resistance of their neighbouring nephron's afferent arteriole resistance. The coupled nephron model exhibits entrainment as observed experimentally. Inhibiting the oscillation in one nephron reduces the amplitude of the oscillation in its neighbour. This result compares well with experiments where the TGF mechanism in one nephron is blocked by the administration of furosemide.

This thesis is concerned with developing and implementing algorithms based upon the geometry of pictures. Spherical pictures have been used in many areas of combinatorial group theory, and particularly, they have shown to be a useful method when studying the second homotopy module, 1T2, of a presentation ([3],[4],[7],[12],[41] and [64]). Computational programs that implement picture theoretical and design algorithms could advance the areas in which picture theory can be used, due to the much faster time taken to derive results than that of manual calculations. A variety of algorithms are presented. A data structure has been devised to represent spherical pictures. A method is given that verifies that a given data structure represents a picture, or set of pictures, over a group presentation. This method includes a new planarity testing algorithm, which can be performed on any graph. A computational algorithm has been implemented that determines if a given presentation defines a group extension. This work is based upon the algorithm of Baik et al. [1] which has been developed using the theory of pictures. A 3-presentation for a group G is given by < P, s >, where P is a presentation for G and s is a set of generators for 1T2. The set s can be described in a number of ways. An algorithm is given that produces a generating set of spherical pictures for 1T2 when s is given in the form of identity sequences. Conversely, if s is given in terms of spherical pictures, then the corresponding identity sequences that describe 1T2 can be determined. The above algorithms are contained in the Spherical PIcture Editor (SPICE). SPICE is a software package that enables a user to manually draw pictures over group presentations and, for these pictures, call the algorithms described above. It also contains a library of generating pictures for the non abelian groups of order at most 30. Furthermore, a method has been implemented that automatically draws a spherical picture from a corresponding identity sequence. Again, this new graph drawing technique can be performed on any arbitrary graph.

This thesis is an account of experiments into the teaching of mathematics to engineering undergraduates which have been conducted over twenty years against a background of changing intake ability, varying output requirements and increasing restrictions on the formal contact time available. The aim has been to improve the efficiency of the teaching-learning process. The main areas of experimentation have been the integration in the syllabus of numerical and analytical methods, the incorporation of case studies into the curriculum and the use of micro-based software to enhance the teaching process. Special attention is paid to courses in Mathematical Engineering and their position in the spectrum of engineering disciplines. A core curriculum in mathematics for undergraduate engineers is proposed and details are provided of its implementation. The roles of case studies and micro-based software are highlighted. The provision of a mathematics learning resource centre is considered a necessary feature of the implementation of the proposed course. Finally, suggestions for further research are made.

Human food production activities can dominate natural systems, altering ecological and evolutionary aspects of the environment. Disease-mediated interactions are of particular concern. For example, parasites may "spill-over'' from farms to wildlife. Parasites isolated on farms can evolve resistance to treatment chemicals , but "spill-back'' from wildlife to farms may alter evolutionary dynamics. Here, we consider exchange of parasites (Lepeophtheirus salmonis) between wild (Oncorhynchus gorbuscha) and farmed salmon. We derive and analyze discrete-time models that implicitly include wild salmon migrations. First, we extend a standard fisheries model to show parasite exchange affects "line-dominance'' in the population ecology of salmon. Second, we extend a classic population genetics model to show that wild salmon can theoretically provide an "ecosystem service'' by delaying the onset of chemical resistance in parasites on farms. This service, however is affected by a nonlinear feedback if farm parasites spill-back to affect wild salmon. / Applied Mathematics