For systems that are in equilibrium, fluctuations can be understood through interactions with external heat reservoirs. For this reason these fluctuations are known as thermal noise, and they usually become vanishingly small in the thermodynamic limit. However, many systems comprising interacting constituents studied by physicists in recent years are both far from equilibrium, and sufficiently small so that they must be considered finite. The finite number of constituents gives rise to an inherent demographic noise in the system, a source of fluctuations that is always present in the stochastic dynamics. This thesis investigates the role of stochastic fluctuations in the macroscopically observable dynamical behaviour of non-equilibrium, finite systems. To facilitate such a study, we construct microscopic models using an individual based modelling approach, allowing the explicit form of the demographic noise to be identified. In many physical systems and theoretical models, absorbing states are a defining feature. Once a system enters one, it cannot leave. We study the dynamics of a system with two symmetric absorbing states, finding that the amplitude of the multiplicative noise can induce a transition between two universal modes of domain coarsening as the system evolves to one of the absorbing states. In biological and ecological systems, cycles are a ubiquitously observed phenomenon, but are di cult to predict analytically from stochastic models. We examine a potential mechanism for cycling behaviour due to the flow of probability currents, induced by the athermal nature of the demographic noise, in a single patch population comprising two competing species. We find that such a current by itself cannot generate macroscopic cycles, but when combined with deterministic dynamics which constrain the system to a closed circular manifold, gives rise to global quasicycles in the population densities. Finally, we examine a spatially extended system comprising many such patch populations, exploring the emergence of synchronisation between the different cycles. By a stability analysis of the global synchronised state, we probe the relationship between the synchronicity of the metapopulation and the magnitude of the coupling between patches due to species migration. In all cases, we conclude that the nature of the demographic noise can play a pivotal role in the macroscopically observed dynamical behaviour of the system.

Stochastic modelling of gene regulatory networks provides an indispensable tool for understanding how random events at the molecular level influence cellular functions. A common challenge of stochastic models is to calibrate a large number of model parameters against the experimental data. Another difficulty is to study how the behaviour of a stochastic model depends on its parameters, i.e. whether a change in model parameters can lead to a significant qualitative change in model behaviour (bifurcation). This thesis addresses such computational challenges by a tensor-structured computational framework. After a background introduction in Chapter 1, Chapter 2 derives the order of convergence in volume size between the stationary distributions of the exact chemical master equation (CME) and its continuous Fokker-Planck approximation (CFPE). It also proposes the multi-scale approaches to address the failure of the CFPE in capturing the noise-induced multi-stability of the CME distribution. Chapter 3 studies the numerical solution of the high-dimensional CFPE using the tensor train and the quantized-TT data formats. In Chapter 4, the tensor solutions are applied to study the parameter estimation, robustness, sensitivity and bifurcation structures of stochastic reaction networks. A Matlab implementation of the proposed methods/algorithms is available at http://www.stobifan.org.

Many biological processes can be described in terms of chemical reactions and diffusion. In this thesis, reaction-diusion mechanisms related to the formation of Turing patterns are studied. Necessary and sufficient conditions under which Turing instability occur is presented. Behaviour of Turing patterns is investigated with a use of deterministic approach, compartment-based stochastic simulation algorithm and molecular-based stochastic simulation algorithm.

In 1985, Pierre Wack, arguably the father of modern scenario thinking, wrote the following: “Forecasts often work because the world does not always change. But sooner or later forecasts will fail when they are needed most: in anticipating major shifts…” (Wack, 1985: 73). The truth of this statement have again become apparent, first as the “food price crisis” played out during 2007 and 2008, and secondly as the current financial and economic crisis are playing out. Respected market commentators and analysts, both internationally and within South Africa, made all sorts of “informed predictions” on topics ranging from oil prices, interest rates, and economic growth rates to input costs and food prices. The problem is that none of these “respected views” and “informed predictions and estimates” became true within the period that was assigned to these predictions. In fact, just the opposite occurred: the unexpected implosion of the global economy and hence commodity markets. The result of the experts “getting it so wrong”, is that questions are being asked about the reliability of risk and uncertainty analysis. Even though the experts used highly advanced analytical techniques in analyzing the risks and uncertainties in order to formulate predictions and outlooks, both the “food price crisis” and the economic implosion were totally unanticipated. The same questions need to be asked in terms of risk and uncertainty analyses in agricultural economics. With agriculture experiencing a period of fundamental changes causing significant uncertainty, risk and uncertainty analyses in agriculture will need to move to the next level in order to ensure that policies and business strategies are robust enough to withstand these newly arising uncertainties. The proposed solution to this problem and therefore the hypothesis offered and tested by this thesis is to work with two techniques in conjunction without combining it when developing a view of the future. The two techniques used, namely intuitive scenario thinking and stochastic modelling are based on two fundamentally different hypotheses namely: the future is like the past and present (stochastic modelling), and the future is not like the past and present but is a result of combining current and unexpectedly new forces or factors (intuitive scenario thinking). The idea behind this stems from the philosophy of Socrates, whereby he postulated that the truth can never be fully known and therefore, when working with the truth, one needs to work with multi-hypotheses about the truth until all but one hypothesis can be discarded. This will then bring one closer to the truth, but never lead you to know the truth in full, since the truth can’t be known in full. Applying this idea means conjunctively using two techniques which are based on the two hypotheses about the future. From a literature review it was realised that two such techniques existed, namely, stochastic modelling and scenario thinking. Stochastic modelling, by its very nature, is based on the assumption that the future is like the past and present since historical data, historical inter-relationships, experience, and modelling techniques are used to develop the model, apply it, and to interpret its results. Scenario thinking on the other hand, and specifically intuitive logics scenario thinking, is based on the notion that the future is not like the past or present, but is rather a combination of existing and new and unknown factors and forces. At first the perceived problem with this idea was thought to exist in the problem of using both techniques in combination, since the two techniques are fundamentally different because of the fundamentally different assumptions on which they are based. The question and challenge was therefore whether these two techniques could be used in combination, and how? However, the solution to this problem was more elementary than what was initially thought. As the two techniques are fundamentally different, it implies that the two techniques can’t be combined because the two underlying assumptions can’t be combined. However, what is possible is to use it in conjunction without adjusting either technique. Rather, one would allow each technique to run its course, which at the same time leads to cross-pollination in terms of ideas and perspectives, where possible and applicable. The cross-pollination of ideas and perspectives will then create a process whereby ideas regarding the two basic assumptions on the future are crystallised and refined through a learning process, hence resulting in clearer perspectives on both hypotheses about whether the future will be like the past and present, or whether the future will be a combination of existing and new but unknown factors and forces. These clearer perspectives provide a framework to the decision-maker whereby the two basic hypotheses on the future can be applied simultaneously to develop strategies and policies that are likely robust enough to be successful in both instances. It also provides a framework whereby reality can be interpreted as it unfolds, which signals to the decision-maker which of the two hypotheses is playing out. This will assist the decision-maker in better perceiving what is in fact happening, hence what the newly perceived truth is in terms of the future, and therefore what needs to be done in order to survive and grow within this newly developing future, reality, or truth. The presentation of three case studies assists in testing the hypothesis of this thesis as presented in chapter one, and concludes that the hypothesis can’t be rejected. Hence, through the presentation of the case studies it is found that using scenario thinking in conjunction with stochastic modelling does indeed facilitate a more complete understanding of the risks and uncertainties pertaining to policy and strategic business decisions in agricultural commodity markets, through fostering a more complete learning experience. It therefore does facilitate better decision-making in an increasingly turbulent and uncertain environment. / Thesis (PhD)--University of Pretoria, 2010. / Agricultural Economics, Extension and Rural Development / unrestricted

Im Endeffekt zielt die Physik darauf ab, die Gesetze und die geltenden Gleichungen aufzudecken, die der Entwicklung eines bestimmten Systems zugrunde liegen: man sammelt eine Messreihe der Eigenschaften, die den Zustand dieses besonderen Systems beschreibet und versucht eine Gleichung abzuleiten, die die Messreihe von Messungen reproduziert. Verhält sich das System gut genug, sollte seine Entwicklung für eine vernünftige lange Zeit vorhersehbar sein. Verhält sich das System nicht gut genug, kann es schwieriger sein dies zu erreichen. Leider ist meinstens so der Fall: natürliche und vom Menschen gemachte Prozesse ergeben sich typischerweise aus dem Wechselspiel zwischen deterministischen und stochastischen Beiträgen, die jeweils unterschiedliche Merkmale aufweisen. Dieses Manuskript behandelt einige Hauptergebnisse aus dem Gebiet der stochastische Modellierung und bildet ein Konvolut aus 23 verschiedenen Arbeiten, die zwischen 2005 und 2018 in Fachzeitschriften veröffentlicht wurden. Der erste Teil behandelt spezifische theoretische Aspekte der stochastischen Modellierung, nämlich die Lösung des Einbettungsproblems in nichtstationären Reihen, die Signaltrennung der stochastischen Signalüberlagerung, die Abschätzung der topologischen globalen Eigenschaften komplexer Netzwerke und die Implementierung von Algorithmen für stochastische Analyse und für zufällige Moleküldynamik. Der zweite Teil behandelt die Annäherung der stochastischen Modellierung von spezifischen Anwendungen in der Physik und in vier weiteren interdisziplinären Bereichen: Geophysik, Energiesystemen, Finanzen und die Gesellschaft. Die spezifischen Anwendungen in der Physik finden Gebrauch des Entropiekonzeptes auf Monsterwelle, das Konzept der kleinsten Wirkung auf stochastische Prozesse angewendet, die Beziehung zwischen Perkolationsübergängen und der laminaren Ablöseblase auf Flügelprofilen, Thermostatistik komplexer Netzwerke, und die Ablagerungsprozesse von Granularpartikeln. Die Anwendungen in den interdisziplinären Themen konzentrieren sich auf die Modellierung der Schadstoffkonzentration in städtischen Zentren (Geophysik), Modellierung der Energieumwandlung, Turmschwingungen und Ermüdungslasten in Windturbinen (Energiesysteme), Stabilität von Finanznetze und nicht stationärer Entwicklung der Volumenpreise (Finanzen) und die menschliche Mobilität und sozialen Netzwerke (Gesellschaft). Abschließend werden zurkünftige Forschungsvorhaben erörtert, wo die hier presentierten Arbeiten von Nutzen sind, nähmlich für Energiesysteme, Finanzen und Neurowissenschaften.

<p>Northern boreal peatlands represent a globally significant carbon pool that are at risk of drying through land-use change and projected future climate change. The current ecohydrological conceptualization of peatland response to persistent water table (WT) drawdown is largely based on short-term manipulation experiments, but where the long-term response may be mediated by vegetation and microtopography dynamics. The objective of this thesis is to examine the changes to peatland evapotranspiration, soil physical properties, and moisture stress in response to a long-term WT manipulation. The energy balance, hydrology, vegetation, and soil properties were examined at three adjacent peatland sites in the southern sub-boreal region which were subjected to WT manipulations on the order of ±10 cm at two treatment sites (WET, and DRY) compared to the reference site (INT) as a result of berm construction in the 1950s.</p> <p>Sites with an increasing depth to WT were found to have greater microtopographic variation and proportion of the surface covered by raised hummocks. While total abundance of the major plant functional groups was altered, species composition and dominant species of vascular and non-vascular species within microforms was unaltered. Changes in vegetation and microtopography lead to differences in albedo, surface roughness, and surface moisture variability. However, total ET was only significantly different at the WET site. Transpiration losses accounted for the majority of ET, where LAI best explained differences in total ET between sites. Surface moisture availability did not appear to be limiting on moss evaporation, where lab results showed similar moisture retention capacity between microforms and sites, and where low surface bulk density was shown to be a strong controlling factor. Modelling results further suggested that, despite dry surface conditions, surface moisture availability for evaporation was often not limited based on several different parameterizations of peat hydraulic structure with depth.</p> / Doctor of Philosophy (PhD)

Transit agencies seeking to improve transit service delivery are increasingly considering the deployment of transit signal priority (TSP). However, the impact of TSP on transit service and on the general traffic stream is a function of many factors, including intersection geometry, signal timings, traffic demands, TSP strategies and parameters, transit vehicle headways, timing when transit vehicles arrive at the intersection, etc. Previous studies have shown that depending on these factors, the net impact of TSP in terms of vehicle or person delay can be positive or negative. Furthermore, due to financial constraints, transit agencies are often able to deploy TSP at only a portion of all of the candidate intersections. Consequently, there is a need to estimate the impact of TSP prior to implementation in order to assist in determining at which intersections TSP should be deployed. Currently, the impacts of TSP are often estimated using microscopic simulation models. However, the application of these models is resource intensive and requires specialized expertise that is often not available in-house to transit agencies. In this thesis, an analytical model was proposed for estimating the delay impacts of green extension and early green (red truncation) TSP strategies. The proposed model is validated with analytical model reported in the literature and microscopic simulation model. This is followed by model sensitivity analysis. A software module is developed using the proposed model. The usefulness of the model is illustrated through its application to estimate the TSP performance. Finally, a prioritization is conducted on sixteen intersections with different geometric and operational traffic strategies. The overall results indicate that the proposed model is suitable for both estimating the pre-deployment and post-deployment TSP performance. The proposed model is suitable for implementation within a spreadsheet and requires considerably less effort, and less technical expertise, to apply than a typical micro-simulation model and therefore is a more suitable tool for transit agencies to use for prioritising TSP deployment.

The present thesis deals with the development and performance analysis ofanisotropy-resolving models for the small, unresolved scales (”sub-grid scales”,SGS) in large eddy simulation (LES). The models are characterised by a descriptionof anisotropy by use of explicit algebraic models for both the subgridscale(SGS) stress tensor (EASSM) and SGS scalar flux vector (EASSFM). Extensiveanalysis of the performance of the explicit algebraic SGS stress model(EASSM) has been performed and comparisons made with the conventionalisotropic dynamic eddy viscosity model (DEVM). The studies include LES ofplane channel flow at relatively high Reynolds numbers and a wide range ofresolutions and LES of separated flow in a channel with streamwise periodichill-shaped constrictions (periodic hill flow) at coarse resolutions. The formersimulations were carried out with a pseudo-spectral Navier–Stokes solver, whilethe latter simulations were computed with a second-order, finite-volume basedsolver for unstructured grids. The LESs of channel flow demonstrate that theEASSM gives a good description of the SGS anisotropy, which in turn gives ahigh degree of resolution independence, contrary to the behaviour of LES predictionsusing the DEVM. LESs of periodic hill flow showed that the EASSMalso for this case gives significantly better flow predictions than the DEVM.In particular, the reattachment point was much better predicted with the EASSMand reasonably well predicted even at very coarse resolutions, where theDEVM is unable to predict a proper flow separation.The explicit algebraic SGS scalar flux model (EASSFM) is developed toimprove LES predictions of complex anisotropic flows with turbulent heat ormass transfer, and can be described as a nonlinear tensor eddy diffusivity model.It was tested in combination with the EASSM for the SGS stresses, and itsperformance was compared to the conventional dynamic eddy diffusivity model(DEDM) in channel flow with and without system rotation in the wall-normaldirection. EASSM and EASSFM gave predictions of high accuracy for meanvelocity and mean scalar fields, as well as stresses and scalar flux components.An extension of the EASSM and EASSFM, based on stochastic differentialequations of Langevin type, gave further improvements. In contrast to conventionalmodels, these extended models are able to describe intermittent transferof energy from the small, unresolved scales, to the resolved large ones.The present study shows that the EASSM/EASSFM gives a clear improvementof LES of wall-bounded flows in simple, as well as in complex geometriesin comparison with simpler SGS models. This is also shown to hold for a widerange of resolutions and is particularly accentuated for coarse resolution. The advantages are also demonstrated both for high-order numerical schemes andfor solvers using low-order finite volume methods. The models therefore havea clear potential for more applied computational fluid mechanics. / <p>QC 20140304</p> / Explicit algebraic sub-grid scale modelling for large-eddy simulations

Transit agencies seeking to improve transit service delivery are increasingly considering the deployment of transit signal priority (TSP). However, the impact of TSP on transit service and on the general traffic stream is a function of many factors, including intersection geometry, signal timings, traffic demands, TSP strategies and parameters, transit vehicle headways, timing when transit vehicles arrive at the intersection, etc. Previous studies have shown that depending on these factors, the net impact of TSP in terms of vehicle or person delay can be positive or negative. Furthermore, due to financial constraints, transit agencies are often able to deploy TSP at only a portion of all of the candidate intersections. Consequently, there is a need to estimate the impact of TSP prior to implementation in order to assist in determining at which intersections TSP should be deployed. Currently, the impacts of TSP are often estimated using microscopic simulation models. However, the application of these models is resource intensive and requires specialized expertise that is often not available in-house to transit agencies. In this thesis, an analytical model was proposed for estimating the delay impacts of green extension and early green (red truncation) TSP strategies. The proposed model is validated with analytical model reported in the literature and microscopic simulation model. This is followed by model sensitivity analysis. A software module is developed using the proposed model. The usefulness of the model is illustrated through its application to estimate the TSP performance. Finally, a prioritization is conducted on sixteen intersections with different geometric and operational traffic strategies. The overall results indicate that the proposed model is suitable for both estimating the pre-deployment and post-deployment TSP performance. The proposed model is suitable for implementation within a spreadsheet and requires considerably less effort, and less technical expertise, to apply than a typical micro-simulation model and therefore is a more suitable tool for transit agencies to use for prioritising TSP deployment.

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