Dynamics of the Stochastic Chemostat Model with Monod-Haldane Response Function

A stochastic chemostat model with an inhibitor and noise independent of population sizes

Noise-induced bifurcations in the stochastic chemostat model with general nutrient uptake functions

In this paper, we formulate and investigate a two‐microorganism and single nutrient chemostat model with Monod‐Haldane response function and random perturbation. First, for the corresponding deterministic system, we introduce the conditions of the stability of the equilibrium points. Then, using Lyapunov function and Itô’s formula, we investigate the existence and uniqueness of the global positive solution of the stochastic chemostat model. Furthermore, we explore and obtain the criterions of the extinction and the permanence for the stochastic model. Finally, numerical simulations are carried out to illustrate our main results.

In this paper, a stochastic chemostat model with variable yield and Contois growth function is investigated. The yield coefficient depends on the limiting nutrient, and the environmental noises are given by independent standard Brownian motions. First, the existence and uniqueness of global positive solution are proved. Second, by using stochastic Lyapunov function, Itô’s formula, and some important inequalities, stochastic characteristics for the stochastic model are studied, including the extinction of micro-organism, the strong persistence in the mean of micro-organism and, the existence of a unique stationary distribution of the stochastic model. Third, the necessary condition of an optimal stochastic control for the stochastic model is established by Hamiltonian function. In addition, some numerical simulations are carried out to illustrate the theoretical results and the influence of the variable yield on the microorganism.

Stochastic characteristics of a chemostat model with variable yield

ABSTRACTIn this paper, we analyse a stochastic chemostat model with distributed delay and degenerate diffusion. We transform the stochastic model with weak kernel case into an equivalent system through the linear chain technique. Since the diffusion matrix is degenerate, the uniform ellipticity condition is not satisfied. The Markov semigroup theory is used to obtain the existence and uniqueness of a stable stationary distribution. We prove the densities of the distributions of the positive solutions can converge in to an invariant density. The existence of a stable stationary distribution implies stochastic persistence of the microorganism.

In this paper, we propose a fractional stochastic volatility jump-diffusion model which extends the Bates(1996) model. Where we model the volatility as a fractional process. Extensive empirical studies show that the distributions of the logarithmic returns of financial asset usually exhibit properties of self-similarity and long-range dependence and since the fractional Brownian motion has these two important properties, it has the ability to capture the behavior of underlying asset price. Further incorporating jumps into the stochastic volatility framework gives further freedom to financial mathematicians to fit both the short and long end of the implied volatility surface. We propose a stochastic model which contains both fractional and jump process. Then we price options using Monte Carlo simulations along with a variance reduction technique(antithetic variates). We use market data from the S&P 500 index and we compare our results with the Heston and Bates model using error measures. The results show our model greatly outperforms previous models in terms of estimation accuracy.

Stochastic wave models play a central role in our present-day wave modelling capabilities. They are frequently used to compute wave statistics, to generate boundary conditions and to include wave effects in coupled model systems. Historically, such models were developed to predict the wave field evolution in deep water where the conditions of Gaussianity generally hold. However, in recent decades, such models have been applied to the shallower coastal environment where the stochastic representation of the dominant wave physics becomes questionable. This is primarily due to the increased influence of wave nonlinearity and the additional depth-induced wave processes that are dominant in this region.<br/><br/>Unfortunately, the two most dominant wave processes in the surf zone: depth-induced wave breaking and nonlinear triad wave-wave interactions are also the least well represented and understood. This is due to both their complexity and the scarcity of analytical solutions for realistic wave fields. As such, they represent a significant obstacle in the accurate modelling of the wave dynamics in the coastal region. Providing accurate representations of these wave processes is essential to answering the questions demanded from stochastic wave models from coastal engineers for coastal management and design. Such advancements are necessary to improve our understanding of wave-induced processes, to reduce costs in managing the coastal environment and to tackle contemporary issues such as uncertainties with respect to increased sea level rise.<br/><br/>Due to the complexity of depth-induced wave breaking, a complete representation of this wave process does not exist for both stochastic and deterministic modelling frameworks. Although there is extensive literature on the subject of parameterizing depth-induced wave breaking in a stochastic sense, these parameterizations are inconsistent with theory, observations and (deterministic) model predictions. In particular, present-day modelling defaults perform poorly over (near-)horizontal bathymetries with over-enhanced wave dissipation of locally-generated waves and insufficient dissipation of swell waves. Equally, nonlinear triad wave-wave interactions are poorly represented in stochastic wave models due to the problem of closure and the impractical computational expense of more accurate representations. In particular, the most commonly applied parameterization in the wave literature incorrectly predicts the evolution of the spectral shape, and the convergence to an equilibrium high-frequency tail deep in the surf zone. Correctly resolving these issues is essential for the management of many of the activities occurring at the coast; from the design of coastal defenses to feasibility studies for wave energy converters, from port operation and availability to vessel navigation, from understanding the ecology at the coast to the fisheries, and from managing leisure and tourism to safety at the coast.<br/><br/> In this work, we investigate the process of depth-induced wave breaking through a comprehensive analysis of the literature and a comparison of modelling performance. Here, we use an extensive set of wave observations representing a large range of wave conditions and bathymetric profiles. The analysis demonstrates that no currently available depth-induced breaking source term is capable of sufficiently representing the process of depth-induced wave breaking. This is shown to be in agreement with the wave literature with parameterizations either over-predicting wave dissipation for locally generated waves or under-predicting wave dissipation for non-locally generated waves over (near-)horizontal bathymetries. To address this issue, a new joint scaling using both local wave and bathymetric conditions is proposed. Using both the normalized characteristic wave number and local bottom slope unifies two approaches prevalent in the wave literature. This is shown to improve the model performance for the dissipation of both locally and non-locally generated waves over (near-)horizontal bathymetries. <br/><br/>Furthermore, the validity of the assumption that wave dissipation can be modelled as analogous to a 1D dissipative bore is explored. Subsequently, a heuristic directional modification is introduced for depth-induced wave breaking dissipation models. This directionally partitions the 2D spectrum into several directional partitions that are assumed to be unidirectional. Model results demonstrate that the effect of the directional partitioning is to reduce the dissipation of wave energy and to enhance the significant wave height; in agreement with field measurements. Not only is this modification shown to be applicable to the joint wave breaking parameterization proposed in this study, but also for well-established parameterizations. <br/><br/>The effects of both the proposed scaling and directional modification are then reviewed from an operational context and are compared to state-of-the-art source terms, field observations and a hypothetical storm representative of Dutch design conditions. Such design conditions are expected to be representative of design conditions found globally. In an environment where storm intensities may be increasing, for example due to global warming, the results of wave breaking models near the coast under such extreme conditions become of greater relevance. The influence of wave breaking models in coupled model systems is anticipated to provide important new insights in understanding the various wave-driven processes along our coasts. <br/><br/>Next, the representation of the nonlinear triad wave-wave interactions in stochastic wave models is reviewed. In particular, the collinear approximation used to transform 1D triad source terms for implementation in 2D stochastic wave models is revisited. These approximations are necessitated by considerations of computational efficiency. The conventional collinear approximation is shown to be inconsistent at the unidirectional limit and to be a primary source of modelling error. Instead of converging to the values predicted by the 1D triad source terms at the unidirectional limit, the energy transfers as computed by stochastic wave models are shown to become unbounded. This results in a dimensional calibration coefficient which is at least an order of magnitude smaller than that found in the wave literature. Consequently, for directional wave conditions, 1D triad source terms implemented with the conventional collinear approximation insufficiently capture the wave evolution. To address this problem, a new collinear approximation is presented which accounts for the wave energy contained within a finite directional bandwidth. This collinear approximation is shown to converge correctly at the unidirectional limit and to agree well with predictions from a second-order accurate deterministic wave model. In particular, better agreement is shown in the modelling prediction of the spectral shape and related integral parameters, e.g. wave period, under idealized wave conditions. Under certain conditions, these error reductions are shown to be more significant than differences between the underlying triad models.<br/><br/>The contribution of this work demonstrates that while the underlying theory underpinning stochastic wave modelling in the coastal environment still remains questionable, the accurate determination of wave statistics in the coastal zone is tenable. With the advancements presented in this study, the new source terms correspond better with the current wave literature and are shown to provide significant steps forward over existing default source terms. The developments presented here are anticipated to form the foundation for future source term research, and to be used for the representation of the dominant wave physics in the coastal environment in operational wave models.

The paper describes a stochastic radiowave propagation model useful for assessing the effects of forests and other vegetation on communications signals in the UHF band. The stochastic electromagnetic theory employed as the basis of this model considers the forest as a planar, stratified, anisotropic, discrete, random medium, bounded above by air and below by ground, of randomly positioned and oriented canonical scatterers representing the principal forest constituents (tree trunks are represented as infinitely long, parallel, circular, dielectric cylinders; branches as short ones; and leaves as flat, circular, dielectric discs). A physically appealing representation for the mean (coherent) field component of the propagating radio wave has been obtained by recognising that the ensemble of discrete scatterers can be replaced by an equivalent continuous medium described by an effective dyadic permittivity e. The associated electromagnetic boundary-value problem is solved to identify the principal contributions to the mean field: the direct wave, the reflected wave and the lateral wave. Because the equivalent continuous medium characterised by the effective dyadic permittivity e is linear, Fourier-transform techniques have been employed to generalise the model so that it accommodates arbitrarily modulated waveforms.

Finance: Longevity Risk Management for Individual Investors Optimal SP-Based Personal Financial Planning with Intermediate and Long Term Goals Intertemporal Surplus Management with Jump Risks Jump-Diffusion Risk-Sensitive Benchmarked Asset Management Dynamic Portfolio Optimization Under Regime-Based Firm Strength Options Portfolio Management as a Chance Constrained Problem Stochastic Models for Optimizing Immunization Strategies in Fixed-Income Security Portfolios Under Some Sources of Uncertainty Stochastic Programming and Optimization in Horserace Betting Production Planning and Logistics: Multi-Stage Stochastic Programming for Natural Gas Infrastructure Design with a Production Perspective Stochastic Programming Model for Optimizing the Production of Farmed Atlantic Salmon Prioritizing Network Interdiction of Nuclear Smuggling Sawmill Production Planning Under Uncertainty: Modelling and Solution Approaches Energy: An Electricity Procurement Model with Energy and Peak Charges Value of Flexibility in Hydroelectric Reservoir Operations Multi-Lag Benders Decomposition for Power Generation Planning with Nonanticipativity Constraints on the Dispatch of LNG Thermal Plants Theory Papers: Stochastic Second-Order Cone Programming in Mobile Ad-Hoc Networks Stochastic Frequency Assignment Problem.

A tropical cyclone (TC) vortex is an immense, coherent, organized-convective system. Beneath this large-scale organization, is a litany of azimuthally asymmetric convective motions that exist on a spectrum of scales. These asymmetries are especially dominant during periods when the vortex undergoes critical transitions in its intensity and structure. However, the precise nature of influence of the organization of asymmetries on TC intensity change remains an enigma. The inherent difficulty in predicting their behavior is because asymmetries may arise due to different external or intrinsic sources and occur at different spatial and temporal scales while several complex mechanisms act near-simultaneously to dictate their evolution in time. As a result, multiple pathways are possible for a TC vortex that is influenced by these asymmetries. Our preliminary investigations using numerical models made it apparent that there wasn't a single, unifying way to address this problem. In this thesis, I outline multiple novel techniques of diagnosing and predicting which of the many pathways are likely for a TC vortex that is influenced by azimuthal asymmetries. First, using three-dimensional numerical simulations of a pair of sheared and non-sheared vortices, I demonstrate the diagnostic potential of the juxtaposition in the azimuthal phasing of: (i) the asymmetrically distributed vertical eddy flux of moist-entropy across the top of the boundary layer, and the radial eddy flux of moist-entropy within the boundary layer; and (ii) eddy relative vorticity, eddy moist-entropy, and vertical velocity throughout the depth of the vortex. Second, I introduce an energetics-based diagnostic framework that computes the energy transactions occurring at asymmetries across various length-scales in the wavenumber domain. By applying it to select cases, this thesis uncovers the relative importance of all the energy pathways that support or disrupt the growth of asymmetries within the vortex. Contrary to the traditional explanations of convective aggregation/disaggregation and axi/asymmetrization through barotropic mean-eddy transactions, my thesis reveals that the growth or disruption of asymmetries are predominantly due to (i) the baroclinic conversion from available potential to kinetic energy at individual scales of asymmetries and (ii) the transactions of kinetic energy across asymmetries of different length scales. Finally, this thesis introduces two further diagnostic frameworks targeted at tackling the problem of real-time forecasting of TC rapid intensity changes. The first is an empirical framework which examines symmetric and asymmetric convection and other state variables within the vortex, and in the environment across a suite of TCs and identifies a set of `important' variables that are significantly different during time periods that precede a rapid intensification as opposed to a rapid weakening. My framework then ranks the variables identified based on how significantly they influence a rapid intensity change in a TC and the amplification factor of any associated variability. We recommend that future observational, and consequent TC modeling and data assimilation efforts prioritize the highest ranked variables identified here. The second is a stochastic model wherein a scale-specific stochastic term is added to the equations describing the energy transactions within the TC vortex. By simulating a stochastic forcing that may arise from any scale, I compute the probability of the vortex transitioning into a rapidly intensifying or a rapidly weakening configuration across an ensemble of scenarios. In summary, this thesis introduces and applies a variety of diagnostic techniques that help determine the impact of azimuthal asymmetries on TC intensity evolution.

Increasing attention has been focused on the analysis of the realized volatility, which can be treated as a proxy for the true volatility. In this paper, we study the potential use of the realized volatility as a proxy in a stochastic volatility model estimation. We estimate the leveraged stochastic volatility model using the realized volatility computed from five popular methods across six sampling-frequency transaction data (from 1-min to 60- min) based on the trust region method. Availability of the realized volatility allows us to estimate the model parameters via the MLE and thus avoids computational challenge in the high dimensional integration. Six stock indices are considered in the empirical investigation. We discover some consistent findings and interesting patterns from the empirical results. In general, the significant leverage effect is consistently detected at each sampling frequency and the volatility persistence becomes weaker at the lower sampling frequency.

The present review discusses recent developments in numerical techniques for the solution of systems with stochastic uncertainties. Such systems are modelled by stochastic partial differential equations (SPDEs), and techniques for their discretisation by stochastic finite elements (SFEM) are reviewed. Also, short overviews of related fields are given, e.g. of mathematical properties of random fields and SPDEs and of techniques for high-dimensional integration. After a summary of aspects of stochastic analysis, models and representations of random variables are presented. Then mathematical theories for SPDEs with stochastic operator are reviewed. Discretisation-techniques for random fields and for SPDEs are summarised and solvers for the resulting discretisations are reviewed, where the main focus lies on series expansions in the stochastic dimensions with an emphasis on Galerkin-schemes.

The spatial variability of geotechnical properties can lead to the uncertainty of settlement for frozen soil foundation around the oil pipeline, and it can affect the stability of permafrost foundation. In this paper, the elastic modulus, cohesion, angle of internal friction and poisson ratio are taken as four independent random fields. A stochastic analysis model for the uncertain settlement characteristic of frozen soil foundation around an oil pipeline is presented. The accuracy of the stochastic analysis model is verified by measured data. Considering the different combinations for the coefficient of variation and scale of fluctuation, the influences of spatial variability of geotechnical properties on uncertain settlement are estimated. The results show that the stochastic effects between elastic modulus, cohesion, angle of internal friction and poisson ratio are obviously different. The deformation parameters have a greater influence on stochastic settlement than the strength parameters. The overall variability of settlement reduces with the increase of horizontal scale of fluctuation and vertical scale of fluctuation. These results can improve our understanding of the influences of spatial variability of geotechnical properties on uncertain settlement and provide a theoretical basis for the reliability analysis of pipeline engineering in permafrost regions.

Although there have been numerous studies on the economics of weed control, relatively few of these studies have focused on natural ecosystems. The aims of this papers are: A) to identify the combination of control options that is optimal for blackberry (Rubus anglocandicans) in Australian natural ecosystems, B) to assess whether an integrated control strategy is superior to chemical-only strategies, C) to evaluate the economic net benefits of specific biologically oriented control methods (a rust and grazing by goats), D) to determine how changes in model parameters affect the optimal control strategy. To address these aims, a stochastic dynamic simulation model and a stochastic dynamic programming model are developed. The results indicate that, while an integrated strategy combining chemical and non-chemical control methods may be optimal in certain circumstances, it is not necessarily superior to a chemical-dominant strategy in all cases. Results show that in most cases a combination of herbicides and one physical control method (mowing) is optimal in the base-case scenario. However, if chemical efficacy is greater than in the base-case, the optimal strategy involves chemicals only. The results indicate that grazing goats for control of blackberry can be optimal despite its stochastic effectiveness and relatively low efficacy.