Staff Publications

Staff Publications

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    'Staff publications' is the digital repository of Wageningen University & Research

    'Staff publications' contains references to publications authored by Wageningen University staff from 1976 onward.

    Publications authored by the staff of the Research Institutes are available from 1995 onwards.

    Full text documents are added when available. The database is updated daily and currently holds about 240,000 items, of which 72,000 in open access.

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The uptake of carbon sources by Aspergillus niger
Sloothaak, Jasper - \ 2017
University. Promotor(en): Vitor Martins dos Santos, co-promotor(en): Peter Schaap; Juan Tamayo Ramos. - Wageningen : Wageningen University - ISBN 9789463432085 - 179
aspergillus niger - carbon - proteomes - glucose - cell membranes - markov processes - organic acids - koolstof - proteomen - celmembranen - markov-processen - organische zuren

Fungi have been used as food and in food fermentations long before written accounts were created and they have been used in folk medicine in ancient cultures. For centuries, species of the genus Aspergillus have been used for the preparation of traditional Asian foodstuffs or together with baker’s yeast in preparation of alcoholic beverages and have therefore been of great economical value. Later, Aspergillus niger has been used for large-scale production of organic acids, enzymes and other food-additives. Today, we aim to harness its saprophytic nature and extraordinary ability to degrade and utilize plant material that is naturally recalcitrant to degradation. To facilitate that ability, the range of sugar transporters employed by this fungus is large even among fungi. This makes it an excellent choice for the identification and characterization of a variety of proteins with different substrate specificities, with potential application in the design of newly engineered cell factories. This thesis was focused on the identification and characterization of previously unknown sugar uptake transporters. Aspergillus niger transporter proteins for the uptake of glucose, xylose, galacturonic acid and rhamnose were identified and characterized.

Bayesian Markov random field analysis for integrated network-based protein function prediction
Kourmpetis, Y.I.A. - \ 2011
University. Promotor(en): Cajo ter Braak, co-promotor(en): Roeland van Ham. - [S.l.] : S.n. - ISBN 9789085859598 - 113
statistiek - bayesiaanse theorie - markov-processen - netwerkanalyse - biostatistiek - toegepaste statistiek - bio-informatica - eiwitten - genen - moleculaire biologie - statistics - bayesian theory - markov processes - network analysis - biostatistics - applied statistics - bioinformatics - proteins - genes - molecular biology

Unravelling the functions of proteins is one of the most important aims of modern biology. Experimental inference of protein function is expensive and not scalable to large datasets. In this thesis a probabilistic method for protein function prediction is presented that integrates different types of data such as sequences and networks. The method is based on Bayesian Markov Random Field (BMRF) analysis. BMRF was initially applied to genome wide protein function prediction using network data in yeast and in also in Arabidopsis by integrating protein domains (i.e InterPro signatures), expressions and protein protein interactions. Several of the predictions were confirmed by experimental evidence. Further, an evolutionary discrete optimization algorithm is presented that integrates function predictions from different Gene Ontology (GO) terms to a single prediction that is consistent to the True Path Rule as imposed by the GO Directed Acyclic Graph. This integration leads to predictions that are easy to be interpreted. Evaluation of of this algorithm using Arabidopsis data showed that the prediction performance is improved, compared to single GO term predictions.

Solving large structured Markov Decision Problems for perishable inventory management and traffic control
Haijema, R. - \ 2008
University of Amsterdam. Promotor(en): J. van der Wal; N.M. van Dijk. - Amsterdam : Thela Thesis - ISBN 9789036101011 - 357
optimalisatie - markov-processen - dynamisch programmeren - inventarisaties - controle - voorraden - bederfelijke producten - bloed - verkeersgeleiding - optimization - markov processes - dynamic programming - inventories - control - stocks - perishable products - blood - controlled traffic
Flexible decision-making in crisis events : discovering real options in the control of foot-and-mouth disease epidemics
Ge, L. - \ 2008
University. Promotor(en): Ruud Huirne, co-promotor(en): A.R. Kristensen; Monique Mourits. - [S.l.] : S.n. - ISBN 9789085049692 - 149
crises - mond- en klauwzeer - epidemieën - besluitvorming - ziektebestrijding - markov-processen - onzekerheid - dynamisch programmeren - bayesiaanse theorie - dynamische modellen - bedrijfseconomie - beslissingsondersteunende systemen - beslissingsmodellen - foot and mouth disease - epidemics - decision making - disease control - markov processes - uncertainty - dynamic programming - bayesian theory - dynamic models - business economics - decision support systems - decision models

Crisis event, foot-and-mouth disease (FMD), epidemic control, real options, decision flexibility, multi-level hierarchic Markov process (MLHMP), uncertainty, decision-support framework, turning moment, dynamic programming, Bayesian forecasting, dynamic models, overreacting, underreacting

This research introduced the real options way of thinking into decision-making in crisis events like animal epidemics, with foot-and-mouth disease (FMD) as a case in point. A unique angle was taken to investigate decision flexibility in choosing optimal control strategies. The main objective was to develop a flexible decision-support framework which corresponds to practice and provides consistent treatment of ongoing uncertainty in controlling animal epidemics. Conceptualisation and operationalisation of decision flexibility were the two main focuses.
A decision analysis revealed the dynamic and sequential nature of decision- making in the control of animal epidemics. The importance of decision flexibility was attributed to the existence of uncertainty and linked decisions in the multi-stage decision process. Timing of control options and the possibility of learning were found to be essential in conceptualising decision flexibility. To operationalise decision flexibility, the main methodological approach was the integration of multi-level hierarchic Markov process (MLHMP) and Bayesian forecasting methods. Based on MLHMP and dynamic generalised linear models (DGLM), a new decision-support framework was developed to investigate the impact of uncertainty and the possibility of learning in choosing the optimal timing of control options over time. The framework paid special attention to the interdependency among strategic, tactical, and operational decisions in managing FMD epidemics. The decision-support framework was shown to be useful in contingency planning for future epidemics.
Addressing the decision flexibility in a dynamic decision process, real options analysis and MLHMP were found to be complementary in developing the flexible decision-support framework. Both required dynamic assessment of future epidemic development and control options. Towards empirical application of the decision-support framework, an integrated epidemic-economic modelling approach was described and illustrated with simulated epidemics. It was shown that, by including decision flexibility in the dynamic decision process of epidemic control, the new modelling approach enabled more realistic estimation of the costs of underreacting or overreacting than the traditional static approaches.

Bayesian statistics for infection experiments
Heres, L. ; Engel, B. - \ 2004
In: Bayesian Statistics and Quality Modelling in the Agro-Food Production Chain / Boekel, van, Stein, A., Bruggen, van, Dordrecht : Kluwer (Wageningen UR Frontis series vol. 3) - ISBN 1402019165 - p. 131 - 139.
bayesiaanse theorie - markov-processen - monte carlo-methode - ziekten overgebracht door voedsel - pathogenen - pluimvee - epidemiologie - bayesian theory - markov processes - monte carlo method - foodborne diseases - pathogens - poultry - epidemiology
To intervene cycles of food-borne pathogens in poultry new intervention methods need to be tested for their effectiveness. In this paper a statistical method is described that was applied to quantify the observed differences between test groups and control groups. Treated chickens and their controls were inoculated with several doses and were daily examined for the shedding of the tested pathogens. For these infection experiments with individually housed chickens and where binary data were available for each individual chicken a Bayesian analysis employing Markov Chain Monte Carlo (MCMC) was applied for the statistical analyses. The Cox’ proportional hazard reflected the typical features of the data, i. e. dependency, waitingtime structure and censoring. The outcomes of the analyses are two measures of difference in susceptibility between the feed groups. The first effect measure is a relative risk of being infected. The second is a difference in waiting time or a difference in inoculation dose to get a comparable proportion of infected animals
Genetic algorithms and Markov Chain Monte Carlo: Differential Evolution Markov Chain makes Bayesian computing easy
Braak, C.J.F. ter - \ 2004
Wageningen : Biometris (Biometris ) - 14
monte carlo-methode - algoritmen - markov-processen - simulatie - optimalisatiemethoden - monte carlo method - algorithms - markov processes - simulation - optimization methods
Differential Evolution (DE) is a simple genetic algorithm for numerical optimization in real parameter spaces. In a statistical context one would not just want the optimum but also its uncertainty. The uncertainty distribution can be obtained by a Bayesian analysis (after specifying prior and likelihood) using Markov Chain Monte Carlo (MCMC) simulation. In this paper the essential ideas of DE and MCMC are integrated into Differential Evolution Markov Chain (DE-MC). DE-MC is a population MCMC algorithm, in which multiple chains are run in parallel. DE-MC solves an important problem in MCMC, namely that of choosing an appropriate scale and orientation for the jumping distribution. In DE-MC the jumps are simply a multiple of the differences of two random parameter vectors that are currently in the population. Simulations and examples illustrate the potential of DE-MC. The advantage of DE-MC over conventional MCMC are simplicity, speed of calculation and convergence, even for nearly collinear parameters and multimodal densities
Computer simulation to support policy-making in Aujeszky's disease control
Buijtels, J.A.A.M. - \ 1997
Agricultural University. Promotor(en): A.A. Dijkhuizen; R.B.M. Huirne; Mart de Jong. - S.l. : Buijtels - ISBN 9789054856528 - 187
ziekte van marek - ziekte van aujeszky - varkens - diergeneeskunde - vaccinatie - immunisatie - immunotherapie - vaccins - investering - kosten-batenanalyse - economische evaluatie - computersimulatie - simulatie - simulatiemodellen - dynamisch programmeren - markov-processen - economie - gebruikswaarde - economische impact - nederland - marek's disease - aujeszky's disease - pigs - veterinary science - vaccination - immunization - immunotherapy - vaccines - investment - cost benefit analysis - economic evaluation - computer simulation - simulation - simulation models - dynamic programming - markov processes - economics - use value - economic impact - netherlands
<p>Aujeszky's disease is a contagious viral disease that affects the central nervous system of pigs. Several eradication programs or measures are available, each of them providing different results. Determining the preferred strategy is to a large extent a matter of economic consideration.<p>Under the EU rules, countries or regions that are Aujeszky-free can ban imports of breeding animals carrying antibodies of the disease; movements to Aujeszky-free areas from other areas of both breeding and rearing pigs are subject to strict conditions and controls, which differ depending on whether or not the area of origin has an EU-approved eradication program. If important import destinations achieve disease-free status, exporting countries that have failed to eradicate the disease will be severely penalized. Therefore, sterner demands are to be expected considering control and eradication of Aujeszky's disease in the Netherlands in the future. To meet these demands, the objective of this study was to develop a computer simulation environment in which "what-if' scenarios can be performed to explore the epidemiological and economic effects of different Aujeszky's disease control programs. The model can be used to support the choice of the optimal eradication program under various conditions, in particular from an epidemiological and economic point of view.<p>First, a flexible economic framework to evaluate Aujeszky's disease eradication programs was developed, and illustrated with an example (Chapter 2). The framework has four elements: changes in percentage of infectious herds, changes in product quantities, changes in product prices and economic integration. Each of these elements is defined as a separate module in the simulation model and has its own input and output data, depending on the control strategy under consideration. With these elements all epidemiological and economic aspects of the disease can be monitored over time.<p>In an illustrative example, probability distributions of the number of infectious herds corresponding to each control strategy were compared and the optimal strategy was chosen, according to the risk attitude of the decision maker. The framework can be considered a standardized approach in comparing and selecting animal health control strategies by integrating technical and economic data and principles.<p>To obtain epidemiological information with respect to the control of Aujeszky's disease virus, an epidemiological state-transition simulation model was constructed to evaluate the spread of the virus (Chapter 3). In the model, the population of herds in the Netherlands is subdivided into four herd types: great-grandparent stock+multiplier, rearing, farrowing and fattening. Every time step, each herd is in one of 32 states per herd type. The states are based on (1) the reproduction ratio R <sub>ind</sub> , which is the number of individuals infected by one infectious individual, (2) the prevalence for each value of R <sub>ind </sub> and (3) the expected number of infectious animals in an infectious herd within each prevalence range of the herds. The different values Of R <sub>ind</sub> are based as much as possible on field data and experiments, where different vaccination strategies were applied.<p>The transition matrix with the probabilities of every possible transition from one state to another was calculated on a weekly base. With this matrix the distribution of herds over states from week to week was derived. To include the non-linearity of the transmission process, the transmission probabilities from non-infectious to either non-infectious or infectious were developed such that they depend on the state vector itself The fraction of herds that becomes infectious equals one minus the fraction of herds that has not been infected by the virus emitted by infectious herds.<p>Calculations revealed that infection in the Dutch pig population would not disappear without vaccination, nor with a vaccination scheme in which sows were vaccinated less than 3 times per year and fattening pigs once per cycle (Chapter 4). The infection, however, would be eradicated within 2 to 3 years, if sows were vaccinated 3 or 4 times per year and fattening pigs twice per cycle. The outcome turned out to be sensitive to the impact of other than animal contacts on the number of new effective virus introductions per time unit.<p>The structure of the production pyramid and herd density in the affected regions were other important factors which influenced the course of infection. To examine the impact of these factors the total number of herds in the Netherlands were further subdivided into four regions (North, East, West-Middle, South).<p>Outcomes showed that the percentage of infectious herds in equilibrium was highest for rearing herds (76.3%) and lowest for great-grandparent stock+multiplier herds (20.0%) if no vaccination was done. The herd type "fattening" had more impact on the effectivity of the different vaccination strategies than the herd type "farrowing". This difference is becoming less if more intensive vaccination strategies are applied. Besides the difference in herd type, also herd density and the percentage of non-vaccinated herds were an important factor in the eradication process.<p>After simulating these epidemiological characteristics of Aujeszky's disease virus, market outcomes and pig producers' returns were simulated under different scenarios with respect to closure of export markets for live piglets and fattened pigs (Chapter 5). If the Netherlands fails to eradicate Aujeszky's disease before its trading partners in these markets, live piglet exports would be banned, reducing industry revenue and export earnings by about 9% and 10% respectively in the medium term. If exports of live fattened pigs are also banned, the reductions are 26 and 32% respectively. The piglet-producing sector would be more<br/>severely affected than the fattening sector. The model also showed that, if export markets for carcasses were also to close for an unspecified food safety reason, capacity of the industry would fall over 50%.<p>Lastly four control strategies to eradicate Aujeszky's disease virus in the Netherlands were compared epidemiologically and economically (Chapter 6). Vaccination decreased the number of cases per production loss. The decrease was largest if vaccination strategy changed from "no vaccination" to the less intensive vaccination. Extra vaccinations under more intensive vaccination strategies, however, still had impact. The attendant costs were highest per dead animal (especially for gilts) and per abortion. Growth delay of gilts and piglets turned out to be of minor importance.<p>The sales distribution on the piglet markets (import, export and on the domestic markets) was particularly influenced by vaccination, but the decreases in revenues were only less than 4.3%. The only exception was the number of piglets and live animals that were imported into the Netherlands, which decreased by more than 15% and about 9% respectively. The accompanying revenues from piglets and fattened pigs were highest if "no vaccination" was done. Compared with the revenue in this strategy, this difference is greatest on the piglet market, as the decrease in revenue was about 3.6%, while the decrease was about 0.55% on the market of fattened pigs.<p>According to the resulting present values over a period of 10 years, "no vaccination" is economically the best solution only if no trade restrictions are to be expected. Economically speaking, however, the most intensive vaccination strategy should be applied, if an export ban of two years on live animals to, for instance, Germany is expected within 10 years after the start of the vaccination strategy. A prolonged export ban makes this strategy even more favourable. From an economic point of view intermediate vaccination strategies are never preferred.<p>The main conclusions of this thesis are:<br/>- State-transition simulation proves to be an appropriate method to evaluate transmission of Aujeszky's disease virus. The epidemiological information obtained can well be used in economic evaluation of different control strategies.<br/>- Aujeszky's disease is only eradicated in the Netherlands if the most intensive vaccination strategy (≥3 times per year) is applied for breeding sows, and fattening pigs are vaccinated at least once per cycle.<br/>- If applying the most intensive vaccination strategy, it takes about 200 weeks for an average herd to become non-infectious.<br/>- The relative impact of other than animal contacts on the number of new virus introductions increases from 4% to 98%, if the vaccination strategy is changed from "no vaccination" to the most intensive vaccination program.<br/>- Subdivision of the total population into herd types and regions is important to enhance insight into transmission of infection in the pig population and to support decision making at regional level.<br/>- Price equilibrium models can well estimate the short-term changes in prices as well as those in the medium term. To accomplish this, it is of importance that sufficient historical data about quantities, prices and the infections occurred are available to estimate the required parameters accurately. A monthly data-set of about 10 years turned out to be sufficient.<br/>- Direct production losses from Aujeszky's disease virus are less than 6% of the vaccination costs when vaccination is carried out. More than 80% of these losses are caused by growth delay of fattening pigs.<br/>- The most intensive vaccination strategy (i.e., sows are vaccinated 3 or 4 times per year and fattening and rearing pigs twice per cycle) is economically preferred if an export ban on part of the live animals is expected during at least 2 years within 10 years after the start of the vaccination program. If this is not to be expected then "no vaccination" turns out to be the best strategy. The risk of an export ban on live animals should justify the eradication of the virus from the population.<br/>- For the current situation in the Netherlands it is economically preferred to start blood sampling all sows and remove the gE-positive animals instead of continuing vaccination, provided that the additional risk of new introductions of the virus is sufficiently limited.
Application of stochastic dynamic programming models in optimization of reservoir operations : a study of algorithmic aspects
He, Q. ; Nandalal, K.D.W. ; Bogardi, J.J.K.M. - \ 1995
Wageningen : Landbouwuniversiteit Wageningen (Rapport / Landbouwuniversiteit, Vakgroep Waterhuishouding 56) - 135
dynamisch programmeren - meren - markov-processen - reservoirs - wateropslag - dynamic programming - lakes - markov processes - water storage
Application of dynamic programming for the analysis of complex water resources systems : a case study on the Mahaweli River basin development in Sri Lanka
Kularathna, M.D.U.P. - \ 1992
Agricultural University. Promotor(en): J.J. Bogardi; P. van Beek. - S.l. : Kularathna - 163
watergebruik - schadepreventie - irrigatie - watervoorziening - waterbeheer - watervoorraden - dynamisch programmeren - markov-processen - sri lanka - modellen - onderzoek - hydrologie - water use - loss prevention - irrigation - water supply - water management - water resources - dynamic programming - markov processes - models - research - hydrology
<p>The technique of Stochastic Dynamic Programming (SDP) is ideally suited for operation policy analyses of water resources systems. However SDP has a major drawback which is appropriately termed as its "curse of dimensionality".<p>Aggregation/Disaggregation techniques based on SDP and simulation are presented to analyze a complex water resources system. The system under consideration serves two major purposes: hydropower generation and irrigation. The identification of subsystems by their functional and physical characteristics was an important first step in the analysis. Subsequently each subsystem is represented by a hypothetical composite reservoir to arrive at an operation policy for the interface point of the subsystems. A more detailed analysis which considers the real configurations of the subsystems is performed by following this operation policy of the interface point. Two approaches: sequential optimization and iterative optimization are presented. In these approaches, each subsystem is individually analyzed using two-reservoir SDP models.<p>The applicability of an Implicit Stochastic Approach in which the operation of the system is optimized for a number of deterministic hydrologic data series is also investigated. To complement the aggregation technique of the Composite Reservoir, subsequent disaggregation techniques are proposed. Three different techniques: (1) A statistical disaggregation, (2) An optimization/simulation-based technique, and (3) The disaggregation of the composite policy in the actual operation by incorporating a single-time-step optimization are tested.<p>The accuracy of the sequential and iterative optimization approaches are evaluated by applying them to a subsystem of three reservoirs in a cascade for which the deterministic optimum pattern is also determined by an Incremental Dynamic Programming (IDP) model. In the case of the Implicit Stochastic Approach, the results are compared with the results of the explicit SDP approach and the deterministic optimum operation pattern, in addition to the historical operation pattern of the system. The results of the Composite Policy Disaggregation techniques are compared to the results obtained by real multireservoir optimizations carried out by the use of explicit SDP models.
Optimaliseringstechnieken: principes en toepassingen. Derde geheel herziene druk.
Hendriks, T.H.B. ; Beek, P. van; Leijster, W. de - \ 1991
Houten : Bohn Stafleu van Loghum - ISBN 9789031312412 - 348
optimalisatie - programmeren - matrices - lineair programmeren - operationeel onderzoek - transport - verkeer - wiskundige modellen - dynamisch programmeren - markov-processen - computer software - computerwiskunde - lineaire algebra - vectorruimten - veeltermen - logistiek - queuing theory - niet-lineair programmeren - optimization - programming - linear programming - operations research - traffic - mathematical models - dynamic programming - markov processes - computational mathematics - linear algebra - vector spaces - polynomials - logistics - nonlinear programming
Analysis of the exit problem for randomly perturbed dynamical systems in applications
Roozen, H. - \ 1990
Agricultural University. Promotor(en): J. Grasman. - S.l. : Roozen - 156
wiskundige modellen - theorie - stochastische processen - dynamisch programmeren - markov-processen - mathematical models - theory - stochastic processes - dynamic programming - markov processes
<p>In the preface of his book entitled 'Theory and applications of stochastic differential equations', Z. Schuss (1980) noticed a gap between the theory of stochastic differential equations and its applications. In addition to the work of Schuss and many others in the field, the present work aims at narrowing this gap.<p>This thesis deals with randomly perturbed dynamical systems. Such systems frequently arise in the modelling of phenomena in biology, mechanics, chemistry, and physics. In some cases random perturbations form a minor aspect of the problem under study. Then a deterministic description can be used. In the present work the behaviour of the dynamical systems depends essentially on the random perturbations. We encounter systems with so-called 'diffusion across the flow' (Chapters 1,2) and systems with 'diffusion against the flow' (Chapters 1,3-5). The stability of equilibria of these systems (and thus, the lifetime, reliability of these systems) is affected by random perturbations.<p>In the study of so-called 'exit problems' we consider a domain in the state space of the dynamical system and try to compute statistical quantities related to escape from this domain, such as the probability density function of the exit-time, the probability density function of exit points on the boundary of the domain (or, less ambitiously, the first few statistical moments of these densities: mean, variance, etc.). The expectation value of the exit-time can be used to express the stochastic stability of the system.<p>We speak of randomly <em>perturbed</em> dynamical <em></em> systems, so we assume that the stochastic fluctuations are small. This is often a realistic assumption. To derive expressions for the statistical quantities mentioned above, we employ asymptotics where the small parameter is related to the intensity of the random perturbations. The asymptotic method used in Chapter 2 is well-established. The asymptotics in Chapters 3-6 are of a formal character. The asymptotic analysis is performed to the lowest order necessary to incorporate the essential effects. In view of the complexity of this simplest approach, we did not carry out higher order calculations.<p>The first chapter forms an introduction to some important topics in the theory of exit problems. We discuss the relevant (initial-) boundary-value problems, the classification of boundaries of domains of stochastic dynamical systems, and we give elementary examples of systems with 'diffusion across the flow' and 'diffusion against the flow' and their asymptotic solution. This chapter facilitates access to literature on exit problems and to the remaining chapters of this thesis. A more detailed treatment of the topics touched upon in this chapter is found in the cited literature.<p>Chapter 2 is concerned with the dynamics of a loaded stiff rod. The load consists of a deterministic part and a small stochastic part. An accumulation of stochastic load fluctuations may drive the energy of the rod across some critical level. The expectation value of the time to reach this critical energy level is a measure for the reliability of the system that contains the rod. According to the directions in which the loads act, various cases are distinguished. We derive expressions for the expectation value of the exit-time and (for some of the cases) of a number of other statistical quantities, as the exit-time density, its moments and cumulants and the probability density function of the square root of the energy (as a function of time). We use an asymptotic method known as the averaging technique. As a matter of fact, the model is a randomly loaded slightly damped oscillator. Since many practical systems near equilibrium behave essentially like a slightly damped oscillator, the. results obtained may be expected to have a wide range of application.<p>In Chapter 3 we study the exit problem for a stochastic dynamical system of interacting biological populations. Exit from the domain (the positive orthant) corresponds with extinction of a population. We start with a birth and death process, having a discrete state space, and subsequently formulate an 'approximate' Fokker- Planck (or forward Kolmogorov) equation in a continuous state space. It is assumed that the deterministic system associated with the stochastic dynamical system has a point attractor in the positive orthant. The biological system will remain for some (probably long) time in a neighbourhood of the attracting point, but after a (rare) succession of random fluctuations, one of the populations will get extinct. Determining the expected time of exit (of whichever of the populations), and of which population will probably get extinct first, requires the numerical solution of a system of so-called 'ray equations' (obtained from the Fokker-Planck equation by the WKB-method). In literature these differential equations are provided with initial conditions, which entails difficulties in the numerical construction of contours in the state space on which the eikonal function attains a constant value (confidence contours). We define boundary conditions instead of initial conditions and thereby resolve these difficulties. The ideas are illustrated by a two-dimensional generalized Lotka-Volterra model. This model allows a nice demonstration of the concepts of deterministic stability and stochastic stability. Numerically constructed confidence contours are shown for predator-prey, mutualism and competition variants of the model. We carry out numerical simulations of birth- death processes to check the results.<p>A discussion of various ways of numerical solution of the system of ray equations is found in Chapter 4. In particular we explain the boundary-value method referred to above. Moreover we give some details on the numerical construction of rays and confidence contours. At the end we present an example with intersecting rays. This phenomenon is investigated analytically in Chapter 6.<p>In Chapter 5 we are concerned again with a stochastic version of the two- dimensional generalized Lotka-Volterra model. The approach differs from that in Chapter 3 in that now we pay attention to what happens near the boundaries of the domain (the positive coordinate axes). The main difficulty is caused by the fact that the normal components of both the drift and the diffusion coefficients vanish near the boundaries, as linear functions of the distance to the boundaries. To obtain expressions for the statistical quantities of interest, we generalize a method of other authors in the study of a similar one-dimensional problem. The asymptotic expressions contain some unknown constants, that can be obtained numerically. Explicit calculations are carried out for a predator-prey system as an example,<p>Applying the WKB-method to the forward Kolmogorov equation, we obtain the ray equations. In the solution of the ray equations one sometimes observes intersecting rays forming caustic surfaces. This phenomenon is studied in Chapter 6. Near locations of intersecting rays, the WKB-approximation does not hold. We derive a uniform asymptotic expansion in terms of new canonical integrals whose validity extends over regions containing caustics. We start with the simple case of a cusp arising in a diffusion problem for which explicit results can be obtained. Subsequently, we generalize it to a formal approach to singularities arising in the forward Kolmogorov equation.<br/>The text of each of the chapters has appeared as a report or a publication in a scientific journal.
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