Conceptual Mathematical Model Research Articles

Cerebral venous blood outflow: a theoretical model based on laboratory simulation.

The cerebrovascular bed and cerebrospinal fluid circulation have been modeled extensively except for the cerebral venous outflow, which is the object of this study. A hydraulic experiment was designed for perfusion of a collapsible tube in a pressurized chamber to simulate the venous outflow from the cranial cavity. The laboratory measurements demonstrate that the majority of change in venous flow can be attributed to either inflow pressure when the outflow is open, or the upstream transmural pressure when outflow is collapsed. On this basis, we propose a mathematical model for pressure distribution along the venous outflow pathway depending on cerebral blood flow and intracranial pressure. The model explains the physiological strong coupling between intracranial pressure and venous pressure in the bridging veins, and we discuss the limits of applicability of the Starling resistor formula to the venous flow rates. The model provides a complementary explanation for ventricular collapse and origin of subdural hematomas resulting from overshunting in hydrocephalus. The noncontinuous pressure flow characteristic of the venous outflow is pinpointed as a possible source of the spontaneous generation of intracranial slow waves. A new conceptual mathematical model can be used to explain the relationship between pressures and flow at the venous outflow from the cranium.

Chaos and regular dynamics in model multi-habitat plankton-fish communities

This work is focused on the role of diffusive interaction between separate habitats in a patchy environment in plankton pattern formation. We demonstrate that conceptual reaction-diffusion mathematical models constitute an appropriate tool for searching and understanding basic mechanisms of plankton pattern formation and complex spatio-temporal plankton dynamics

Semantic approaches in the philosophy of science

In this article I give an overview of some recent work in philosophy of science dedicated to analysing the scientific process in terms of (conceptual) mathematical models of theories and the various semantic relations between such models, scientific theories, and aspects of reality. In current philosophy of science, the most interesting questions centre around the ways in which writers distinguish between theories and the mathematical structures that interpret them and in which they are true, i.e. between scientific theories as linguistic systems and their non-linguistic models. In philosophy of science literature there are two main approaches to the structure of scientific theories, the statement or syntactic approach—advocated by Carnap, Hempel and Nagel—and the non-statement or semantic approach—advocated, among others, by Suppes, the structuralists, Beth, Van Fraassen, Giere, Wójcicki. In conclusion, I briefly review some of the usual realist inspired questions about the possibility and character of relations between scientific theories and reality as implied by the various approaches I discuss in the course of the article. The models of a scientific theory should indeed be adequate to the phenomena, but if the theory is ‘adequate’ to (true in) its conceptual (mathematical) models as well, we have a model-theoretic realism that addresses the possible meaning and reference of ‘theoretical entities’ without relapsing into the metaphysics typical of the usual scientific realist approaches.

Air emissions from exposed, contaminated sediment and dredged materials: 2. Diffusion from laboratory-spiked and aged field sediments

The mass transfer of three polycyclic aromatic hydrocarbons (naphthalene, phenanthrene and pyrene) and a heterocyclic aromatic hydrocarbon (dibenzofuran) from sediment to air was studied in a large-area flux chamber. A laboratory-spiked local (University Lake, UL) sediment and an aged contaminated field (Indiana Harbor Canal, IHC) sediment was used. The effects of initial sediment moisture content, and changing air relative humidity were investigated. For high moisture conditions in the UL sediment, the flux remained large whereas for low moisture conditions, there was a sharp decrease in flux as a result of surface drying of the sediment. Under similar air velocities and moisture conditions, the flux from the aged IHC sediment was considerably smaller than from the laboratory-spiked UL sediment. Whereas, the flux from laboratory-spiked UL sediment was predicted satisfactorily by a conceptual mathematical model, that from the aged IHC sediment did not agree with the model predictions. It was concluded that only a portion of the contaminant was available for desorption from the aged sediment due to the differences in the sorption characteristics of relatively fresh and aged contaminated sediments.

The role of organic attenuation in saturated clay barrier system

A review of literature finds that advection, diffusion, and retardation are the main processes that govern the migration of organic contaminants through compacted clay landfill liners. However, much emphasis is placed on the hydraulic conductivity in the specification for landfill liner design. It is misunderstood that if the hydraulic conductivity of the clay is low, then the liner must provide an adequate barrier for groundwater pollution prevention. Unfortunately, even the advection is minimal, contaminants can migrate through clay by simple Fickian diffusion at a rate that can be significant. The process of diffusion is mainly dependent on the concentration gradient between the leachate and the groundwater. If a clay lining system is installed, the only way to reduce the effect of diffusion is to reduce the concentration gradient by pollutant retardation. In this paper, the relative importance of molecular diffusion and advection, and the effect of pollutant retardation on the advective and diffusive transport are discussed using a conceptual-mathematical model. Based on a review of organic contaminant attenuation by clay liners, a guideline on the development of a high organic attenuation engineered barrier as a second line of defence for containment sites is proposed.

The role of organic attenuation in saturated clay barrier system

A review of literature finds that advection, diffusion, and retardation are the main processes that govern the migration of organic contaminants through compacted clay landfill liners. However, much emphasis is placed on the hydraulic conductivity in the specification for landfill liner design. It is misunderstood that if the hydraulic conductivity of the clay is low, then the liner must provide an adequate barrier for groundwater pollution prevention. Unfortunately, even the advection is minimal, contaminants can migrate through clay by simple Fickian diffusion at a rate that can be significant. The process of diffusion is mainly dependent on the concentration gradient between the leachate and the groundwater. If a clay lining system is installed, the only way to reduce the effect of diffusion is to reduce the concentration gradient by pollutant retardation. In this paper, the relative importance of molecular diffusion and advection, and the effect of pollutant retardation on the advective and diffusive transport are discussed using a conceptual-mathematical model. Based on a review of organic contaminant attenuation by clay liners, a guideline on the development of a high organic attenuation engineered barrier as a second line of defence for containment sites is proposed.

Mathematical models and understanding in ecology

Abstract Of all possible mathematical models, only “conceptual” models, i.e., models which are understandable, manageable, and capable of being fully explored, can be of help in attaining an understanding of ecological systems and processes. However, the hitherto promulgated “philosophy” of conceptual mathematical modelling was not in a position to convince empiricists of the possibilities afforded by mathematical models on the one hand, and on the other to prevent theorists from becoming so engrossed in their theories as to neglect the testability of their hypotheses. The goal of this paper is to reveal the shortcomings of this “philosophy” of mathematical modelling and to outline a more promising strategy for creating models. It recommends choosing “patterns” actually existing in ecological systems, rather than questions of a general nature, as a point of departure for conceptual models. In this way, the advantages of conceptual models can be utilized without having to relinquish the testability of the hypotheses proposed with the aid of the models.

Calibration of conceptual models for rainfall-runoff simulation

Abstract Conceptual mathematical models are a useful tool for rainfallrunoff modelling of a basin. The calibration of such models has attracted the attention of a number of hydrologists since unique and optimal parameters are difficult to obtain. The calibration of a conceptual model is discussed through a simple conceptual model whose parameters are determined using a search technique. It is shown that the optimization algorithm converges to a global optimum even when the errors in the initial parameters are quite significant and the input environment is noisy.

Modeling of meltwater infiltration in subfreezing snow

A mathematical model which incorporates the processes that influence water flow and heat transfer in subfreezing snow was developed. Among the aspects of snow included are density and grain‐size heterogeneities, capillary‐pressure gradients, meltwater refreezing, time dependent hydraulic and thermal parameters, and heat conduction. From this conceptual mathematical model a numerical model of two‐dimensional meltwater infiltration was developed. Results from various test cases show which data are most important to measure accurately in the field, in order to determine how the snowpack will respond to an introduction of meltwater. These simulations also show the importance of the orientation of the various layers which make up the snowpack and how randomly distributed heterogeneities can produce two‐dimensional flow of meltwater under unsaturated conditions. Finally, it is demonstrated that various assumptions related to density and porosity variations, dimensionality of flow, capillary effects, etc., which have been made by past investigators for ideal situations may not be valid under many circumstances, and several suggestions are made for improving predictions of meltwater behavior. Sensitivity analysis showed that the model is most sensitive to changes in bulk density, residual saturation of wet snow and meltwater supply rates, whereas changes in snow temperature and mean grain size had less marked effect.

A simple, conceptual mathematical model for the activated sludge process and its variants

A simple structured kinetic model is applied to the activated sludge process. The objective is less to predict exact process performance than to illustrate some of the possibilities and difficulties in producing a comprehensive model for all the process variants. The rate equations are chosen so as to reduce to the Monod equation during balanced growth. Because these rate equations are linear, the cell growth and substrate uptake in a stirred tank can be defined exactly in terms of the average composition of the biomass. It is shown that this is not valid for other forms of rate equations. The stored substrate to protoplasm ratio in the flocs is found to decrease with increasing mean cell residence time. If extracellular biopolymers are included in the stored substrate this corresponds qualitatively to observations of poor flocculation in extended aeration. The model is also applied to the contact stabilization process and is found to be in agreement with the essential process variables.

PCB transport into lake sediments. Conceptual model and laboratory simulation

A laboratory simulation study was performed to observe the transport of a polychlorinated biphenyl (PCB) in lake bottom sediment. A conceptual mathematical model, which describes PCB fate due to the combination of molecular diffusion in pore water and partitioning onto sediment solids, was assessed. An experimental apparatus was designed to simulate a lake bottom sediment/water interface condition, in which Aroclor 1242 was introduced. Sampling and analytical methods were devised, so that bed depth-PCB concentration profiles could be determined from the simulation. Very thin slices, 0.02-0.1 cm, allowed the construction of concentration profiles to quantify the time variation of penetration of {sup 14}C-labeled components into bottom sediment. The consistent behavior of effective diffusivities extracted from profiles suggests the model may adequately describe chemical transport in sediment.

An evaluation of the impact of persistent water level changes on the areal extent of Georgian Bay/North Channel marshlands

A conceptual mathematical model has recently been devised to assist environmental managers in predicting the impact on coastal marsh areas of long-term changes in water levels. The model considers such impact solely in terms of the geometry of the confining basin, the change in ambient water level, and the maximum depth for which bottom-rooted emergent vegetation is present. This model is applied to 17 shoreline marshes of various shapes in the Georgian Bay/North Channel region of the Great Lakes. Model outputs of predicted maximum and minimum marsh area subsequent to changes in long-term levels are compared to marsh areas measured from available historical air photos dating from 1935 to 1985. The results of such comparisons indicate that such a geometric model, despite its neglect of the biological complexities of marsh ecology, can serve as a valuable tool for assessing the range of impacts of both natural and man-made changes in long-term ambient water levels on shoreline marshes.

A conceptual model of unit-mass response function for nonpoint source pollutant runoff

A conceptual mathematical model of unit-mass response function (UMRF) of nonpoint source pollutants (NPSP) is presented. The physical aspects of the mathematics involved yield explanation of some basic mechanisms observed in nature for various environmental responses, like inherent hysteretical effects (rising-falling flow dependence) and linear pollutant load/discharged water volume correlations. The rationale of certain empirical relationships is discussed, and reasonable explanations are given for frequently occurring cases of lack of correlation for water quality field data. The mathematical model substantiates screening or planning modelling techniques proposed in the literature, provided that reference weighted pollutant concentrations in the runoff are used. A conceptual form of such an average concentration arises also from the mass transfer and balance relationships investigated. The model is designed to portray consistent NPSP balances (with interphase mass transfer) via an extension of Nash' conceptual scheme of rainfall-runoff transformations. The limited number of parameters and their clear physical meaning make this approach suitable to NPSP simulation and forecasting. The derived form of UMRF is simple, thus calling for microcomputer software applications. Although the proposed model structure consists of a lumped parameter formulation, the approach may serve as an alternative to empirical routing procedures often implemented in distributed parameter codes. The techniques adopted make use of standard mathematics of hydrological and chemical models, and therefore the related high degree of sophistication developed for simulation and forecasting can be exploited by the present approach for ecological modelling. Results of applications to the case study of the Venice Lagoon (Italy) are presented and discussed.