Stability of Some Models in Mathematical Biology

Lately there has been an increasing awareness of the adverse side effect from the use of pesticides on the environment and on human health. As an alternative solution attention has been directed to the so-called "Biological Control" where pests are removed from the environment by the use of another living but harmless organism. A detailed study of biological control requires a clear understanding on the types of interaction between the species involved. We have to know exactly the conditions under which the various species achieve stability and live in coexistence. It is here that mathematics can contribute in understanding and solving the problem. A number of models for single species are presented as an introduction to the study of two species interaction. Specifically the following interactions are studied: -Competition -Predation -Symbiosis. All the above interactions are modelled based on ordinary differential equations. But such models ignore many complicating factors. The presence of delays is one such factor. In the usual models it is tacitly assumed that the coefficients of change for a given species depend only on the instantaneous conditions. However biological processes are not temporally isolated, and the past influences the present and the future. In the real world the growth rate of a species does not respond immediately to changes in the population of interacting species, but rather will do so after a time lag. This concept should be taken into account, and this leads to the study of delay differential equations. However the mathematics required for the detailed analysis of the behaviour of such a model can be formidable, especially for biologists who share the subject. By the aid of computer and using Mathematica software (version 3.0), the main properties of the solutions of many models related to the various interactions can be clarified

Shake table testing of the dynamic interaction between two and three adjacent buildings (SSSI)

The dynamic interaction of adjacent buildings in cities and urban areas through the soil medium is inevitable. This fact has been confirmed by various analytical and numerical studies. However, very little research is available on the physical modelling of the Structure-Soil-Structure Interaction (SSSI) problem and its effect on the dynamics of adjacent structures. In this paper, a series of shaking table tests was conducted at the Earthquake and Large Structures Laboratory (EQUALS) at the University of Bristol to examine the effects of SSSI on the response of a model building when bordered by up to two other model buildings under dynamic excitation. The results indicated that depending on their height, the presence of one or two adjacent building could positively or negatively alter seismic power and peak acceleration responses of a building in comparison to when it is tested in isolation.</p

Priority of Applying Stochastic Differential Equations over their Ordinary Counterparts in Accurately Predicting Stock Prices

Background and aims. Despite the abundance of deterministic models designed to study applied issues in biology, physics, economics, and other fields, which have undergone continuous development and modification to achieve satisfactory future predictions, it is of utmost importance to note that prediction is crucial in the field of economics. It allows sellers and buyers to determine the optimal time for buying and selling. However, as known, market movements are not stable but are subject to numerous and continuous random fluctuations. Therefore, any mathematical model dealing with financial issues must take this into consideration. The aim of this work is to study the solution behavior of one of the most important stochastic economic models, the Black-Scholes model, to demonstrate its superiority over its deterministic counterpart by comparing their results with actual prices. Methods. Prior to studying the stochastic model, its deterministic counterpart was solved, which is represented by an ordinary differential equation. Since calculus of stochastic functions is not very common and differs completely from the known calculus, a series of definitions were provided to explain how to handle them, leading to one of the most important formulas, known as the Itô formula, which was extensively utilized to obtain an explicit solution for the Black-Scholes equation. Additionally, the forms and nature of the solutions were clarified using the MATLAB program. Results. Based on realistic data of a commodity's prices over a four-year period, the necessary coefficients values were calculated, and then substituted into the solution formula for both the deterministic and stochastic models to predict the expected price in both cases after a quarter of a year. The actual price was then revealed, and the results were recorded. Conclusion. By comparing the results presented in the table, it is evident that the expected price using the stochastic model is much closer to the actual price than that predicted by the deterministic model, indicating its superiority in prediction accuracy

Evaluation of rocking and coupling rotational linear stiffness coefficients of adjacent foundations

This paper presents closed-form expressions for rocking spring stiffnesses and coupling interaction rotational spring stiffnesses for a set of closely spaced footings. Substructuring is used to derive analytically the exact reduced order spring models of the system. The stiffness coefficients of this reduced order model are determined by using (1) an extended, novel application of Boussinesq's surface displacement of a point-loaded half-space and (2) an empirically derived formulation that makes use of both finite-element and experimental results. Further validation suggests that, within the scope of epistemic uncertainty present in the physical world, the interaction formulas between two footings are sufficient for more general multifooting interaction cases.</p

Damage-based seismic planar pounding analysis of adjacent symmetric buildings considering inelastic structure-soil-structure interaction

In cities and urban areas, building structures located at close proximities inevitably interact under dynamic loading by direct pounding and indirectly through the underlying soil. Majority of the previous adjacent buildings pounding studies that have taken the Structure-Soil-Structure Interaction (SSSI) problem into account have used simple lumped mass-spring-dashpot models under plane-strain conditions. In this research, the problem of SSSI-included pounding problem of two adjacent symmetric in plan buildings resting on a soft soil profile excited by uniaxial earthquake loadings is investigated. To this end, a series of SSSI models considering one-directional nonlinear impact elements between adjacent co-planar stories and using a method for direct FE modeling of 3D inelastic underlying soil volume have been developed to accurately study the problem. An advanced inelastic structural behavior parameter, the seismic damage index, has been considered in this study as the key nonlinear structural response of adjacent buildings. Based on the results of SSSI and fixed-base cases analyses presented herein, two main problems are investigated, namely, the minimum building separation distance for pounding prevention and seismic pounding effects on structural damage in adjacent buildings. The final results show that at least three times the IBC 2009 minimum distance for building separation recommended value is required as a clear distance for adjacent symmetric buildings to prevent the occurrence of seismic pounding. At the IBC recommended distance, adjacent buildings experienced severe seismic pounding and therefore significant variations in storey shear forces and damage indices. <br clear="none"/

Influence of foundation type on seismic response of low-rise structures in liquefiable soil

The 2010-2011 Canterbury Earthquake Sequence (CES) caused extensive damage to low-rise structures in the city of Christchurch, New Zealand, mainly due to liquefaction-induced effects including settlement and angular distortion. This paper will present the results of dynamic centrifuge tests comparing the effects of liquefaction on the seismic performance of isolated structures with different types of shallow foundations (strips or a raft), and the effect of being situated adjacent to a heavier neighbouring structure of the same foundation type (i.e. considering structure-soilstructure interaction, SSSI). Performance will be evaluated under a sequence of successive earthquakes from the 2010-2011 CES and 2011 Tohoku Earthquake, Japan, to permit study under ground motions and aftershocks generating full liquefaction either extensively or to only a limited depth below ground level. The results show firstly that lower intensity ground shaking occurs at the ground surface when liquefaction occurs and that this can be estimated as a function of the degree of liquefaction using a simple estimation method. When subjected to these ground motions, using strip foundations for isolated structures can result in a reduction in structural demand, especially when the soil is extensively liquefied. When a neighbouring structure with the same foundation type is present, the effects of SSSI within liquefied soil result in changes to natural period and damping such that raft-founded structures exhibited lower structural demands. In either case (isolated or adjacent), a reduction in structural demand is accompanied by an increase in post-earthquake permanent foundation deformation

Model Testing of Recovery Loads for Decommissioning Skirted Subsea Structures

Decommissioning of subsea structures founded on shallow foundations currently requires the complete re-moval of the structure, but there are challenges in predicting the associated recovery loads. This work exam-ines the extraction behaviour of shallow, skirted, square foundations in soft clay by means of a series of dis-placement-controlled small scale model tests. Skirt length, uplift rate, surface drainage condition and pre-load were all varied. Extraction behaviour is seen to be different depending on whether the foundation re-tains a soil plug which in turn seems to depend on the kinematics rather than rate effects. Prediction of re-covery loads based on assuming plugged behaviour in all cases provided an upper bound without excessive conservatism. Prediction of recovery loads based on model preload was only successful under certain condi-tions, and thus should be considered as an estimate only

BADANIE TRYBÓW ROZRUCHU JEDNOFAZOWYCH SILNIKÓW ASYNCHRONICZNYCH PRZY ZMIANIE PARAMETRÓW UZWOJEŃ STOJANA, KONDENSATORA PRZESUWAJĄCEGO FAZĘ I NAPIĘCIA ZASILANIA

Single-phase induction motors (SPIM) are widely used in household appliances, agriculture, trade, medicine and other areas where a cheap unregulated electric drive powered by a single-phase AC network is required. They are produced in millions of pieces per year. Therefore, significant attention has always been paid to research aimed at reducing resource consumption in the production and operation of these engines, improving the initial characteristics and increasing their competitiveness. The article conducted a study of the starting torque when starting single-phase induction motors depending on the initial phase of the voltage of the single-phase network supplying the stator winding, on the phase angles of the network voltage at fixed phase angles of the starting winding. It is also analysed how the active resistance and inductive leakage resistance of the stator winding and the equivalent rotor winding, and the resistance of the capacitor capacitance affect the starting characteristics of the engine.Jednofazowe silniki asynchroniczne znajdują szerokie zastosowanie w sprzęcie AGD, rolnictwie, handlu, medycynie i innych dziedzinach, gdzie wymagany jest tani nieregulowany napęd elektryczny zasilany z jednofazowej sieci prądu przemiennego. Produkowane są w milionach sztuk rocznie. Dlatego zawsze dużą wagę przywiązywano do badań mających na celu zmniejszenie zużycia zasobów w produkcji i eksploatacji tych silników, poprawę parametrów wyjściowych i zwiększenie ich konkurencyjności. W artykule przeprowadzono badania momentu rozruchowego przy rozruchu jednofazowych silników asynchronicznych w zależności od fazy początkowej napięcia sieci jednofazowej zasilającej uzwojenie stojana, od kątów fazowych napięcia sieciowego przy ustalonych kątach fazowych rozruchu. meandrowy. Analizowano także wpływ rezystancji czynnej i indukcyjnej rezystancji uzwojenia stojana i zastępczego uzwojenia wirnika oraz rezystancji pojemności kondensatora na charakterystykę rozruchową silnika

Nonlinear soil-structure interaction analysis of multistorey building

The interaction among structures, their foundations and the soil medium below the foundations alter the actual behaviour of the structure considerably than what is obtained from the consideration of the structure alone. Conventionally, superstructures are usually analyzed by assuming the structure to be fixed at the foundation level. Such an analysis neglects the flexibility of foundation and compressibility of soil mass. It is also assumed, conventionally, that the soil is behaving linearly neglecting that fact that it is nonlinear in nature. In this study an attempt has been made to carry out a two dimensional linear and nonlinear analysis of the problem of a multistory building incorporating soil-structure interaction with respect to nonhomogenous soil properties in Malaysia. Two techniques of analysis have been carried out, in the first, linear stress strain relationship is assumed for the soil where finite element method has been employed in modelling the superstructure members and foundation beam while Winkler’s springs have been attached to the foundation beam to represent the soil layer below foundation, and then a linear coupled finite infinite element modelling is done. Three noded isoparametric beam bending element with three degrees of freedom that takes into account of the transverse shear forces and axial flexural interaction, this element is used to represent the frame members in all types of analyses. Eight noded isoparametric quadrilateral finite element is used to represent the near filed of soil while the far field is represented by using five noded isoparametric infinite element. In the second analysis, the same coupled finite-infinite element modelling is used, the that, the soil is considered to behave nonlinear and a hyperbolic model is used to take this nonlinearity into account. The result showed the importance of taking soil structure interaction into account,results obtained from each analysis have been obtained and comparison among various analyses has been stated