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

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