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.

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