Analysis of Stability of Some Population Models with Harvesting

Applied mathematics, which means application of mathematics to problems, is a wonderful and exciting subject. It is the essence of the theoretical approach to science and engineering. It could refer to the use of mathematics in many varied areas. Mathematical model is applied to predict the behaviour of the system. This behaviour is then interpreted in terms of the word model so that we know the behaviour of the real situation. We can apply mathematical languages to transform ecology's phenomena into mathematical model, including changes of popUlations and how the populations of one system can affect the population of another. The model is expected to give us more information about the real situation and as a tool to make a decision. Some models that constitute autonomous differential equations are presented; Malthusian and logistic model for single population; two independent populations, competing model, and prey-predator model for two populations; and extension of prey-predator model involving three populations. In this thesis we will study the effect of harvesting on models. The models are based on Lotka-Volterra model. All models involve harvesting problem and some stable equilibrium points related to maximum profit or maximum sustainable yield problem. The objectives of this thesis are to analyse, to investigate the stability of equilibrium point of the models and to control the exploitation efforts such that the population will not vanish forever although being exploited. The methods used are linearization method, eigenvalues method, qualitative stability test and Hurwitz stability test. Some assumptions are made to avoid complexity. Maple V software release 4 is used to determine the equilibrium points of the model and also to plot the trajectories and draw the surface. The single population model is solved analytically.We found that in single population model, the existence of population depends on the initial population and harvesting rate. In model that involves two and three populations, the populations can live in coexistence although harvesting is applied. The level of harvesting, however, must be strictly controlled.

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