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- Analysis of Stability of Some Population Models with Harvesting
Analysis of Stability of Some Population Models with Harvesting
Thesis Abstract:
Applied mathematics, which means application of mathematics to problems, is a wonderful and exciting subject. It is the essence of the theoretical approach toscience and engineering. It could refer to the use of mathematics in many varied areas. Mathematical model is applied to predict the behavior of the system. This behavior is then interpreted in terms of the word model so that we know the behavior of the real situation.
Mathematical languages can be applied to transform ecology’s phenomena into mathematical model, including changes of populations and how the population of one system can affect the population of another. The model is expected to give more information about the real situation and as a tool in decision making.
Some models that constitute autonomous differential equations were presented: (1) Malthusian and logistic model for single population; (2) two independent populations, competing model and prey-predator model of two populations; and (3) extension of prey-predator model involving three populations. This research will study the effect of harvesting on models.
The models were based on Lotka-Volterra model. All models involved harvesting problem and some equilibrium points related to maximum profit or maximum sustainable yield problem. The objectives of this research were 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 were linearization method, eigenvalues method, qualitative stability test, and Hurwitz stability test. Some assumptions were made to avoid complexity. Maple V software release 4 was used to determine the equilibrium points of the model and to plot the trajectories and draw the surface. The single population model was solved analytically.
Results showed that in single population model, the existence of population depended on the initial population and harvesting rate. In the model that involved two and three populations, the population can live in coexistence although harvesting was applied. The level of harvesting, however, must be strictly controlled.