Deterministic mathematical model for fish harvesting

Differential equations have been used to create mathematical models of real world systems in which rates of change are involved, for example in the study of how population grows or shrinks. One of the earliest models by Thomas Malthus has been found to be unrealistic since it predicts that population will grow exponentially and without bound – a prospect that defies physical limitations. Verhulst in his logistic population model developed a generalized version of the Malthusian model but added that natural factors such as disease and predators also affect mortality rate, and hence the rate of population growth. The Lokta – Voltera, which is a two species model, describes interactions between a predator and a prey in an ecosystem. The model forms the basis of many models used today in analysis of population dynamics, but unfortunately, in its original form, it lacks a stable equilibrium point. In mid 1980’s Ragozin and Brown, established the existence of a steady state and described a unique approach to it for a predator Prey System. Nile perch is a predator since it feeds on other small fish e.g. haplochromines, but it is also a prey because it is harvested for food. Rabuor and Polovina showed that there is a decline in the volume of Nile Perch and they attributed this to harvesting. So far, none of the models cited above considered harvesting as a factor affecting population growth. The need to develop a model for fish harvesting in general and in particular, the Lates Niloticus (Nile Perch) was Born out of this gap. The main objective of this study was to develop a model which can predict the amount of Nile perch harvested at any time t. The procedure in this study involved using differential equations to yield specific insights into the management of complex ecological systems e.g. population outbreaks. We have analyzed the existing logistics model for equilibrium solutions and stability. To test on the stability of the model, we obtain secondary data from the Kenya Marine and Fisheries Institute (KMFRI) and Lake Basin Development Authority (LBDA). The data so obtained has been analysed using the concept of stability in order to develop the required model. We have managed in this thesis to construct three different models viz: i) The Constant –rate harvesting model ii) The Proportional – rate harvesting model iii) A model depicting when harvesting is a function of P2(t) where P(t) represents the amount of Nile Perch harvested at any time t. Stability analysis and verification of the models revealed that the proportional – rate harvesting model is more reliable as compared to the other two. This is because its solutions are closer to actual values of the Nile perch harvested for the period under investigation. It is hoped that the model would help economic and social planners in controlling the population of Lates Niloticus. The result will also give more insight in research in mathematical modeling.