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Analysis and Simulation of an Optimal Control Model of an Oyster Population Displaying an Allee Effect

By Timothy Raymond McDade


During the last century, the oyster population of the Chesapeake Bay area has diminished greatly due to overfishing, pollution and climate change. Our Optimal Control model finds a sustainable solution that balances oyster harvesting with the health of the population. We wish to find the value of our Effort (control) function that harvests the most oysters possible without fishing the population to extinction. We create a Hamiltonian function and apply Bang-Bang Control in order to find a singular E* between 0 and Emax such that E* will balance out with the natural growth rate of the population to form a constant, stable population. Our model uses analytical and numerical solutions to determine the optimal sustainable population N* and E* for a Bang-Bang Control model. The analytical model also solves for times T1 and T2 at which the piecewise Heaviside eff ort function switches values of E(t). In marine population study, there has not been extensive use of mathematics, especially optimal control theory. Consequently, as seen in our Future Work section, there is much room for expansion upon current scholarship regarding optimal control theory. Only by incorporating several environmental factors can one succeed in using mathematics to develop a successful harvesting strategy

Topics: Optimal Control, Differential Equations, Oysters, Allee Effect, Runge-Kutta
Publisher: College of William and Mary
Year: 2012
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