Abstract:
Modeling the dynamics of commercial fisheries is an active field at the crossroads of mathematics, ecology, and economics. Population levels are one of the most basic statistics used for all conservation. In all natural resource management, predictions for future populations are required to predict the impacts of management strategies. Collapses of commercial fish stocks have been reported in many regions \cite{Bret,Deet,Hil}. Accurate understanding of both ecological and behavioral responses to commercial exploitation in fisheries is an important tool to combat population collapse. In this dissertation, we study two population models with applications in commercial fisheries. First, we study the ratio-dependent predator-prey model with constant harvests, a continuous model that has been proposed, but for which previous analysis has not been conducted. We explore the dynamics of the model in general, and then conduct a search for limit cycles that persist when the predator harvest changes. We find that stable, persistent limit cycles are not very common, especially when there is also a harvest of the prey species. We apply the model to the oyster and black drum system in Louisiana and find that a variety of parameters show a near-best fit, a track of increasing oyster populations to a level some 20-50\% higher than its 23-year average. Next, we study a new discrete model that we created. This model seeks to predict harvest changes in a tractable way, and we provide stability analysis of the model. We expand the model to incorporate several agents and apply the model from several initial conditions where overfishing is occurring. We then propose a simple management rule that, in our simulations, improves performance of the model.
