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Osei, Bonsu Mensah

Mathematics

2005

PhD

A general modeling framework for biological invasion is formulated. Both Reaction Diffusion (RD) and Integro- Difference (ID) equations are used to develop this framework. First, a generalization of the reaction component to include both linear and non-linear density dependence and the Allee Effect is derived. The diffusion component of the RD model framework is generalized to include dispersal, advection and habitat heterogeneity. A similar generalization of the ID model framework is developed by generalizing the redistribution kernel, which plays a crucial role in ID formulation. Fluctuations in model parameters can have big impact on the dynamics of a system. A general fluctuation methodology for growth equations are developed for the derived generalized model. It has been experimentally shown that two species can both invade a habitat and profit from the invasion. Models to illustrate this phenomenon are developed. Analysis of the system of RD equations that describe this phenomenon leads to the stochastic Burgers equation, which can be shown to have applications in ecology. Analysis of the ID model framework due to fluctuations is formulated.
An Evolutionary Algorithm, in particular Genetic Algorithm, approach to model selection is applied to the generalized RD and ID equations using zebra mussel's data from Lake Champlain which demarcates the states of New York and Vermont. We show that even though this method is feasible, the field data used might need to be interpolated and smoothed. We found that at 90% averaging, our model very accurately predicted the behavior of the zebra mussels. At 70% averaging, good approximations could be made. At 50% and lower, the model was unable to accurately predict mussel density over time. Based on these simulations we estimate the speed of invasion for zebra mussels in Lake Champlain to be 5 - 9 km / year well below the empirically data calculated value of 35 km / year.