| Invasive species are defined as any species capable of propagating into a nonnative environment. If established, they can be extremely difficult to eradicate, or even manage. In this thesis, we developed and analyzed a mathematical model of biological control to prevent or attenuate the explosive increase of an invasive species population, that functions as a top predator, in a three species food chain. We incorporated controls that were shown to drive down the invasive population growth and, in certain cases, eliminated explosion. Hence, the population did not reach an uncontrollable level. We referred to these new controls as "ecological damping", as their inclusion dampens the invasive species population growth. We again investigate the effect of random environmental fluctuation on such explosive growth of the invasive species. We find that environmental noise alone, cannot prevent explosive growth in invasive populations. We interpret these results as showing that invasive species can strongly adapt to a randomly changing environment.;We investigated an important phenomenon known as Allee effect in a three species food chain model. Allee effect is phenomenon characterized by positive density dependence, that is a positive correlation between population density and individual fitness. However, the effect is not well studied in multi-level trophic food chains. We therefore considered a ratio dependent spatially explicit three species food chain model, where the top predator is subjected to a strong Allee effect. Next, we numerically investigated the decay rate to a target attractor. We found decay estimates that are O(mgamma), where gamma was found explicitly to be close to 1.0. Furthermore, we provided overexploitation theorems for the food chain model, showing that overexploitation has to be driven by the middle predator. In particular overexploitation is not possible without an Allee effect in place. We also uncovered a rich class of Turing patterns in the model which depend significantly on the Allee threshold parameter m.. |
