| Čoahkkáigeassu: | We introduce the spatial Lambda-Fleming-Viot model for natural selection within a population distributed on a spatial continua. We then analyse the behaviour of this model when one considers the appearance of a rare mutation. We investigate how the mechanism for reproduction will effect the resulting limiting behaviour. In particular, we not only show convergence to classical superBrownian motion but also, under suitable assumptions, to superBrownian motion with stable branching mechanism. We then look at the effect of a randomly fluctuating environment on the spatial Lambda-Fleming-Viot model. This is motivated by the changing effects of natural selection driven by the environment such as the El Niño effect. The final part of the thesis is devoted to an investigation of the genealogies within a population evolving under sweepstakes reproduction. This is related to the stable branching mechanism discussed earlier, but with a focus on the practical implications when sampling from a population one might expect to be modeled with a stable branching mechanism.
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