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Fallopia japonica (Houtt. Ronse Decraene) (syn. Reynoutria japonica syn. Polygonum cuspidatum, Japanese knotweed) is an aggressively invasive alien weed in the United Kingdom (UK) and throughout its introduced range. Its presence can herald considerable costs, both in terms of its ecological impact as a threat to biodiversity and economically due to the physical damage caused to property and the associated costs of treatment and disposal of the plant. There is therefore increasing interest in eradicating this alien species and as a result many different management techniques have been applied to try and control its spread. It is important to ascertain which of these are most appropriate in any given situation and so tools that can test the impact and efficiency of these techniques both quickly and cheaply would be extremely useful. In this thesis mathematical models are developed for the spatial spread of F. japonica on a local scale in the UK.