For a better understanding of vegetation patterns the idea of a description via systems of partial differential equations (PDE) - also with non-local terms - came up. This type of PDE had become more popular over the last decade, however, analysis results are still rare. In my Master Thesis I studied a special system and have shown existence and uniqueness results for a larger class of problems. I also established an approximation of their non-local term, which is specially geared to the transition from non-local to local PDE. In the end a numerical simulation was indeed capable of showing that so called “fairy circles” (a unexplainable vegetation pattern) do appear within the given vegetation-model.