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To summarize, he's interested in modeling the spatial dynamics of vegetation patterns in desert areas using nonlinear reaction-diffusion equations. These equations can show threshold collapse behavior, where variations in grazing or precipitation can lead to an abrupt and (nearly) irreversible disappearance of vegetation, i.e. total desertification. He would like to identify early warning signals of an impending collapse. Unfortunately he has not yet progressed to finding such indicators, but he did lots of math to characterize the system dynamics.