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We study versions of the contact process with three states, and withinfections occurring at a rate depending on the overall infection density.Motivated by a model described in [17] for vegetation patterns in aridlandscapes, we focus on percolation under invariant measures of such processes.We prove that the percolation transition is sharp (for one of our models thisrequires a reasonable assumption). This is shown to contradict a form of'robust critical behaviour' with power law cluster size distribution for arange of parameter values, as suggested in [17].

Sharpness versus robustness of the percolation transition in 2D contact processes