We study versions of the contact process with three states, and with infections occurring at a rate depending on the overall infection density. Motivated by a model described in Kéfi et al. (2007) for vegetation patterns in arid landscapes, we focus on percolation under invariant measures of such processes. We prove that the percolation transition is sharp (for one of our models this requires a reasonable assumption). This is shown to contradict a form of 'robust critical behaviour' with power law cluster size distribution for a range of parameter values, as suggested in Kéfi et al. (2007).
Van Den Berg, J., Björnberg, J. E., & Heydenreich, M. (2015). Sharpness versus robustness of the percolation transition in 2d contact processes. Stochastic Processes and Their Applications, 125(2), 513–537. https://doi.org/10.1016/j.spa.2014.09.010