The terminology of renewable resource management becomes to span not only prototype harvesting problems but also various new types such as invasive species one. In all of these problems, process uncertainty of stock growth associated with environmental variability is one of the critical factors that significantly affects the management efficiency. While it may seem that a series of past researches fully examine optimal policy under process uncertainty, the case of convex unit harvest costs has not been fully characterized yet. Focusing on such a case, this paper addresses how the degree of process uncertainty affects optimal escapement level. The result suggests that optimal escapement level does not monotonically vary with process uncertainty. In many plausible cases, it should be adapted in a U-shaped manner, which is in contrast with the conventional wisdom.