The role of the branch of the autonomic nervous system mediates control of organ processes when the body is essentially .

parasympathetic; at rest

Explanation: The parasympathetic nervous system branch of the autonomic nervous system is largely responsible for the relaxation of the body especially at rest where it undoes the activities of the sympathetic branch by decreasing respiration, heart rate and then increasing the body's digestion rate. As such, this branch is also responsible for digestion response in a relaxed, resting, or feeding states. The role therefore of the parasympathetic branch of the autonomic nervous system mediates control of organ processes when the body is essentially at rest.