  We derive the explicit solution to a singular stochastic control problem of
the monotone follower type with an expected ergodic criterion as well as to its
counterpart with a pathwise ergodic criterion. These problems have been
motivated by the optimal sustainable exploitation of an ecosystem, such as a
natural fishery. Under general assumptions on the diffusion coefficients and
the running payoff function, we show that both performance criteria give rise
to the same optimal long-term average rate as well as to the same optimal
strategy, which is of a threshold type. We solve the two problems by first
constructing a suitable solution to their associated Hamilton-Jacobi-Bellman
(HJB) equation, which takes the form of a quasi-variational inequality with a
gradient constraint.
