  We study the optimal sustainable harvesting of a population that lives in a
random environment. The novelty of our setting is that we maximize the
asymptotic harvesting yield, both in an expected value and almost sure sense,
for a large class of harvesting strategies and unstructured population models.
We prove under relatively weak assumptions that there exists a unique optimal
harvesting strategy characterized by an optimal threshold below which the
population is maintained at all times by utilizing a local time push-type
policy. We also discuss, through Abelian limits, how our results are related to
the optimal harvesting strategies when one maximizes the expected cumulative
present value of the harvesting yield and establish a simple connection and
ordering between the values and optimal boundaries. Finally, we explicitly
characterize the optimal harvesting strategies in two different cases, one of
which is the celebrated stochastic Verhulst Pearl logistic model of population
growth.
