  This paper develops near-optimal sustainable harvesting strategies for the
predator in a predator-prey system. The objective function is of long-run
average per unit time type. To date, ecological systems under environmental
noise are usually modeled as stochastic differential equations driven by a
Brownian motion. Recognizing that the formulation using a Brownian motion is
only an idealization, in this paper, it is assumed that the environment is
subject to disturbances characterized by a jump process with rapid jump rates.
Under broad conditions, it is shown that the systems under consideration can be
approximated by a controlled diffusion system. Based on the limit diffusion
system, control policies of the original systems are constructed. Such an
approach enables us to develop sustainable harvesting policies leading to near
optimality. To treat the underlying problems, one of the main difficulties is
due to the long-run average objective function. This in turn, requires the
handling of a number of issues related to ergodicity. New approaches are
developed to obtain the tightness of the underlying processes based on the
population dynamic systems.
