  A new stochastic control problem of population dynamics under partial
observation is formulated and analyzed both mathematically and numerically,
with an emphasis on environmental and ecological problems. The decision-maker
can only randomly and time-discretely observe and impulsively intervene the
population dynamics governed by a regime-switching stochastic differential
equation. The hybrid nature of the problem leads to an optimality equation
containing an integro-differential equation and a static optimization problem.
It is therefore different from the conventional Hamilton-Jacobi-Bellman
equations. Existence and solvability issues of this optimality equation are
analyzed in a viscosity sense. Its exact solution to a reduced but still
nontrivial model is derived as well. The model is finally applied to a
realistic environmental management problem in a river using a finite difference
scheme.
