  We analyze the optimal harvesting problem for an ecosystem of species that
experience environmental stochasticity. Our work generalizes the current
literature significantly by taking into account non-linear interactions between
species, state-dependent prices, and species injections. The key generalization
is making it possible to not only harvest, but also `seed' individuals into the
ecosystem. This is motivated by how fisheries and certain endangered species
are controlled. The harvesting problem becomes finding the optimal
harvesting-seeding strategy that maximizes the expected total income from the
harvest minus the lost income from the species injections. Our analysis shows
that new phenomena emerge due to the possibility of species injections.
  It is well-known that multidimensional harvesting problems are very hard to
tackle. We are able to make progress, by characterizing the value function as a
viscosity solution of the associated Hamilton-Jacobi-Bellman (HJB) equations.
Moreover, we provide a verification theorem, which tells us that if a function
has certain properties, then it will be the value function. This allows us to
show heuristically, as was shown in Lungu and $\O$ksendal (Bernoulli '01), that
it is almost surely never optimal to harvest or seed from more than one
population at a time.
  We approximate the continuous-time systems by Markov chains and show that the
optimal harvesting-seeding strategies of the Markov chain approximations
converge to the correct optimal harvesting strategy. This is used to provide
numerical approximations to the optimal harvesting-seeding strategies and is a
first step towards a full understanding of the intricacies of how one should
harvest and seed interacting species. In particular, we look at three examples:
one species modeled by a Verhulst-Pearl diffusion, two competing species and a
two-species predator-prey system.
