  We analyze the long term behavior of interacting populations which can be
controlled through harvesting. The dynamics is assumed to be discrete in time
and stochastic due to the effect of environmental fluctuations. We present
extinction and coexistence criteria when there are one or two interacting
species. We then use these tools in order to see when harvesting leads to
extinction or persistence of species, as well as what the optimal harvesting
strategies, which maximize the expected long term yield, look like. For single
species systems, we show under certain conditions that the optimal harvesting
strategy is of bang-bang type: there is a threshold under which there is no
harvesting, while everything above this threshold gets harvested.
  The second part of the paper is concerned with the analysis of ecosystems
that have two interacting species which can be harvested. In particular, we
carefully study predator-prey and competitive Ricker models when there are two
species. In this setting we show how to find the optimal proportional
harvesting strategy. If the system is of predator-prey type the optimal
proportional harvesting strategy is, depending on the interaction parameters
and the price of predators relative to prey, either to harvest the predator to
extinction and maximize the asymptotic yield of the prey or to not harvest the
prey and to maximize the asymptotic harvesting yield of the predators. If the
system is competitive, in certain instances it is optimal to drive one species
extinct and to harvest the other one. In other cases it is best to let the two
species coexist and harvest both species while maintaining coexistence.
  In the setting of the competitive Ricker model we show that if one competitor
is dominant and pushes the other species to extinction, the harvesting of the
dominant species can lead to coexistence.
