  We consider the harvesting of a population in a stochastic environment whose
dynamics in the absence of harvesting is described by a one dimensional
diffusion. Using ergodic optimal control, we find the optimal harvesting
strategy which maximizes the asymptotic yield of harvested individuals. To our
knowledge, ergodic optimal control has not been used before to study harvesting
strategies. However, it is a natural framework because the optimal harvesting
strategy will never be such that the population is harvested to extinction --
instead the harvested population converges to a unique invariant probability
measure.
  When the yield function is the identity, we show that the optimal strategy
has a bang-bang property: there exists a threshold $x^*>0$ such that whenever
the population is under the threshold the harvesting rate must be zero, whereas
when the population is above the threshold the harvesting rate must be at the
upper limit. We provide upper and lower bounds on the maximal asymptotic yield,
and explore via numerical simulations how the harvesting threshold and the
maximal asymptotic yield change with the growth rate, maximal harvesting rate,
or the competition rate.
  We also show that, if the yield function is $C^2$ and strictly concave, then
the optimal harvesting strategy is continuous, whereas when the yield function
is convex the optimal strategy is of bang-bang type. This shows that one cannot
always expect bang-bang type optimal controls.
