Harvesting of populations in stochastic environments
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.
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.