# Asymptotic harvesting of populations in random environments

## Abstract

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.

## Keywords

Ergodic control Stochastic harvesting Ergodicity Stochastic logistic model Stochastic environment## Mathematics Subject Classification

92D25 60J70 60J60## Notes

### Acknowledgements

We thank two anonymous referees for very insightful comments and suggestions that led to major improvements.

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