### Abstract

We study the (Ito) stochastic differential equation dX(t) = rX(t)(K - X(t))dt + aX(t)(K - X(t))dB(t), X_{0} = x > 0 as a model for population growth in a stochastic environment with finite carrying capacity K > 0. Here r and a are constants and B(t) denotes Brownian motion. If r ≤ 0, we show that this equation has a unique strong global solution for all x > 0 and we study some of its properties. Then we consider the following problem: What harvesting strategy maximizes the expected total discounted amount harvested (integrated over all future times)? We formulate this as a stochastic control problem. Then we show that there exists a constant optimal 'harvest trigger value' x* ε (0, K) such that the optimal strategy is to do nothing if X(t) < x* and to harvest X(t) - x* if X(t) > x*. This leads to an optimal population process X(t) being reflected downward at x*. We find x* explicitly.

Original language | English |
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Pages (from-to) | 47-75 |

Number of pages | 29 |

Journal | Mathematical Biosciences |

Volume | 145 |

Issue number | 1 |

DOIs | |

Publication status | Published - Oct 1 1997 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Modelling and Simulation
- Biochemistry, Genetics and Molecular Biology(all)
- Immunology and Microbiology(all)
- Agricultural and Biological Sciences(all)
- Applied Mathematics

### Cite this

*Mathematical Biosciences*,

*145*(1), 47-75. https://doi.org/10.1016/S0025-5564(97)00029-1