Abstract
We study the (Ito) stochastic differential equation dX(t) = rX(t)(K - X(t))dt + aX(t)(K - X(t))dB(t), X0 = 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 |
|---|---|
| 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
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Optimal harvesting from a population in a stochastic crowded environment. / Lungu, E. M.; Øksendal, B.
In: Mathematical Biosciences, Vol. 145, No. 1, 01.10.1997, p. 47-75.Research output: Contribution to journal › Article
TY - JOUR
T1 - Optimal harvesting from a population in a stochastic crowded environment
AU - Lungu, E. M.
AU - Øksendal, B.
PY - 1997/10/1
Y1 - 1997/10/1
N2 - We study the (Ito) stochastic differential equation dX(t) = rX(t)(K - X(t))dt + aX(t)(K - X(t))dB(t), X0 = 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.
AB - We study the (Ito) stochastic differential equation dX(t) = rX(t)(K - X(t))dt + aX(t)(K - X(t))dB(t), X0 = 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.
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UR - http://www.scopus.com/inward/citedby.url?scp=0030807378&partnerID=8YFLogxK
U2 - 10.1016/S0025-5564(97)00029-1
DO - 10.1016/S0025-5564(97)00029-1
M3 - Article
VL - 145
SP - 47
EP - 75
JO - Mathematical Biosciences
JF - Mathematical Biosciences
SN - 0025-5564
IS - 1
ER -