Optimal harvesting from a population in a stochastic crowded environment

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Author list: Lungu EM, Oksendal B

Publisher: Elsevier

Place: NEW YORK

Publication year: 1997

Journal: Mathematical Biosciences (0025-5564)

Journal acronym: MATH BIOSCI

Volume number: 145

Issue number: 1

Start page: 47

End page: 75

Number of pages: 29

ISSN: 0025-5564

eISSN: 1879-3134

Languages: English-Great Britain (EN-GB)


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Abstract

We study the (Ito) stochastic differential equationdX(t) = rX(t)(K-X,)dt + alpha X-t(K - X-t)dB(t), X-0 = x > 0as a model for population growth in a stochastic environment with finite carrying capacity K > 0. Here r and alpha are constants and B-t denotes Brownian motion. If r greater than or equal to 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* is an element of (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. (C) 1997 Elsevier Science Inc.


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