Optimal harvesting from a population in a stochastic crowded environment

• Published in: Mathematical biosciences 1997-10, Vol.145 (1), p.47-75
• Main Authors: Lungu, E.M., Øksendal, B.
• Format: Article
• Language: English
• Subjects: Biological and medical sciences; Computerized, statistical medical data processing and models in biomedicine; Environment; Mathematics; Medical sciences; Medical statistics; Models, Theoretical; Population Growth; Stochastic Processes
• ISSN: 0025-5564; 1879-3134
• DOI: 10.1016/S0025-5564(97)00029-1

We study the (Ito) stochastic differential equation dX t = rX t(K−X t)dt+αX 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 α 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 ∗ ∈ (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.