Optimal harvesting from a population in a stochastic crowded environment

We study the (Ito) stochastic differential equation dXt = rXt(K−Xt)dt+αXt(k−Xt)dBt, X0 = 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 Bt 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 Xt < x∗ and to harvest Xt − x∗ if Xt > x∗. This leads to an optimal population process Xt being reflected downward at x∗. We find x∗ explicitly.