Optimization model to start harvesting in stochastic aquaculture system

Abstract

Establishment of cost-effective management strategy of aquaculture is one of the most important issues in fishery science, which can be addressed with bio-economic mathematical modeling. This paper deals with the aforementioned issue using a stochastic process model for aquacultured non-renewable fishery resources from the viewpoint of an optimal stopping (timing) problem. The goal of operating the model is to find the optimal criteria to start harvesting the resources under stochastic environment, which turns out to be determined from the Bellman equation (BE). The BE has a separation of variables type structure and can be simplified to a reduced BE with a fewer degrees of freedom. Dependence of solutions to the original and reduced BEs on parameters and independent variables is analyzed from both analytical and numerical standpoints. Implications of the analysis results to management of aquaculture systems are presented as well. Numerical simulation focusing on aquacultured Plecoglossus altivelis in Japan validates the mathematical analysis results. Bus. Ind. 2017, 33 476-493 The quantity W * is related to Ψ that solves the reduced BE (12), and therefore, Ψ and W * have to be simultaneously solved. The curve C is therefore a free boundary. Assuming the unique existence of such C, the one-dimensional domain 0 < w < +∞ for each t < T can be separated into the two regions: the waiting region S 1 (t) = {w; Ψ (t, w) > w} and the Appl. Stochastic Models Bus. Ind. 2017, 33 476-493 Appl. Stochastic ModelsProof of Proposition 3.3 J ,t with p = ( p) i (i = 1, 2) is denoted as J ,t,i . For each ∈ t,T , the inequalitydirectly follows from (5). This leads to

Optimal harvesting policy of an inland fishery resource under incomplete information

“…The parameter 0 ≤ ω < 1 modulates the sensitivity of the decision‐maker on the biomass N s X s : larger ω would lead to higher sensitivity of the performance index on the integrands for smaller N s X s . The present performance index is a generalization of that in the works of Yoshioka and Yaegashi …”

“…On the basis of the system of SDEs and the performance index, the dynamic programming principle (see section 5 in the work of Fleming and Soner) leads to the three‐dimensional HJB governing Φ, ie, for 0 < t < T , n > 0, 0 < x < 1, 0 < π < 1, with the generator L c defined for generic sufficiently smooth F = F ( t , n , x , π ) as The three‐dimensional HJB equation is reduced to a two‐dimensional degenerate parabolic partial differential equation assuming the solution of the separation of variables type as in the works of Yoshioka and Yaegashi, ie, …”

“…Following the works of Yoshioka and Yaegashi 43,44 and Lungu and Øksendal, 46 the population dynamics of the fishery resource is assumed to be described with Itô SDEs…”

“…The body growth rate of the fishery resource is assumed to be unknown for the decision‐maker. This point distinguishes the proposed mathematical model from the conventional models operated under complete information . In addition, the population dynamics is assumed to be subject to regime switching of the body growth rate.…”

“…This point distinguishes the proposed mathematical model from the conventional models operated under complete information. [42][43][44] In addition, the population dynamics is assumed to be subject to regime switching of the body growth rate. We consider that the growth rate can be estimated through a direct observation of noisy temporal fish growth.…”