Optimal harvesting policy of an inland fishery resource under incomplete information

Abstract

A new mathematical model for finding the optimal harvesting policy of an inland fishery resource under incomplete information is proposed in this paper. The model is based on a stochastic control formalism in a regime‐switching environment. The incompleteness of information is due to uncertainties involved in the body growth rate of the fishery resource: a key biological parameter. Finding the most cost‐effective harvesting policy of the fishery resource ultimately reduces to solving a terminal and boundary value problem of a Hamilton‐Jacobi‐Bellman equation: a nonlinear and degenerate parabolic partial differential equation. A simple finite difference scheme for solving the equation is then presented, which turns out to be convergent and generates numerical solutions that comply with certain theoretical upper and lower bounds. The model is finally applied to the management of Plecoglossus altivelis, a major inland fishery resource in Japan. The regime switching in this case is due to the temporal dynamics of benthic algae, the main food of the fish. Model parameter values are identified from field measurement results in 2017. Our computational results clearly show the dependence of the optimal harvesting policy on the river environmental and biological conditions. The proposed model would serve as a mathematical tool for fishery resource management under uncertainties.

Cost‐efficient monitoring of continuous‐time stochastic processes based on discrete observations

“…A finite difference scheme for numerical discretization of the problem (30)‐(34) is presented. Our scheme is based on the previously established monotone and stable numerical schemes, relying on exact solutions to local boundary value problems 7,42,51 . Similar numerical schemes can also be found in and Mickens et al 59 and Gyulov et al 60 Our numerical contribution is to demonstrate that the scheme is stable, monotone, and can potentially approximate the viscosity solution to the problem (30)‐(34) if it exists.…”

“…Appropriate boundary conditions have to be specified when necessary in a standard setting, but we assume that the PDE (30) is satisfied over Ω for each t > 0 as in some degenerate parabolic PDEs 55 . This assumption is satisfied in our applications where the target SDEs have degenerate diffusion and/or drift coefficients 7,42,51 such that the PDE (30) is naturally satisfied over Ω for each t > 0. This is also true in some other problems in biology and ecology 50,56 .…”

“…The first model application is monitoring biological growth of a fishery resource. The SDE of the biological growth of the fish Plecoglossus altivelis altivelis in Japan 7 with the parameter values r = 0.028 (1/day) and σ = 0.062 (1/day 1/2 ) is used for this computational example. Here, 0 ≤ X t ≤ 1 is the normalized body weight.…”

“…For example, environmental monitoring is essential in analyzing greenhouse gas emissions 1 and transport of organic and inorganic matters 2,3 . Tracking biological phenomena in natural environment requires monitoring population dynamics of target species 4‐7 . Monitoring financial time series is of importance in business activities 8 .…”

On a non-standard two-species stochastic competing system and a related degenerate parabolic equation

“…The nuisance filamentous algae, such as the periphyton Cladophora glomerata, are weak against turbulent river flows found in natural rivers, but can persist under low-flow conditions occurring in rivers where humans regulate the flow conditions through dam operations. In addition, aquatic species like Plecoglossus altivelis altivelis eat but cannot digest them [18]. On the other hand, non-filamentous algae, such as diatoms serving as staple foods of the fish, are more tolerant against turbulence but possibly have smaller intrinsic growth rates (Schmidt et al [16] for periphyton and Morin et al [12] for diatoms.).…”