Received: 10 January 2017 Revised: 24 September 2017 Accepted: 13 November 2017
Mathematical analysis for management of released fish
Faculty of Life and Environmental
Science, Shimane University, Matsue,
Graduate School of Agriculture, Kyoto
University, Kyoto, Japan
Hidekazu Yoshioka, Faculty of Life and
Environmental Science, Shimane
University, Matsue 690-8504, Japan.
JSPS, Grant/Award Number: 15H06417;
WEC Applied Ecology Research,
Grant/Award Number: 2016-02
This short paper presents an optimal control model for cost-effective and ecolog-
ically conscious management of released fish. Finding an optimal management
strategy reduces to solving a Hamilton-Jacobi-Bellman equation having a solu-
tion by separation of variables. This paper analyzes the behavior of the solution
and the associated optimal management strategy. The latter is verified with an
actual strategy for Plecoglossus altivelis in Japan.
bioeconomic model, Hamilton-Jacobi-Bellman equation, inland fishery, optimal control
Fish in inland waters are one of humanity's main sources of protein. A current severe issue facing inland fisheries is the
significant decline in fish catches.
This is not simply an economic but also an ecological issue as many of the fish are
important for aquatic ecosystems.
They are also culturally important for humans; an example is Plecoglossus altivelis
(Ayu) in Japan.
Finding an appropriate management strategy of the fish is thus an urgent issue for inland fishery.
Bioeconomic models have been utilized for analyzing fish population dynamics.
In application, fish population is often
nonrenewable. The example is P. altivelis released into a river in spring that grows up until the coming autumn in the
same river. Because released P. altivelis have been reported as not contributing to reproduction,
released members of
this species may be modeled as nonrenewable. Recently, Japanese inland fishery seeks a cost-effective and ecologically
conscious management strategy for P. altivelis, which is the main motivation of this paper.
The purpose here is to present and analyze a bioeconomic model for management of nonrenewable fish population. The
model is based on controlled population dynamics, and finding their optimal management strategy reduces to solving a
Hamilton-Jacobi-Bellman equation (HJBE).
The HJBE turns out to have a solution by separation of variables governed
by an ordinary differential equation (ODE). This paper explores the behavior of the solution and the associated optimal
management strategy with particular emphasis on the application to released P. altivelis.
2 MATHEMATICAL MODEL
The growth period of nonrenewable fish population is set as [0, T] with the release time t = 0 and the terminal time t = T.
The total number of individuals and their representative body weight at the time t are denoted as N
∶[0, T] →
∶[0, T] →
, respectively. N
= R + k
Optim Control Appl Meth. 2018;39:1141–1146. wileyonlinelibrary.com/journal/oca Copyright © 2017 John Wiley & Sons, Ltd. 1141