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A Mixed Optimal Control Approach for Upstream Fish Migration

Journal of Sustainable Development of Energy, Water and Environment Systems
Volume 7, Issue 1, pp 101-121
DOI: http://dx.doi.org/10.13044/j.sdewes.d6.0221 (registered soon)
Hidekazu Yoshioka1 , Tomoyuki Shirai2, Daisuke Tagami2
1 Faculty of Life and Environmental Science, Shimane University, Nishikawatsu-cho 1060, Matsue 690-8504, Japan
2 Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka 819-0395, Japan

Abstract

This paper proposes a simple mathematical model forupstream fish migration along rivers. The model describes the fish migration along a river based on a mixed optimal control approach having swimming velocity, school size, and stopping time of migration as control variables. The optimization problem reduces to a variational inequality. Its explicit “viscosity” solution is presented with the dependence of the fish migration on river environment. To prove uniqueness of the solution to the variational inequality requires a constructive argument not based on the conventional theorems. A novel finite difference scheme for solving the variational inequality is also proposed with its convergence results. An application example of the model discusses the upstream migration of Plecoglossus altivelis (Ayu) in Japan, which evaluates the dependence of the fish migration on the habitat quality and provides recommendations for managing river environment. This is an interdisciplinary research between environmental and mathematical fields.

Keywords: Upstream fish migration, Variational inequality, Viscosity solution, Finite difference scheme, Plecoglossus altivelis (Ayu).

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