OPTIMAL SHAPES OF WEIRS FOR TRAPPING MIGRATORY FISH

Abstract

Use of fishing weirs can be seen throughout the world to trap migratory fish in rivers. In rivers flowing into Lake Biwa, Shiga prefecture, Japan, a peculiar type of traditional fishing weir has been operated, targeting anadromous fish species such as Plecoglossus altivelis and Oncorhynchus masou rhodurus. That type of fishing weir has the advantage of catching fish alive, adding an extra value to the fish. Hence the structure of such a weir is considered to be historically optimized so that physiological damages to the fish are minimum. In this study, assuming that the horizontal shape is designed to minimize the traveling time of fish from any point along the downstream side of the weir to the gate of no return situated at the bank, as well as to minimize the size of structure, a mathematical problem is formulated in the framework of dynamic programming to determine the optimal shape. Geometric consideration results in the traveling time as a functional of the shape, whose slope of the tangent is dealt with as the control variable. The value function and the optimal control solve the Hamilton-Jacobi-Bellman equation, which represents the principle of optimality. The system of the Hamilton-Jacobi-Bellman equation is finally reduced to an ordinary differential equation with an initial condition. Some computational results are in good agreement with the actual shapes of the fishing weirs installed across the rivers flowing into Lake Biwa. This mathematical approach is also applicable to other problems such as optimal design of fish ladders.