Identifying the Cost Function for Upstream Migration of Individual Fishes in 1-D Open Channels Based on an Optimal Control Theory

Abstract

This paper verifies a control-theoretic approach for assessing upstream migration of individual fishes in 1-D open channels. This approach uses a solution to a Hamilton-Jacobi-Bellman equation (HJBE) for the optimal swimming velocity of individual fishes as the minimizer of the cost to migrate upstream. With the help of an analytically derived formula from the solution to the HJBE, the cost function, which is the core of the control-theoretic approach, is identified from the observed data sets of swimming speed of migratory fishes in laboratorial and natural open channels. Both advantages and disadvantages of using the identified cost function are discussed and its sensitivity analysis is carried out. Applicability of the presented mathematical model with the identified cost function to practical problems is also mentioned to indicate its potential applicability to practical problems.

A simple game-theoretic model for upstream fish migration

“…1. The cost function f in (4) shows that the fish is more high speed-averse for larger n, because it more rapidly grows for larger n. The cost function f in (4) is convex and increasing with respect to u; which is in good accordance with the conventional experimental results on swimming behavior of fishes (Brodersen et al 2008;Cucco et al 2012;Mori et al 2015;Roche et al 2013;Svendsen et al 2010;Yoshioka et al 2016a). The objective function is rewritten with (2) as…”

“…On the parameter n, Yoshioka et al (2016a) showed Oð10 0 Þ for a variety of migratory fish species. Identification of the order of the parameter k is more difficult than that for the parameters n and m because of its lumped nature.…”

“…The variables to be optimized in the present model contain the swimming speed and school size that should be chosen by the fish school, so that the net cost of migration is minimized. The present model is an extended counterpart of the model for upstream migration of isolated individuals (Yoshioka and Shirai 2015;Yoshioka et al , 2016a. Assuming that certain simplifications reduce finding the optimizer to solving a self-consistency equation.…”

Cost-Minimizing Upstream Migration Strategy of Isolated and Schooling Fishes in 1-D Open Channel Flows

“…The objective function consists of the two independent terms; the first term represents the hydrodynamic cost and the other represents the non-hydrodynamic cost. Here, f represents the hydraulic cost per time, which is a convex and increasing function of u 15) . It is reasonable to assume that no swimming cost has to be paid when the fishes do not swim ( 0…”

“…respectively, where the condition * max V u u < < is certainly satisfied in Eq. (15). The above candidate of minimizer satisfies a second-order necessary condition 21)…”