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

Abstract

Fish migration in open channels, such as rivers and drainage canals, is a fundamental research topic of civil and environmental engineering. This paper establishes a new and simple mathematical approach for efficiently analyzing upstream fish migration in 1-D open channel flows based on an optimization theory considering advantages and disadvantages of forming a school. Upstream migration of isolated and schooling fishes follows a differential equation where swimming speed and total number of individuals are optimized, so that a biological cost function is minimized. The optimal swimming speed of a fish school and its total number of individuals are exactly derived assuming realistic functional shapes of the cost function. The optimal swimming velocity is validated with experimental results of upstream fish migration of three migratory fish species. Reasonable ranges of several model parameters involved in the cost function are inferred from theoretical consideration. The obtained results of this paper employing a new mathematical approach can be potentially useful for considering swimming behavior of isolated and schooling fishes.

A Mixed Optimal Control Approach for Upstream Fish Migration

“…is the swimming velocity of the individual against the current with the speed Vflow. Correlation between the observed and modelled uswim is 0.80 [27], showing its reasonable accuracy. The cost function…”

“…This is because these effects are possibly interacting and difficult to mathematically describe in detail. Therefore, the lumped approach [27] is employed:…”

“…Following the literature for viscosity solutions [27], the definition of viscosity solutions to the eq. ( 6) is stated as follows.…”

“…The cost function f for P. altivelis is firstly presented, which is used in the simulation below. Yoshioka et al [27] collected the observed datasets of swimming behavior of individual P. altivelis against water currents, and found the concave formula:…”