Abstract
The swimming cost is one of the most important quantities for comprehension and assessment of fish migration. We propose a generalized exact formula for the swimming cost per unite time during upstream migration of isolated fish. The formula assumes a quadratic relationship between the flow speed and the swimming speed, which turns out to reasonably handle the qualitatively different experimental observation results of the fish swimming behavior available in the literature. Starting from the quadratic relationship, with the help of an ordinary differential equation derived from an optimality principle, the exact formula of the swimming cost is derived with the help of an elementary calculus technique. The derived swimming cost is an increasing and convex function of the swimming speed, which agrees well with the reported results for a wide range of fish species. Applicability and limitation of the present formula are discussed with an emphasis on biological meaning of its strict positivity and the range of swimming speed. The obtained results of this paper give new insights into modeling and analysis of upstream fish migration.
