# A model against the flow

A mathematical model uncovers the hydrodynamic mechanism that causes fish to swim against the flow of current.

One fascinating and perplexing fact about fish is that they tend to orient themselves and swim against the flow, rather than with it. This phenomenon is called rheotaxis, and it has countless examples, from salmon migrating upstream to lay their eggs to trout drift-foraging in a current. Yet, despite over a century of experimental studies, the mechanisms underlying rheotaxis remain poorly understood. There is general consensus that fish rely on water- and body-motion cues to vision, vestibular, tactile, and other senses. However, several questions remain unanswered, including how blind fish can perform rheotaxis or whether a passive hydrodynamic mechanism can support the phenomenon. One aspect that has been overlooked in studies of rheotaxis is the bidirectional hydrodynamic interaction between the fish and the surrounding flow, that is, how the presence of the fish alters the flow, which, in turn, affects the fish.

To address these open questions about rheotaxis, Porfiri, Zhang and Peterson wanted to develop a mathematical model of fish swimming, one that could help understand the passive hydrodynamic pathway that leads to swimming against a flow. Unlike experiments on live animals, a mathematical model offers the ability to remove cues to certain senses without interfering with animal behavior.

Porfiri, Zhang and Peterson modeled a fish as a pair of vortices located infinitely close to each other, rotating in opposite directions with the same strength. The vortex pair could freely move through an infinitely long channel with an imposed background flow, devoid of all sensory information expect of that accessed through the lateral line. Analyzing the resulting system revealed that there is a critical speed for the background flow above which the fish successfully orients itself against the flow, resulting in rheotaxis. This critical speed depends on the width of the channel the fish is swimming in. Depriving the fish of sensory information received through the lateral line does not preclude rheotaxis, indicating that rheotaxis could emerge in a completely passive manner.

The finding that the critical speed for rheotaxis depends on channel width could improve the design of experiments studying the phenomenon, since this effect could confound experiments where fish are confined in narrow channels. In this vein, Porfiri, Zhang and Peterson’s model could assist biologists in designing experiments detailing the multisensory nature of rheotaxis. Evidence of the importance of bidirectional hydrodynamic interactions on fish orientation may also inform modeling research on fish behavior.