Geometric methods for efficient planar swimming of copepod nauplii

TL;DR

This paper models the planar swimming of copepod nauplii using geometric control theory, demonstrating controllability, analyzing singular configurations, and numerically exploring optimal swimming strategies.

Contribution

It introduces a mathematically tractable model of copepod nauplius swimming and applies optimal control theory to analyze energy-efficient movement strategies.

Findings

Model reproduces observed copepod behavior

System is controllable with singular configurations

Numerical simulations reveal interesting extremal behaviors

Abstract

Copepod nauplii are larval crustaceans with important ecological functions. Due to their small size, they experience an environment of low Reynolds number within their aquatic habitat. Here we provide a mathematical model of a swimming copepod nauplius with two legs moving in a plane. This model allows for both rotation and two-dimensional displacement by periodic deformation of the swimmer's body. The system is studied from the framework of optimal control theory, with a simple cost function designed to approximate the mechanical energy expended by the copepod. We find that this model is sufficiently realistic to recreate behavior similar to those of observed copepod nauplii, yet much of the mathematical analysis is tractable. In particular, we show that the system is controllable, but there exist singular configurations where the degree of non-holonomy is non-generic. We also partially characterize the abnormal extremals and provide explicit examples of families of abnormal curves. Finally, we numerically simulate normal extremals and observe some interesting and surprising phenomena.