Microswimmers lose by swimming selfishly

Abstract

At low Reynolds number, microswimmers are coupled by the flows they create, even when they are apart. The best way for each swimmer to push itself forward then depends on its neighbors, turning swimming together into a game. To study this, we optimize the shape changes of soft swimmers, alone and in pairs, with a differentiable Stokes solver. From random initial membrane motions, a lone swimmer converges to a traveling membrane wave. In a pair, the stroke of a swimmer affects its neighbor far more than the neighbor affects its own best stroke. For swimmers in line, this influence decays as $1/\ell$ with their separation $\ell$, whereas the incentive to respond decays as $1/\ell^2$. Selfish swimmers, which maximize their own payoff, respond only to the weak incentive. The cooperative gain they miss decays only as $1/\ell^2$, and their selfish adaptations leave the pair worse off than if both had kept their solitary strokes. Related swimmers, which also value the payoff of their neighbor, recover this gain, whereas competitors pay to hinder each other, both following a law that we derive analytically. Hydrodynamics alone thus makes swimming in company a social dilemma, whose outcome depends on how much the swimmers value each other.