  Living microorganisms are capable of a tactic response to external stimuli by
swimming towards or away from the stimulus source; they do so by adapting their
tactic signal transduction pathways to the environment. Their self-motility
thus allows them to swim against a traveling tactic wave, whereas a simple
fore-rear asymmetry argument would suggest the opposite. Their biomimetic
counterpart, the artificial microswimmers, also propel themselves by harvesting
kinetic energy from an active medium, but, in contrast, lack the adaptive
capacity. Here we investigate the transport of artificial swimmers subject to
traveling active waves and show, by means of analytical and numerical methods,
that self-propelled particles can actually diffuse in either direction with
respect to the wave, depending on its speed and waveform. Moreover, chiral
swimmers, which move along spiraling trajectories, may diffuse preferably in a
direction perpendicular to the active wave. Such a variety of tactic responses
is explained by the modulation of the swimmer's diffusion inside traveling
active pulses.
