Nonreciprocity as a generic route to traveling and oscillatory states*

Time periodic patterns occur ubiquitously in nature. Most mathematical models that capture such spatiotemporal dynamics, including reaction–diffusion equations, excitable systems, and prey–predator equations, are unified by the fact that the dynamical variables are nonconserved fields. Here, we demonstrate that traveling and oscillatory patterns can arise in multicomponent systems described by purely diffusive conserved fields from nonreciprocal interactions between species. The appearance of time periodic states in a purely diffusive system with no apparent external forcing is unexpected and defies intuition. Our work suggests that nonreciprocity provides a generic mechanism for the establishment of time periodic states, which might be relevant to both biological systems and synthetic systems with nonreciprocal effective interactions mediated by non-equilibrium environments, such as mixtures of active colloids.