Nonreciprocal McKean-Vlasov Equations: From Stationary Instabilities to Travelling Waves
Nonreciprocal interactions, in which action-reaction symmetry is broken, provide a powerful route to collective dynamics that cannot be captured by equilibrium free-energy minimisation. Here, we introduce and analyse a two-species nonreciprocal McKean-Vlasov equation derived from an underlying system of interacting stochastic particles. Combining linear stability analysis, weakly nonlinear arguments, pseudo-spectral simulations, and Langevin particle dynamics, we show that the structure of nonreciprocity controls the onset and nature of collective order. For spatially uniform weak nonreciprocity, asymmetry shifts the critical diffusion threshold but produces only stationary instabilities, indicating that uniform imbalance alone is insufficient to generate sustained time-dependent motion. In contrast, spatially modulated nonreciprocity fundamentally enriches the dynamics: depending on its symmetry and coupling to the interaction potential, the homogeneous state can lose stability through Hopf bifurcations, giving rise to standing and travelling wave states. We identify both subcritical and supercritical Hopf transitions, relate the selected patterns to Landau saturation coefficients, and show that travelling waves can emerge even in the weak-nonreciprocity regime without explicit microscopic run-and-chase rules. Direct Langevin simulations confirm that these oscillatory and travelling states persist at the particle level and are not artefacts of the continuum mean-field description. Our results establish nonreciprocal McKean-Vlasov equations as a minimal framework for understanding how spatially structured asymmetric interactions generate self-organized motion, dynamical phase transitions, and nonequilibrium collective order.