Bifurcation theory captures band formation in the Vicsek model of flock formation

Collective behavior occurs ubiquitously in nature and it plays a key role in bacterial colonies, mammalian cells or flocks of birds. Here, we examine the average density and velocity of self-propelled particles, which are described by a system of partial differential equations near the flocking transition of the Vicsek model. This agent-based model illustrates the trend towards flock formation of animals that align their velocities to an average of those of their neighbors. Near the flocking transition, particle density and velocity obey partial differential equations that include a parameter $\epsilon$ measuring the distance to the bifurcation point. We have obtained analytically the Riemann invariants in one and two spatial dimensions for the hyperbolic ($\epsilon=0$) and parabolic ($\epsilon\neq 0$) system and, under periodic initial-boundary value conditions, we show that the solutions include wave trains. Additionally, we have found wave trains having oscillation frequencies that agree with those predicted by a linearization approximation and that may propagate at angles depending on the initial condition. The wave amplitudes increase with time for the hyperbolic system but are stabilized to finite values for the parabolic system. To integrate the partial differential equations, we design a basic numerical scheme which is first order in time and space. To mitigate numerical dissipation and ensure good resolution of the wave features, we also use a high order accurate WENO5 reconstruction procedure in space and a third order accurate Runge-Kutta scheme in time. Comparisons with direct simulations of the Vicsek model confirm these predictions.