Bistable Waves and Propagation Success on Hexagonal Lattice

Inspired by propagation phenomena occurring on media with hexagonal structures, we propose an idealized system of bistable reaction-diffusion equations posed on a hexagonal lattice, and investigate the existence and speed of traveling wave solutions. We establish a sharp criterion for propagation success (nonzero wave speed) and failure (zero wave speed), where the criterion is determined by an angle-dependent periodic threshold function and the ratio of two parameters in the nonlinearity. Moreover, the sign of the wave speeds is governed by the angle and the parameter ratio. Our results imply that the hexagonal structure and bistability significantly affect the propagation dynamics.