Velocity diagram of traveling waves for discrete reaction-diffusion equations

We consider a discrete version of reaction-diffusion equations. A typical example is the fully overdamped Frenkel-Kontorova model, where the velocity is proportional to the force. We also introduce an additional exterior force denoted by s. For general discrete and fully nonlinear dynamics, we study traveling waves of velocity c = c(s) depending on the parameter s. Under certain assumptions, we show properties of the velocity diagram c(s) for s. [s-, s +]. We show that the velocity c is nondecreasing in s. (s-, s+) in the bistable regime, with vertical branches c = c+ for s = s+ and c = c- for s = s- in the monostable regime.