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 $\sigma$. For general discrete and fully nonlinear dynamics, we study traveling waves of velocity $c=c(\sigma )$ depending on the parameter $\sigma$. Under certain assumptions, we show properties of the velocity diagram $c(\sigma )$ for $\sigma \in [{\sigma }^{-},{\sigma }^{+}]$. We show that the velocity $c$ is nondecreasing in $\sigma \in ({\sigma }^{-},{\sigma }^{+})$ in the bistable regime, with vertical branches $c\ge c^{+}$ for $\sigma ={\sigma }^{+}$ and $c\le c^{-}$ for $\sigma ={\sigma }^{-}$ in the monostable regime.