Linear spreading speed in non-monotone population models

For a broad class of discrete-time, finite-range interacting particle systems on $\mathbb{Z}$, we establish a linear spreading speed and a one-dimensional shape theorem on the event of survival, without assuming monotonicity or attractiveness of the dynamics. The method requires that the system admits a coupling with supercritical oriented percolation on a coarse-grained lattice. The central technical step is an approximate subadditivity property for the hitting times, obtained through a `shifted coupling’ that compensates for the absence of monotonicity. As a concrete application, we show that a discrete-time branching annihilating random walk fits into this framework, and consequently exhibits a linear spreading speed.