  This paper is devoted to the study of propagation dynamics for a large class
of non-monotone evolution systems. In two directions of the spatial variable,
such a system has two limiting systems admitting the spatial translation
invariance. Under the assumption that each of these two limiting systems has
both leftward and rightward spreading speeds, we establish the spreading
properties of solutions and the existence of nontrivial fixed points, steady
states, traveling waves for the original systems. We also apply the developed
theory to a time-delayed reaction-diffusion equation with a shifting habitat
and a class of asymptotically homogeneous reaction-diffusion systems.
