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This paper is devoted to the study of propagation dynamics for a large classof non-monotone evolution systems. In two directions of the spatial variable,such a system has two limiting systems admitting the spatial translationinvariance. Under the assumption that each of these two limiting systems hasboth leftward and rightward spreading speeds, we establish the spreadingproperties of solutions and the existence of nontrivial fixed points, steadystates, traveling waves for the original systems. We also apply the developedtheory to a time-delayed reaction-diffusion equation with a shifting habitatand a class of asymptotically homogeneous reaction-diffusion systems.

Spatio-temporal dynamics for non-monotone semiflows with limiting systems having spreading speeds