Spatio-temporal dynamics for a class of monotone evolution systems

In this paper, under an abstract setting we establish the spreading properties and the existence, non-existence and global attractivity of spatially heterogeneous steady states for a large class of monotone evolution systems without the translational monotonicity under the assumption that one limiting system has both leftward and rightward spreading speeds and the other one has the uniform asymptotic annihilation. Then we apply the developed theory to study the global dynamics of asymptotically homogeneous integro-difference equations, and provide a counter-example to show that the value of the nonlinear function at the finite range of location may give rise to nontrivial fixed points.