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Summary: A ${C}^{1}$-map $f$ of a compact manifold $M$ is a Morse-Smale endomorphism if the nonwandering set of $f$ is finite and hyperbolic and the local stable and global unstable manifolds of periodic points intersect transversally. Morse-Smale endomorphisms appear naturally in the dynamics of the evolution operator on the set of traveling wave solutions for lattice models of unbounded media. The main result of this paper is the openness of the set of Morse-Smale endomorphisms in the space ${C}^{1}(M,M)$ of ${C}^{1}$-maps of $M$ into itself. The usual order relation on $f$ (given by the intersections of local stable and global unstable manifolds) is used to describe the orbit structure of $f$ and its small ${C}^{1}$-perturbations.