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This paper studies traveling fronts to cooperation–diffusion systems in R N for N ≥ 3 . We consider ( N − 2 ) -dimensional smooth surfaces as boundaries of strictly convex compact sets in R N − 1 , and define an equivalence relation between them. We prove that there exists a traveling front associated with a given surface and show its stability. The associated traveling fronts coincide up to phase transition if and only if the given surfaces satisfy the equivalence relation. [ABSTRACT FROM AUTHOR]