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We establish one of the most important assumptions of the strong persistencetheory for dynamical systems associated to cross diffusion systems of $m$equations ($m\ge2$): the stable sets of semi-trivial steady cannot intersectthe interior of the positive cone of $C(\Omega,\mathbb{R}^m)$. Many exampleswill be provided to show the effects of the cross diffusion.

Strong Persistence of a Class of Strongly Coupled Parabolic Systems of $m$ Equations