Global distribution of the limit-sets for competitive flows

We give the global distribution of the limit sets for competitive dynamical systems in $ \mathbb{R}^n $. The main result is that, for dissipative competitive dynamical systems in $ \mathbb{R}^n $, there is a countable family of disjoint unordered invariant open $ (n-1) $-cells attracting all non-convergent trajectories, which is a natural extension of Hirsch's work [8].