Abstract
In this paper, we study mathematical models of three dimensional competitive systems originated from the represillator and their variants. There are two parts for the mathematical results. In part I, we first prove the uniqueness of the positive equilibrium $ (x^{*}, y^{*}, z^{*}) $. Then we present necessary and sufficient conditions for their local asymptotic stability and instability. In part II, we present sufficient conditions for the global asymptotic stability of $ (x^{*}, y^{*}, z^{*}) $ provided $ (x^{*}, y^{*}, z^{*}) $ is locally asymptotic stable.
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