Permanence and universal classification of discrete-time competitive systems via the carrying simplex

Abstract

<p style='text-indent:20px;'>We study the permanence and impermanence for discrete-time Kolmogorov systems admitting a carrying simplex. Sufficient conditions to guarantee permanence and impermanence are provided based on the existence of a carrying simplex. Particularly, for low-dimensional systems, permanence and impermanence can be determined by boundary fixed points. For a class of competitive systems whose fixed points are determined by linear equations, there always exists a carrying simplex. We provide a universal classification via the equivalence relation relative to local dynamics of boundary fixed points for the three-dimensional systems by the index formula on the carrying simplex. There are a total of <inline-formula><tex-math id="M1">\begin{document}$ 33 $\end{document}</tex-math></inline-formula> stable equivalence classes which are described in terms of inequalities on parameters, and we present the phase portraits on their carrying simplices. Moreover, every orbit converges to some fixed point in classes <inline-formula><tex-math id="M2">\begin{document}$ 1-25 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ 33 $\end{document}</tex-math></inline-formula>; there is always a heteroclinic cycle in class <inline-formula><tex-math id="M4">\begin{document}$ 27 $\end{document}</tex-math></inline-formula>; Neimark-Sacker bifurcations may occur in classes <inline-formula><tex-math id="M5">\begin{document}$ 26-31 $\end{document}</tex-math></inline-formula> but cannot occur in class <inline-formula><tex-math id="M6">\begin{document}$ 32 $\end{document}</tex-math></inline-formula>. Based on our permanence criteria and the equivalence classification, we obtain the specific conditions on parameters for permanence and impermanence. Only systems in classes <inline-formula><tex-math id="M7">\begin{document}$ 29, 31, 33 $\end{document}</tex-math></inline-formula> and those in class <inline-formula><tex-math id="M8">\begin{document}$ 27 $\end{document}</tex-math></inline-formula> with a repelling heteroclinic cycle are permanent. Applications to discrete population models including the Leslie-Gower models, Atkinson-Allen models and Ricker models are given.