Il’ichev, V. G. Local and global properties of nonautonomous dynamical systems and their application to competition models. (Russian, English) Zbl 1030.92023 Sib. Mat. Zh. 44, No. 3, 622-635 (2003); translation in Sib. Math. J. 44, No. 3, 490-499 (2003). Summary: We develop an inheritance principle for local properties by the global Poincaré mapping of nonautonomous dynamical systems. Namely, if a semigroup property is uniformly locally universal then it is shared by the global Poincaré mapping. In studying the global dynamics of competitors in a periodic medium, a crucial role is played by the multiplicative semigroup of the so-called sign-invariant matrices. We give geometric criteria for stability of equilibria (periodic solutions) in competition models. Cited in 1 Document MSC: 92D25 Population dynamics (general) 34D23 Global stability of solutions to ordinary differential equations 37N25 Dynamical systems in biology Keywords:universality; coarseness; sign-invariant matrices; global stability PDFBibTeX XMLCite \textit{V. G. Il'ichev}, Sib. Mat. Zh. 44, No. 3, 622--635 (2003; Zbl 1030.92023); translation in Sib. Math. J. 44, No. 3, 490--499 (2003) Full Text: EuDML EMIS