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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.

MSC:

92D25 Population dynamics (general)
34D23 Global stability of solutions to ordinary differential equations
37N25 Dynamical systems in biology
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