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Inheritance principle in dynamical systems and an application in ecological models. (English. Russian original) Zbl 1237.37021
Differ. Equ. 47, No. 9, 1259-1270 (2011); translation from Differ. Uravn. 47, No. 9, 1247-1257 (2011).
The author develops a general principle of inheritance for a number of properties by the Poincaré map in order to obtain information on general nonautonomous models. It roughly states that, provided some local property is rough and semigroup (we refer to the paper for precise definitions), then the global map has the same property. For the example of competition models, key inherited properties (e.g. sign-invariant matrices) are specified.

This approach is used to derive global stability of periodic models for various nonlinear ecological models.
MSC:
37B55 Topological dynamics of nonautonomous systems
37N25 Dynamical systems in biology
92D40 Ecology
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[1] Il’ichev, V.G., Local and Global Properties of Nonautonomous Dynamical Systems and Their Application in Competition Models, Sibirsk. Mat. Zh., 2003, vol. 44, no. 3, pp. 622–635.
[2] Arnol’d, V.I., Obyknovennye differentsial’nye uravneniya (Ordinary Differential Equations), Moscow: Nauka, 1984.
[3] Il’ichev, V.G., Hereditary Properties of Nonautonomous Dynamical Systems and Their Application in Competition Models, Izv. Vyssh. Uchebn. Zaved. Mat., 2002, no. 6, pp. 26–36.
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[5] Il’ichev, V.G. and Il’icheva, O.A., Sign-Invariant Structures of Matrices, and Discrete Models, Diskret. Mat., 1999, vol. 11, no. 4, pp. 89–100. · doi:10.4213/dm397
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