Heritable properties of nonautonomous dynamical systems and their applications in competition models. (English. Russian original) Zbl 1035.37046

Il’ichev, V. G.

Russ. Math. 46, No. 6, 24-34 (2002); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 2002, No. 6, 26-36 (2002).

For nonautonomous dynamical systems, the author substantiates the inheritance principle of their local properties by the Poincaré mapping. It means that, if a certain property is rough, locally universal and semigroup, then the global Poincaré mapping has the same property. Heritable properties are investigated both for discrete and continuous dynamical systems. An application is given to competition models between two participants, described by the system \(\dot x_1=x_1 f_1(x_1,x_2,t)\), \(\dot x_2=x_2 f_2(x_1,x_2,t)\) with smooth functions \(f_j\), \(j=1,2\), decreasing in \(x_i\) and \(T\)-periodic in \(t\).

Reviewer: Boris V. Loginov (Ulyanovsk)

MSC:

37N25	Dynamical systems in biology
34C25	Periodic solutions to ordinary differential equations
92D25	Population dynamics (general)

Keywords:

discrete and continuous dynamical systems; Poincaré mapping; inheritable properties