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Abstract:
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In this dissertation is performed an analysis of the stability of equilibriums of the
two-dimensional autonomous competitive Lotka–Volterra dynamical system. Necessary and suf-
ficient conditions are determined for equilibriums (without the origin) to be asymptotically sta-
ble or unstable in [0, +∞)2, as well as necessary and sufficient conditions so that the observed
dynamical system has no equilibriums in (0, +∞)2. The appropriate ecological interpretati-
on is also given. It is also obtained that four transcritical bifurcations occur in the observed
dynamical system if it is analyzed on R2.
It is also investigated an analysis of the stability of a three-dimensional autonomous com-
petitive Lotka–Volterra dynamical system. It is also performed an analysis of the stability of
equilibriums lying on the axes or in the interior of the planes of 𝑛-dimensional autonomous
competitive Lotka–Volterra dynamical system. The appropriate ecological interpretation is al-
so given. Conditions for equilibriums lying on the axes or in the interior of the planes to be
asymptotically stable or unstable in [0, +∞)𝑛 are established.
The system of the Friedmann equations, that describes dynamics of the ΛCDM model of
the universe, firstly represented and analyzed as a three-dimensional, and afterwards as an
𝑛-dimensional autonomous dynamical system of a class of Lotka–Volterra dynamical systems,
with density parameters of the universe’s constituents as dependent variables, is investigated.
The appropriate physical interpretation, including observing the evolution of the universe in
the frame of the linear stability theory, is presented. Analytical solutions of these systems
represent new parametrizations of density parameters of the universe’s constituents regarding
to the scale expansion factor of the universe. |