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Dynamics of a family of Lotka-Volterra systems in \(\mathbb{R}^3\). (English) Zbl 1451.37025

Summary: We provide the phase portraits of the 3-dimensional Lotka-Volterra systems \[ \dot{x} = x(y + az), \dot{y} = y(x + z), \dot{z} = bz (-ax + y), \] for all the values of the parameters \(a\) and \(b\), in the finite region and in the infinity region through the Poincaré compactification. We also study the integrability of the system.

MSC:

37C10 Dynamics induced by flows and semiflows
37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
34C07 Theory of limit cycles of polynomial and analytic vector fields (existence, uniqueness, bounds, Hilbert’s 16th problem and ramifications) for ordinary differential equations
70K05 Phase plane analysis, limit cycles for nonlinear problems in mechanics
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