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Rings of constants of generic 4D Lotka-Volterra systems. (English) Zbl 1289.13017

Summary: We show that the rings of constants of generic four-variable Lotka-Volterra derivations are finitely generated polynomial rings. We explicitly determine these rings, and we give a description of all polynomial first integrals of their corresponding systems of differential equations. Besides, we characterize cofactors of Darboux polynomials of arbitrary four-variable Lotka-Volterra systems. These cofactors are linear forms with coefficients in the set of nonnegative integers. Lotka-Volterra systems have various applications in such branches of science as population biology and plasma physics, among many others.

MSC:

13N15 Derivations and commutative rings
12H05 Differential algebra
92D25 Population dynamics (general)
34A34 Nonlinear ordinary differential equations and systems
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References:

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