Rings of constants of four-variable Lotka-Volterra systems

Janusz Zieliński

Open Mathematics (2013)

  • Volume: 11, Issue: 11, page 1923-1931
  • ISSN: 2391-5455

Abstract

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Lotka-Volterra systems appear in population biology, plasma physics, laser physics and derivation theory, among many others. We determine the rings of constants of four-variable Lotka-Volterra derivations with four parameters C 1, C 2, C 3, C 4 ∈ k, where k is a field of characteristic zero. Thus, we give a full description of polynomial first integrals of the respective systems of differential equations.

How to cite

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Janusz Zieliński. "Rings of constants of four-variable Lotka-Volterra systems." Open Mathematics 11.11 (2013): 1923-1931. <http://eudml.org/doc/269582>.

@article{JanuszZieliński2013,
abstract = {Lotka-Volterra systems appear in population biology, plasma physics, laser physics and derivation theory, among many others. We determine the rings of constants of four-variable Lotka-Volterra derivations with four parameters C 1, C 2, C 3, C 4 ∈ k, where k is a field of characteristic zero. Thus, we give a full description of polynomial first integrals of the respective systems of differential equations.},
author = {Janusz Zieliński},
journal = {Open Mathematics},
keywords = {Lotka-Volterra derivation; Polynomial constant; Polynomial first integral; ring of constants},
language = {eng},
number = {11},
pages = {1923-1931},
title = {Rings of constants of four-variable Lotka-Volterra systems},
url = {http://eudml.org/doc/269582},
volume = {11},
year = {2013},
}

TY - JOUR
AU - Janusz Zieliński
TI - Rings of constants of four-variable Lotka-Volterra systems
JO - Open Mathematics
PY - 2013
VL - 11
IS - 11
SP - 1923
EP - 1931
AB - Lotka-Volterra systems appear in population biology, plasma physics, laser physics and derivation theory, among many others. We determine the rings of constants of four-variable Lotka-Volterra derivations with four parameters C 1, C 2, C 3, C 4 ∈ k, where k is a field of characteristic zero. Thus, we give a full description of polynomial first integrals of the respective systems of differential equations.
LA - eng
KW - Lotka-Volterra derivation; Polynomial constant; Polynomial first integral; ring of constants
UR - http://eudml.org/doc/269582
ER -

References

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  1. [1] Almeida M.A., Magalhães M.E., Moreira I.C., Lie symmetries and invariants of the Lotka-Volterra system, J. Math. Phys., 1995, 36(4), 1854–1867 http://dx.doi.org/10.1063/1.531362 Zbl0822.34006
  2. [2] Bogoyavlenskiĭ O.I., Algebraic constructions of integrable dynamical systems - extension of the Volterra system, Russian Math. Surveys, 1991, 46(3), 1–64 http://dx.doi.org/10.1070/RM1991v046n03ABEH002801 Zbl0774.35013
  3. [3] Cairó L., Llibre J., Darboux integrability for 3D Lotka-Volterra systems, J. Phys. A, 2000, 33(12), 2395–2406 http://dx.doi.org/10.1088/0305-4470/33/12/307 Zbl0953.34037
  4. [4] Hegedűs P., The constants of the Volterra derivation, Cent. Eur. J. Math., 2012, 10(3), 969–973 http://dx.doi.org/10.2478/s11533-012-0024-8 Zbl1241.13022
  5. [5] Kuroda S., Fields defined by locally nilpotent derivations and monomials, J. Algebra, 2005, 293(2), 395–406 http://dx.doi.org/10.1016/j.jalgebra.2005.06.011 Zbl1131.13022
  6. [6] Moulin Ollagnier J., Nowicki A., Polynomial algebra of constants of the Lotka-Volterra system, Colloq. Math., 1999, 81(2), 263–270 Zbl1004.12004
  7. [7] Nowicki A., Polynomial Derivations and their Rings of Constants, Uniwersytet Mikołaja Kopernika, Torun, 1994 Zbl1236.13023
  8. [8] Nowicki A., Zielinski J., Rational constants of monomial derivations, J. Algebra, 2006, 302(1), 387–418 http://dx.doi.org/10.1016/j.jalgebra.2006.02.034 Zbl1119.13021
  9. [9] Ossowski P., Zielinski J., Polynomial algebra of constants of the four variable Lotka-Volterra system, Colloq. Math., 2010, 120(2), 299–309 http://dx.doi.org/10.4064/cm120-2-9 Zbl1207.13016
  10. [10] Zielinski J., Factorizable derivations and ideals of relations, Comm. Algebra, 2007, 35(3), 983–997 http://dx.doi.org/10.1080/00927870601117639 Zbl1171.13013
  11. [11] Zielinski J., The five-variable Volterra system, Cent. Eur. J. Math., 2011, 9(4), 888–896 http://dx.doi.org/10.2478/s11533-011-0032-0 Zbl1236.13024
  12. [12] Zielinski J., Ossowski P., Rings of constants of generic 4D Lotka-Volterra systems, Czechoslovak Math. J., 2013, 63(138)(2), 529–538 http://dx.doi.org/10.1007/s10587-013-0035-z Zbl1289.13017

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