# Rings of constants of four-variable Lotka-Volterra systems

Open Mathematics (2013)

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

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topJanusz 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

top- [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] 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] 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] 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] 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] Moulin Ollagnier J., Nowicki A., Polynomial algebra of constants of the Lotka-Volterra system, Colloq. Math., 1999, 81(2), 263–270 Zbl1004.12004
- [7] Nowicki A., Polynomial Derivations and their Rings of Constants, Uniwersytet Mikołaja Kopernika, Torun, 1994 Zbl1236.13023
- [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] 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] 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] 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] 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