# Polynomial algebra of constants of the Lotka-Volterra system

Jean Moulin Ollagnier; Andrzej Nowicki

Colloquium Mathematicae (1999)

- Volume: 81, Issue: 2, page 263-270
- ISSN: 0010-1354

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topMoulin Ollagnier, Jean, and Nowicki, Andrzej. "Polynomial algebra of constants of the Lotka-Volterra system." Colloquium Mathematicae 81.2 (1999): 263-270. <http://eudml.org/doc/210738>.

@article{MoulinOllagnier1999,

abstract = {Let k be a field of characteristic zero. We describe the kernel of any quadratic homogeneous derivation d:k[x,y,z] → k[x,y,z] of the form $d = x(Cy+z)\frac\{∂\}\{∂x\} + y(Az+x)\frac\{∂\}\{∂y\} + z(Bx+y)\frac\{∂\}\{∂z\}$, called the Lotka-Volterra derivation, where A,B,C ∈ k.},

author = {Moulin Ollagnier, Jean, Nowicki, Andrzej},

journal = {Colloquium Mathematicae},

language = {eng},

number = {2},

pages = {263-270},

title = {Polynomial algebra of constants of the Lotka-Volterra system},

url = {http://eudml.org/doc/210738},

volume = {81},

year = {1999},

}

TY - JOUR

AU - Moulin Ollagnier, Jean

AU - Nowicki, Andrzej

TI - Polynomial algebra of constants of the Lotka-Volterra system

JO - Colloquium Mathematicae

PY - 1999

VL - 81

IS - 2

SP - 263

EP - 270

AB - Let k be a field of characteristic zero. We describe the kernel of any quadratic homogeneous derivation d:k[x,y,z] → k[x,y,z] of the form $d = x(Cy+z)\frac{∂}{∂x} + y(Az+x)\frac{∂}{∂y} + z(Bx+y)\frac{∂}{∂z}$, called the Lotka-Volterra derivation, where A,B,C ∈ k.

LA - eng

UR - http://eudml.org/doc/210738

ER -

## References

top- [1] B. Grammaticos, J. Moulin Ollagnier, A. Ramani, J.-M. Strelcyn and S. Wojciechowski, Integrals of quadratic ordinary differential equations in ${\mathbb{R}}^{3}$: the Lotka-Volterra system, Phys. A 163 (1990), 683-722. Zbl0714.34005
- [2] J. Moulin Ollagnier, Polynomial first integrals of the Lotka-Volterra system, Bull. Sci. Math. 121 (1997), 463-476. Zbl0887.34002
- [3] J. Moulin Ollagnier, Rational integration of the Lotka-Volterra system, ibid. 123 (1999), 437-466.
- [4] J. Moulin Ollagnier, A. Nowicki and J.-M. Strelcyn, On the non-existence of constants of derivations: the proof of a theorem of Jouanolou and its development, ibid. 119 (1995), 195-233. Zbl0855.34010
- [5] A. Nowicki, Polynomial derivations and their rings of constants, N. Copernicus University Press, Toruń, 1994.
- [6] A. Nowicki, On the non-existence of rational first integrals for systems of linear differential equations, Linear Algebra Appl. 235 (1996), 107-120. Zbl0843.34013
- [7] A. Nowicki and M. Nagata, Rings of constants for k-derivations in $k[{x}_{1},...,{x}_{n}]$, J. Math. Kyoto Univ. 28 (1988), 111-118. Zbl0665.12024
- [8] A. Nowicki and J.-M. Strelcyn, Generators of rings of constants for some diagonal derivations in polynomial rings, J. Pure Appl. Algebra 101 (1995), 207-212. Zbl0832.12002

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