## Abstract

Abstract The Lotka–Volterra system of autonomous differential equations consists in three homogeneous polynomial equations of degree 2 in three variables. This system, or the corresponding vector field V(A,B,C) , depends on three non-zero parameters and writes V(A,B,C)=V x ∂ x +V y ∂ y +V z ∂ z where V x =x(Cy+z), V y =y(Az+x), V z =z(Bx+y). Similar systems of equations have been studied by Volterra in his mathematical approach of the competition of species. For us, V(A,B,C) is a normal form of a factorisable quadratic system and the study of its first integrals of degree 0 is of great mathematical interest. A first integral is a non-constant function f which satisfies the identity V x ∂f ∂x +V y ∂f ∂y +V z ∂f ∂z =0. As V(A,B,C) is homogeneous, there is a foliation whose leaves are homogeneous surfaces in the three-dimensional space (or curves in the corresponding projective plane), such that the trajectories of the vector field are completely contained in a leaf. A first integral of degree 0 is then a function on the set of all leaves of the previous foliation. In the present paper, we give all values of the triple (A,B,C) of parameters for which V(A,B,C) has an homogeneous rational first integral of degree 0 . Our proof essentially relies on ideas of algebra and combinatorics, especially in proving that some conditions are necessary.

## Full Text

### Topics from this Paper

- First Integral Of Degree
- System Of Autonomous Differential Equations
- Ideas Of Algebra
- Rational Integral
- Lotka Volterra + Show 5 more

Create a personalized feed of these topics

Get Started#### Talk to us

Join us for a 30 min session where you can share your feedback and ask us any queries you have

Schedule a call### Similar Papers

- Journal of Pure and Applied Algebra
- Dec 1, 2001

- Qualitative Theory of Dynamical Systems
- Sep 1, 2001

- Journal of Physics A: Mathematical and Theoretical
- May 30, 2007

- Rendiconti del Circolo Matematico di Palermo
- Dec 1, 2010

- Nonlinear Analysis: Theory, Methods & Applications
- Jul 1, 2007

- Journal of Mathematical Analysis and Applications
- May 1, 2015

- Nonlinear Analysis: Theory, Methods & Applications
- May 1, 2009

- Aug 26, 2019

- Journal of Geometry and Physics
- Nov 1, 2018

- Ukrainian Mathematical Journal
- Sep 1, 1999

- Journal of Physics A: Mathematical and General
- May 21, 1994

- Electronic Journal of Differential Equations
- May 3, 2021

- Applied Mathematics and Computation
- Jan 1, 2015

- Qualitative Theory of Dynamical Systems
- Sep 1, 2004

- arXiv: Mathematical Physics
- Jan 13, 2020

### Bulletin des Sciences Mathématiques

- Bulletin des Sciences Mathématiques
- Dec 1, 2023

- Bulletin des Sciences Mathématiques
- Dec 1, 2023

- Bulletin des Sciences Mathématiques
- Dec 1, 2023

- Bulletin des Sciences Mathématiques
- Dec 1, 2023

- Bulletin des Sciences Mathématiques
- Dec 1, 2023

- Bulletin des Sciences Mathématiques
- Dec 1, 2023

- Bulletin des Sciences Mathématiques
- Dec 1, 2023

- Bulletin des Sciences Mathématiques
- Dec 1, 2023

- Bulletin des Sciences Mathématiques
- Nov 1, 2023

- Bulletin des Sciences Mathématiques
- Nov 1, 2023