Rational integration of the Lotka–Volterra system

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.