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Liouvillian integration of the Lotka-Volterra system

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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 fieldLV(A, B, C), depends on three nonzero (complex) parameters and may be written asLV(A, B, C)=V x∂x+V y∂y+V z∂zwhereV x=x(Cy+z), Vy=y(Az+x), Vz=z(Bx+y). Similar systems of equations have been studied by Volterra in his mathematical approach of the competition of species and this is the reason why this name has been given to such systems.

In fact,LV(A, B, C) can be chosen as a normal form for most of factored quadratic systems; the study of its first integrals of degree 0 is thus of great mathematical interest.

Given a homogeneous vector field, 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 non-zero parameters for whichLV(A, B, C) has a homogeneous liouvillian first integral of degree 0. We also discuss the corresponding problem of liouvillian integration for quadratic factored vector fields that cannot be put in Lotka-Volterra normal form.

Our proof essentially relies on combinatorics and elementary algebraic geometry, especially in proving that some conditions are necessary.

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  1. Gage, UMS CNRS 658 Medicis, École Polytechnique, F 91128, Palaiseau Cedex, France

    Jean Moulin Ollagnier

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Ollagnier, J.M. Liouvillian integration of the Lotka-Volterra system. Qual. Th. Dyn. Syst 2, 307–358 (2001). https://doi.org/10.1007/BF02969345

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