On the first integrals of Lotka-Volterra systems
type: Item
instance of: scholarly article
title: On the first integrals of Lotka-Volterra systems
zbMATH Open document ID: 0948.34025
DOI: 10.1016/S0375-9601(00)00011-6
author: F. G. Gascon
author: Daniel Peralta-Salas
published in: Physics Letters. A
publication date: 2000-07-03T00:00:00Z
review text: The authors consider the construction of time-independent first integrals for three-dimensional Lotka-Volterra systems. They propose a new approach based on a generalized notion of symmetry: a vector field \(S_1\) is a generalized symmetry of the vector field \(X\), if \(S_1\) and \(X\) span a \(2\)-distribution. Assume furthermore that an external symmetry \(S_2\) of this distribution is known and denote by \(w_1\) the contraction of the volume form with \(X\) and \(S_1\). The authors show that this one-form is integrable and that an integrating factor is given by the inverse of the contraction of the one-form with \(S_2\). As a concrete application, the authors take for \(X\) a Lotka-Volterra vector field, for \(S_2\) the dilatation vector field and make a linear ansatz for \(S_1\). With this approach they identify nine integrable classes of Lotka-Volterra equations. Only for one special case the integrating factor is given explicitly, in order to demonstrate that in general the results are only of a local nature.
Mathematics Subject Classification ID: 34C14
Mathematics Subject Classification ID: 34C07
Mathematics Subject Classification ID: 92D25
Mathematics Subject Classification ID: 34A25
zbMATH DE Number: 1468411
zbMATH Keywords: Lotka-Volterra systems
zbMATH Keywords: first integrals
zbMATH Keywords: symmetry
reviewed by: Werner M. Seiler
MaRDI profile type: Publication
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full work available at URL: s0375-9601(00)00011-6
OpenAlex ID: W2061232139
description: scientific article; zbMATH DE number 1468411