On the first integrals of Lotka-Volterra systems
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.
scientific article; zbMATH DE number 1468411
scientific article