Hypergraphs and Lotka-Volterra systems with linear Darboux polynomials

Abstract

We associate parametric classes of $n$-component Lotka-Volterra systems which admit $k$ additional linear Darboux polynomials, with admissible loopless hypergraphs of order $n$ and size $k$. We study the equivalence relation on admissible hypergraphs induced by linear transformations of the associated LV-systems, for $n\le 5$. We present a new 13-parameter 5-component superintegrable Lotka-Volterra system, i.e. one that is not equivalent to a so-called tree-system. We conjecture that tree-systems associated with nonisomorphic trees are not equivalent, which we verified for $n<9$.

20 pages, 17 figures, 7 tables. Version 2: corrected a minus sign in equation (6) and related expressions. Version 3: changed title, revised Lemma 2, added Propositions 3-5, reference [7], and the extension to nonhomogeneous cases. Version 4: most notable changes are that the notion of admissible refers to general classes of LV-systems and the disclaimer that we do not study special subclasses