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<p>We associate parametric classes of <i>n</i>-component Lotka-Volterra systems which admit <i>k</i> additional linear Darboux polynomials, with admissible loopless hypergraphs of order <i>n</i> and size <i>k</i>. We study the equivalence relation on admissible hypergraphs induced by linear transformations of the associated LV-systems, for <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10884_2025_10446_Article_IEq1.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(n\le 5\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≤</mo> <mn>5</mn> </mrow> </math></EquationSource> </InlineEquation>. 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 <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10884_2025_10446_Article_IEq2.gif" Format="GIF" Height="13" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(n&lt;9\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>&lt;</mo> <mn>9</mn> </mrow> </math></EquationSource> </InlineEquation>.</p>

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Hypergraphs and Lotka-Volterra Systems with Linear Darboux Polynomials

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  • 发表日期：

    2025年7月10日

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  • Peter H. van der Kamp

摘要

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 \(n\le 5\) n ≤ 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 \(n<9\) n < 9 .