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Liouville integrable Lotka-Volterra systems

Peter H. van der Kamp, David I. McLaren, G. R. W. Quispel

Abstract

We present $\frac{m^{2}}{4}+\frac{m}{2}+\frac{1-\left(-1\right)^{m}}{8}$ homogeneous $(3m-2)$-parameter families of Liouville integrable $(2m)$- and $(2m-1)$-dimensional Lotka-Volterra systems. We also study inhomogeneous versions of these systems.

Liouville integrable Lotka-Volterra systems

Abstract

We present homogeneous -parameter families of Liouville integrable - and -dimensional Lotka-Volterra systems. We also study inhomogeneous versions of these systems.

Paper Structure

This paper contains 27 sections, 15 theorems, 99 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Homogeneous Liouville integrable LV systems
  3. Even-dimensional homogeneous Liouville integrable LV systems
  4. The even-dimensional homogeneous type $[j,0,0]$ LV system
  5. The even-dimensional homogeneous type $[0,k,0]$ LV system
  6. The even-dimensional homogeneous type $[0,k,l]$ LV system
  7. The even-dimensional homogeneous type $[j,k,l]$ LV system
  8. Odd-dimensional homogeneous Liouville integrable LV systems
  9. The odd-dimensional homogeneous $[j,0,0]$ LV system
  10. The odd-dimensional homogeneous $[0,k,0]$ LV system
  11. The odd-dimensional homogeneous $[0,k,l]$ LV system
  12. The odd-dimensional homogeneous $[j,k,l]$ LV system
  13. Nonhomogeneous Liouville integrable LV systems
  14. Even-dimensional inhomogeneous Liouville integrable LV systems
  15. The even-dimensional inhomogeneous $[j,0,0]$ LV system
  16. ...and 12 more sections

Key Result

Proposition 2

The homogeneous $2(j+1)$-dimensional Lotka-Volterra system LV with matrix is Liouville integrable, with $j$ pairwise Poisson commuting integrals, $\blacktriangleleft$$\blacktriangleleft$

Figures (5)

  • Figure 1: The hypergraph on $6$ vertices of size 2, associated to the LV system with matrix \ref{['fa']}.
  • Figure 2: A tower of hypergraphs. Each hypergraph has a number of edges of degree 2 and at most two sets of nested hyperedges of odd degrees $3,5,\ldots$.
  • Figure 3: A forest on $2(j+1)$ vertices.
  • Figure 4: A hypergraph of order $2(k+1)$ and size $k$.
  • Figure 5: A hypergraph of order $2(k+l+1)$ and size $k+l$.

Theorems & Definitions (28)

  • Definition 1
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Proposition 4
  • proof
  • Proposition 5
  • proof
  • Proposition 6
  • ...and 18 more