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Measure preservation and integrals for Lotka--Volterra tree-systems and their Kahan discretisation

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

Abstract

We show that any Lotka--Volterra tree-system associated with an $n$-vertex tree, as introduced in Quispel et al., J. Phys. A 56 (2023) 315201, preserves a rational measure. We also prove that the Kahan discretisation of these tree-systems factorises and preserves the same measure. As a consequence, for the Kahan maps of Lotka--Volterra systems related to the subclass of tree-systems corresponding to graphs with more than one $n$-vertex subtree, we are able to construct rational integrals.

Measure preservation and integrals for Lotka--Volterra tree-systems and their Kahan discretisation

Abstract

We show that any Lotka--Volterra tree-system associated with an -vertex tree, as introduced in Quispel et al., J. Phys. A 56 (2023) 315201, preserves a rational measure. We also prove that the Kahan discretisation of these tree-systems factorises and preserves the same measure. As a consequence, for the Kahan maps of Lotka--Volterra systems related to the subclass of tree-systems corresponding to graphs with more than one -vertex subtree, we are able to construct rational integrals.
Paper Structure (3 sections, 9 theorems, 87 equations, 6 figures)

This paper contains 3 sections, 9 theorems, 87 equations, 6 figures.

Table of Contents

  1. Measure preservation for tree-systems
  2. Measure preservation for the Kahan map of tree-systems
  3. Integrals for Kahan maps of LV-systems on graphs

Key Result

Proposition 1.1

Let $T$ be a tree on $n$ vertices, and let $m_i$ be the degree of (number of edges which meet at) vertex $i$. The Lotka--Volterra $T$-system Tsy is measure preserving with density where $P_j$ is the DP associated with edge $e_j$, given by Pi.

Figures (6)

  • Figure 1: The bushy tree on 4 vertices.
  • Figure 2: Graph on $4$ vertices.
  • Figure 3: Three subgraphs that are trees.
  • Figure 4: The 2 graphs on 4 vertices associated with distinct classes of Lotka--Volterra $G$-systems.
  • Figure 5: The 6 graphs on 5 vertices associated with distinct classes of Lotka--Volterra $G$-systems.
  • ...and 1 more figures

Theorems & Definitions (25)

  • Proposition 1.1
  • proof
  • Example 1
  • Proposition 2.1
  • proof
  • Example 2
  • Proposition 2.2
  • proof
  • Example 3
  • Lemma 2.3
  • ...and 15 more