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The combinatorial structure of Lotka–Volterra equations and the Koopman operator

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  • Caravelli, Francesco
  • Lin, Yen Ting

Abstract

We study an approach to obtaining the exact formal solution of the 2-species Lotka–Volterra equation based on combinatorics and generating functions. By employing a combination of Carleman linearization and Mori–Zwanzig reduction techniques, we transform the nonlinear equations into a linear system, allowing for the derivation of a formal solution. The Mori–Zwanzig reduction reduces to an expansion which we show can be interpreted as a directed and weighted lattice path walk, which we use to obtain a representation of the system dynamics as walks of fixed length. The exact solution is then shown to be dependent on the generator of weighted walks. We show that the generator can be obtained by the solution of PDE which in turn is equivalent to a particular Koopman evolution of nonlinear observables.

Suggested Citation

  • Caravelli, Francesco & Lin, Yen Ting, 2025. "The combinatorial structure of Lotka–Volterra equations and the Koopman operator," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 670(C).
    • Handle: RePEc:eee:phsmap:v:670:y:2025:i:c:s0378437125001360
    DOI: 10.1016/j.physa.2025.130484
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    References listed on IDEAS

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    1. Jacopo Grilli & György Barabás & Matthew J. Michalska-Smith & Stefano Allesina, 2017. "Higher-order interactions stabilize dynamics in competitive network models," Nature, Nature, vol. 548(7666), pages 210-213, August.
    2. repec:ecr:col031:4957 is not listed on IDEAS
    3. Dennis Huisman & Raf Jans & Marc Peeters & Albert P.M. Wagelmans, 2005. "Combining Column Generation and Lagrangian Relaxation," Springer Books, in: Guy Desaulniers & Jacques Desrosiers & Marius M. Solomon (ed.), Column Generation, chapter 0, pages 247-270, Springer.
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