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

Francesco Caravelli and Yen Ting Lin

Physica A: Statistical Mechanics and its Applications, 2025, vol. 670, issue C

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.

Keywords: Lotka–Volterra equations; Generating functions; Koopman evolution; Formal solution; Carleman linearization; Mori–Zwanzig (search for similar items in EconPapers)
Date: 2025
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Persistent link: https://EconPapers.repec.org/RePEc:eee:phsmap:v:670:y:2025:i:c:s0378437125001360

DOI: 10.1016/j.physa.2025.130484

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