On the combinatorics of Lotka-Volterra equations

TL;DR

This paper develops a combinatorial approach to solving the 2-species Lotka-Volterra equations by transforming nonlinear dynamics into linear systems using advanced mathematical techniques, enabling exact solutions through lattice path representations.

Contribution

It introduces a novel method combining Carleman linearization and Mori-Zwanzig reduction to derive exact solutions for Lotka-Volterra equations via combinatorial lattice path analysis.

Findings

Exact formal solution expressed as weighted lattice path walks

Reduction to PDEs related to Koopman evolution

Representation of dynamics through walk generators

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.