Conservative Replicator and Lotka-Volterra Equations in the context of Dirac$\backslash$ big-isotropic Structures

TL;DR

This paper introduces a Dirac geometry-inspired algorithm to identify constants of motion in replicator equations, extending the class of conservative Lotka-Volterra systems and allowing complex predator-prey interactions.

Contribution

It presents a novel algorithm based on Dirac big-isotropic structures for finding constants of motion in replicator equations and broadens the scope of conservative Lotka-Volterra models.

Findings

Enlarged set of conservative LV equations.

Algorithm for constants of motion in replicator systems.

Interaction between different predators and preys in models.

Abstract

We introduce an algorithm to find possible constants of motion for a given replicator equation. The algorithm is inspired by Dirac geometry and a Hamiltonian description for the replicator equations with such constants of motion, up to a time re-parametrization, is provided using Dirac$\backslash$big-isotropic structures. Using the equivalence between replicator and Lotka-Volterra (LV) equations, the set of conservative LV equations is enlarged. Our approach generalizes the well-known use of gauge transformations to skew-symmetrize the interaction matrix of a LV system. In the case of predator-prey model, our method does allow interaction between different predators and between different preys.