Conservative replicator and Lotka-Volterra equations in the context of Dirac\big-isotropic structures

Alishah Hassan Najafi

doi:10.3934/jgm.2020008

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2020, Volume 12, Issue 2: 149-164. Doi: 10.3934/jgm.2020008

This issue Previous Article Invariant structures on Lie groups Next Article Nonholonomic and constrained variational mechanics

Conservative replicator and Lotka-Volterra equations in the context of Dirac\big-isotropic structures

Alishah Hassan Najafi

Departamento de Matemática, Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, Belo Horizonte, 31270-901, Brazil

Received: August 2019

Early access: March 2020

Published: June 2020

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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.

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Mathematics Subject Classification: Primary: 70H33, 92D25; Secondary: 37J15.

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