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

Abstract

<p style='text-indent:20px;'>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<inline-formula><tex-math id="M2">\begin{document}$ \backslash $\end{document}</tex-math></inline-formula>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.