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

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\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.

Asymptotic dynamics of Hamiltonian polymatrix replicators

“…In [5], another characterization of conservative polymatrix replicators, using quadratic forms, is provided. Furthermore, in [3] the concept of conservative replicator equations (where p = 1) is generalized using Dirac structures.…”