Non-commutative phase-space Lotka-Volterra dynamics: the quantum analogue

TL;DR

This paper explores how quantum mechanics can be applied to classical Lotka-Volterra predator-prey models using non-commutative phase-space methods, revealing quantum effects on ecological dynamics.

Contribution

It introduces a novel quantum phase-space framework for Lotka-Volterra systems, bridging classical and quantum dynamics with explicit quantum corrections.

Findings

Quantum modifications alter classical phase-space trajectories.

Wigner currents describe quantum effects in the system.

Gaussian ensembles provide exact quantum-classical comparisons.

Abstract

The Lotka-Volterra (LV) dynamics is investigated in the framework of the Weyl-Wigner (WW) quantum mechanics (QM) extended to one-dimensional Hamiltonian systems, $\mathcal{H}(x,\,k)$, constrained by the $\partial^2 \mathcal{H} / \partial x \, \partial k = 0$ condition. Supported by the Heisenberg-Weyl non-commutative algebra, where $[x,\,k] = i$, the canonical variables $x$ and $k$ are interpreted in terms of the LV variables, $y = e^{-x}$ and $z = e^{-k}$, eventually associated with the number of individuals in a closed competitive dynamics: the so-called prey-predator system. The WW framework provides the ground for identifying how classical and quantum evolution coexist at different scales, and for quantifying {\it quantum analogue} effects. Through the results from the associated Wigner currents, (non-)Liouvillian and stationary properties are described for thermodynamic and gaussian quantum ensembles in order to account for the corrections due to quantum features over the classical phase-space pattern yielded by the Hamiltonian description of the LV dynamics. In particular, for gaussian statistical ensembles, the Wigner flow framework provides the exact profile for the quantum modifications over the classical LV phase-space trajectories so that gaussian quantum ensembles can be interpreted as an adequate Hilbert space state configuration for comparing quantum and classical regimes. The generality of the framework developed here extends the boundaries of the understanding of quantum-like effects on competitive microscopical bio-systems.