Toda-like Hamiltonian as a probe for quantized prey-predator dynamics

TL;DR

This paper explores a Toda-like Hamiltonian model to analyze quantum effects on prey-predator dynamics, revealing how quantum distortions influence classical ecological models and demonstrating quantum stability in such systems.

Contribution

It introduces a Toda-like Hamiltonian framework for prey-predator dynamics, incorporating quantum distortions and providing analytical solutions for quantum and classical coexistence.

Findings

Quantum distortions can be computed non-perturbatively for Gaussian ensembles.

Classical prey-predator cycles are stable under quantum effects.

Quantum stability extends classical Lotka-Volterra dynamics.

Abstract

Phase-space features of a reduced version of the Toda-like Hamiltonian, $\mathcal{H}(x,\,k)$, written in a form constrained by the condition $\partial^2 \mathcal{H} / \partial x \partial k = 0$, with $x$ and $k$ as canonically conjugate variables, are analyzed in terms of Wigner currents. For Wigner currents convoluted with either thermodynamic or Gaussian ensembles, the underlying Hamiltonian dynamics admits analytic corrections due to quantum distortions over the classical phase-space pattern, computed and interpreted through quantifiers of quantumness and stationarity. Notably, while emulating the Lotka-Volterra (LV) dynamics that describe ecological competition systems, the Toda-like classical dynamics allows for analytical solutions with computable periods corresponding to closed phase-space orbits of isotropic prey-predator population distributions. The essential conditions for understanding how classical and quantum evolution can coexist are provided at different scales of quantumness, driven by the associated convoluting ensemble parameter. In the case of Gaussian statistical ensembles, the exact profile of the quantum distortions over classical prey-predator phase-space trajectories is obtained non-perturbatively. Our results indicate that, besides the classical stability admitted by LV models, the Toda-like patterns also exhibit quantum stability. Therefore, this can be regarded as the first step as a predictive theoretical framework towards more robust descriptions of quantum patterns in competitive microscopic biosystems.